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Exercise 8.1.2 (The maximum modulus principle) If \( f \) is as in the previous exercise, show that \( \left| {f\left( z\right) }\right| < M \) for all interior points \( z \in R \) , unless \( f \) is constant. Theorem 8.1.3 (Phragmén - Lindelöf) Suppose that \( f\left( s\right) \) is entire in the region \[ S\left( {a, b}\right) = \{ s \in \mathbb{C} : a \leq \operatorname{Re}\left( s\right) \leq b\} \] and that as \( \left| t\right| \rightarrow \infty \) , \[ \left| {f\left( s\right) }\right| = O\left( {e}^{{\left| t\right| }^{\alpha }}\right) \] for some \( \alpha \geq 1 \) . If \( f\left( s\right) \) is bounded on the two vertical lines \( \operatorname{Re}\left( s\right) = a \) and \( \operatorname{Re}\left( s\right) = b \), then \( f\left( s\right) \) is bounded in \( S\left( {a, b}\right) \) . Proof. We first select an integer \( m > \alpha, m \equiv 2\left( {\;\operatorname{mod}\;4}\right) \) . Since arg \( s \rightarrow \) \( \pi /2 \) as \( t \rightarrow \infty \), we can choose \( {T}_{1} \) sufficiently large so that \[ \left| {\arg s - \pi /2}\right| < \pi /{4m} \] Then for \( \left| {\operatorname{Im}\left( s\right) }\right| \geq {T}_{1} \), we find that \( \arg s = \pi /2 - \delta = \theta \) (say) satisfies \[ \cos {m\theta } = - \cos {m\delta } < - 1/\sqrt{2}. \] Therefore, if we consider \[ {g}_{\epsilon }\left( s\right) = {e}^{\epsilon {s}^{m}}f\left( s\right) \] then \[ \left| {{g}_{\epsilon }\left( s\right) }\right| \leq K{e}^{{\left| t\right| }^{\alpha }}{e}^{-\epsilon {\left| s\right| }^{m}/\sqrt{2}}. \] Thus, \( \left| {{g}_{\epsilon }\left( s\right) }\right| \rightarrow 0 \) as \( \left| t\right| \rightarrow \infty \) . Let \( B \) be the maximum of \( f\left( s\right) \) in the region \[ a \leq \operatorname{Re}\left( s\right) \leq b,\;0 \leq \left| {\operatorname{Im}\left( s\right) }\right| \leq {T}_{1} \] Let \( {T}_{2} \) be chosen such that \[ \left| {{g}_{\epsilon }\left( s\right) }\right| \leq B \] for \( \left| {\operatorname{Im}\left( s\right) }\right| \geq {T}_{2} \) . Thus, \[ \left| {f\left( s\right) }\right| \leq B{e}^{-\epsilon {\left| s\right| }^{m}\cos \left( {m\arg s}\right) } \leq B{e}^{\epsilon {\left| s\right| }^{m}} \] for \( \left| {\operatorname{Im}\left( s\right) }\right| \geq {T}_{2} \) . Applying the maximum modulus principle to the region \[ a \leq \operatorname{Re}\left( s\right) \leq b,\;0 \leq \left| {\operatorname{Im}\left( s\right) }\right| \leq {T}_{2}, \] we find that \( \left| {f\left( s\right) }\right| \leq B{e}^{\epsilon {\left| s\right| }^{m}} \) . This estimate holds for all \( s \) in \( S\left( {a, b}\right) \) . Letting \( \epsilon \rightarrow 0 \) yields the result. Corollary 8.1.4 Suppose that \( f\left( s\right) \) is entire in \( S\left( {a, b}\right) \) and that \( \left| {f\left( s\right) }\right| \) \( = O\left( {e}^{{\left| t\right| }^{\alpha }}\right) \) for some \( \alpha \geq 1 \) as \( \left| t\right| \rightarrow \infty \) . If \( f\left( s\right) \) is \( O\left( {\left| t\right| }^{A}\right) \) on the two vertical lines \( \operatorname{Re}\left( s\right) = a \) and \( \operatorname{Re}\left( s\right) = b \), then \( f\left( s\right) = O\left( {\left| t\right| }^{A}\right) \) in \( S\left( {a, b}\right) \) . Proof. We apply the theorem to the function \( g\left( s\right) = f\left( s\right) /{\left( s - u\right) }^{A} \) , where \( u > b \) . Then \( g \) is bounded on the two vertical strips, and the result follows. Exercise 8.1.5 Show that for any entire function \( F \in \mathcal{S} \), we have \[ F\left( s\right) = O\left( {\left| t\right| }^{A}\right) \] for some \( A > 0 \), in the region \( 0 \leq \operatorname{Re}\left( s\right) \leq 1 \) . ## 8.2 Basic Properties We begin by stating the following theorem of Selberg: Theorem 8.2.1 (Selberg) For any \( F \in \mathcal{S} \), let \( {N}_{F}\left( T\right) \) be the number of zeros \( \rho \) of \( F\left( s\right) \) satisfying \( 0 \leq \operatorname{Im}\left( \rho \right) \leq T \), counted with multiplicity. Then \[ {N}_{F}\left( T\right) \sim \left( {2\mathop{\sum }\limits_{{i = 1}}^{d}{\alpha }_{i}}\right) \frac{T\log T}{2\pi } \] as \( T \rightarrow \infty \) . Proof. This is easily derived by the method used to count zeros of \( \zeta \left( s\right) \) and \( L\left( {s,\chi }\right) \) as in Theorem 7.1.7 and Exercise 7.4.4. Clearly, the functional equation for \( F \in \mathcal{S} \) is not unique, by virtue of Legendre's duplication formula. However, the above theorem shows that the sum of the \( {\alpha }_{i} \) ’s is well-defined. Accordingly, we define the degree of \( F \) by \[ \deg F \mathrel{\text{:=}} 2\mathop{\sum }\limits_{{i = 1}}^{d}{\alpha }_{i} \] Lemma 8.2.2 (Conrey and Ghosh) If \( F \in S \) and \( \deg F = 0 \), then \( F = 1 \) . Proof. We follow [CG]. A Dirichlet series can be viewed as a power series in the infinitely many variables \( {p}^{-s} \) as we range over primes \( p \) . Thus, if \( \deg F = 0 \), we can write our functional equation as \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{\left( \frac{{Q}^{2}}{n}\right) }^{s} = {wQ}\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\overline{{a}_{n}}}{n}{n}^{s} \] where \( \left| w\right| = 1 \) . Thus, if \( {a}_{n} \neq 0 \) for some \( n \), then \( {Q}^{2}/n \) is an integer. Hence \( {Q}^{2} \) is an integer. Moreover, \( {a}_{n} \neq 0 \) implies \( n \mid {Q}^{2} \), so that our Dirichlet series is really a Dirichlet polynomial. Therefore, if \( {Q}^{2} = 1 \), then \( F = 1 \), and we are done. So, let us suppose \( q \mathrel{\text{:=}} {Q}^{2} > 1 \) . Since \( {a}_{1} = \) 1, comparing the \( {Q}^{2s} \) term in the functional equation above gives \( \left| {a}_{q}\right| = Q \) . Since \( {a}_{n} \) is multiplicative, we must have for some prime power \( {p}^{r}\parallel q \) that \( \left| {a}_{{p}^{r}}\right| \geq {p}^{r/2} \) . Now consider the \( p \) -Euler factor \[ {F}_{p}\left( s\right) = \mathop{\sum }\limits_{{j = 0}}^{r}\frac{{a}_{{p}^{j}}}{{p}^{js}} \] with logarithm \[ \log {F}_{p}\left( s\right) = \mathop{\sum }\limits_{{j = 0}}^{\infty }\frac{{b}_{{p}^{j}}}{{p}^{js}} \] Viewing these as power series in \( x = {p}^{-s} \), we write \[ P\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{r}{A}_{j}{x}^{j} \] \[ \log P\left( x\right) = \mathop{\sum }\limits_{{j = 0}}^{\infty }{B}_{j}{x}^{j} \] where \( {A}_{j} = {a}_{{p}^{j}},{B}_{j} = {b}_{{p}^{j}} \) . Since \( {a}_{1} = 1 \), we can factor \[ P\left( x\right) = \mathop{\prod }\limits_{{j = 1}}^{r}\left( {1 - {R}_{i}x}\right) \] so that \[ {B}_{j} = - \mathop{\sum }\limits_{{i = 1}}^{r}\frac{{R}_{i}^{j}}{j} \] We also know that \[ \mathop{\prod }\limits_{{i = 1}}^{r}\left| {R}_{i}\right| \geq {p}^{r/2} \] so that \[ \mathop{\max }\limits_{{1 \leq i \leq r}}\left| {R}_{i}\right| \geq {p}^{1/2} \] But \[ {\left| {b}_{{p}^{j}}\right| }^{1/j} = {\left| {B}_{j}\right| }^{1/j} = {\left| \mathop{\sum }\limits_{{i = 1}}^{r}\frac{{R}_{i}^{j}}{j}\right| }^{1/j} \] tends to \( \mathop{\max }\limits_{{1 \leq i \leq r}}\left| {R}_{i}\right| \) as \( j \rightarrow \infty \), which is greater than or equal to \( {p}^{1/2} \) . This contradicts the condition that \( {b}_{n} = O\left( {n}^{\theta }\right) \) with \( \theta < 1/2 \) . Therefore, \( Q = 1 \) and hence \( F = 1 \) . We can now prove the following basic result: Theorem 8.2.3 (Selberg) If \( F \in \mathcal{S} \) and \( F \) is of positive degree, then \( \deg F \geq 1 \) . Proof. We follow [CG]. Consider the identity \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{e}^{-{nx}} = \frac{1}{2\pi i}{\int }_{\left( 2\right) }F\left( s\right) {x}^{-s}\Gamma \left( s\right) {ds}. \] Because of the Phragmen - Lindelöf principle and the functional equation, we find that \( F\left( s\right) \) has polynomial growth in \( \left| {\operatorname{Im}\left( s\right) }\right| \) in any vertical strip. Thus, moving the line of integration to the left, and taking into account the possible pole at \( s = 1 \) of \( F\left( s\right) \) as well as the poles of \( \Gamma \left( s\right) \) at \( s = 0, - 1, - 2,\ldots \), we obtain \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{e}^{-{nx}} = \frac{P\left( {\log x}\right) }{x} + \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{F\left( {-n}\right) {\left( -1\right) }^{n}{x}^{n}}{n!}, \] where \( P \) is a polynomial. The functional equation relates \( F\left( {-n}\right) \) to \( F\left( {n + 1}\right) \) with a product of gamma functions. If \( 0 < \deg F < 1 \) , we find by Stirling's formula that the sum on the right-hand side converges for all \( x \) . Moreover, \( P\left( {\log x}\right) \) is analytic in \( \mathbb{C} \smallsetminus \{ x \leq 0 : x \in \) \( \mathbb{R}\} \) . Hence the left-hand side is analytic in \( \mathbb{C} \smallsetminus \{ x \leq 0 : x \in \mathbb{R}\} \) . But since the left-hand side is periodic with period \( {2\pi i} \), we find that \[ f\left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{e}^{-{nz}} \] is entire. Thus, for any \( x \) , \[ {a}_{n}{e}^{-{nx}} = {\int }_{0}^{2\pi }f\left( {x + {iy}}\right) {e}^{iny}{dy} \ll {n}^{-2} \] by integrating by parts. Choosing \( x = 1/n \) gives \( {a}_{n} = O\left( {1/{n}^{2}}\right) \) . Hence the Dirichlet series \[ F\left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{{n}^{s}} \] converges absolutely for \( \operatorname{Re}s > - 1 \) . However, relating \( F\left( {-1/2 + {it}}\right) \) to \( F\left( {3/2 - {it}}\right) \) by the functional equation and using Stirling’s formula, we find that \( F\left( {-1/2 + {it}}\right) \) is not bounded. This contradiction forces \( \deg F \geq 1 \) . An element \( F \in \mathcal{S} \) will be called primitive if \( F \neq 1 \) and \( F = {F}_{1}{F}_{2} \) with \( {F}_{1},{F}_{2} \in \mathcal{S} \) implies \( {F}_{1} = 1 \) or \( {F}_{2} = 1 \) . Thus, a primitive function cannot be factored nontrivially in \( \

(Theorem 8.1.3 (Phragmén - Lindelöf)) Suppose that \( f\left( s\right) \) is entire in the region \[ S\left( {a, b}\right) = \{ s \in \mathbb{C} : a \leq \operatorname{Re}\left( s\right) \leq b\} \] and that as \( \left| t\right| \rightarrow \infty \) , \[ \left| {f\left( s\right) }\right| = O\left( {e}^{{\left| t\right| }^{\alpha }}\right) \] for some \( \alpha \geq 1 \) . If \( f\left( s\right) \) is bounded on the two vertical lines \( \operatorname{Re}\left( s\right) = a \) and \( \operatorname{Re}\left( s\right) = b \), then \( f\left( s\right) \) is bounded in \( S\left( {a, b}\right) \) .

We first select an integer \( m > \alpha, m \equiv 2\left( {\;\operatorname{mod}\;4}\right) \) . Since arg \( s \rightarrow \) \( \pi /2 \) as \( t \rightarrow \infty \), we can choose \( {T}_{1} \) sufficiently large so that \[ \left| {\arg s - \pi /2}\right| < \pi /{4m} \] Then for \( \left| {\operatorname{Im}\left( s\right) }\right| \geq {T}_{1} \), we find that \( \arg s = \pi /2 - \delta = \theta \) (say) satisfies \[ \cos {m\theta } = - \cos {m\delta } < - 1/\sqrt{2}. \] Therefore, if we consider \[ {g}_{\epsilon }\left( s\right) = {e}^{\epsilon {s}^{m}}f\left( s\right) \] then \[ \left| {{g}_{\epsilon }\left( s\right) }\right| \leq K{e}^{{\left| t\right| }^{\alpha }}{e}^{-\epsilon {\left| s\right| }^{m}/\sqrt{2}}. \] Thus, \( \left| {{g}_{\epsilon }\left( s\right) }\right| \rightarrow 0 \) as \( \left| t\right| \rightarrow \infty \) . Let \( B \) be the maximum of \( f\left( s\right) \) in the region \[ a \leq \operatorname{Re}\left( s\right) \leq b,\;0 \leq \left| {\operatorname{Im}\left( s\right) }\right| \leq {T}_{1} \] Let \( {T}_{2} \) be chosen such that

Exercise 8.3.7 Let \( K \) be a quadratic field of discriminant \( d \) . Let \( {P}_{0} \) denote the group of principal fractional ideals \( \alpha {\mathcal{O}}_{K} \) with \( \alpha \in K \) satisfying \( {N}_{K}\left( \alpha \right) > 0 \) . The quotient group \( {H}_{0} \) of all nonzero fractional ideals modulo \( {P}_{0} \) is called the restricted class group of \( K \) . Show that \( {H}_{0} \) is a subgroup of the ideal class group \( H \) of \( K \) and \( \left\lbrack {H : {H}_{0}}\right\rbrack \leq 2 \) . Exercise 8.3.8 Given an ideal \( \mathfrak{a} \) of a quadratic field \( K \), let \( {\mathfrak{a}}^{\prime } \) denote the conjugate ideal. If \( K \) has discriminant \( d \), write \[ \left| d\right| = {p}_{1}^{{\alpha }_{1}}{p}_{2}\cdots {p}_{t} \] where \( {p}_{1} = 2,{\alpha }_{1} = 0,2 \), or 3 and \( {p}_{2},\ldots ,{p}_{t} \) are distinct odd primes. If we write \( {p}_{i}{\mathcal{O}}_{K} = {\wp }_{i}^{2} \) show that for any ideal \( \mathfrak{a} \) of \( {\mathcal{O}}_{K} \) satisfying \( \mathfrak{a} = {\mathfrak{a}}^{\prime } \) we can write \[ \mathfrak{a} = r{\wp }_{1}^{{a}_{1}}\cdots {\wp }_{t}^{{a}_{t}} \] \( r > 0,{a}_{i} = 0,1 \) uniquely. Exercise 8.3.9 An ideal class \( C \) of \( {H}_{0} \) is said to be ambiguous if \( {C}^{2} = 1 \) in \( {H}_{0} \) . Show that any ambiguous ideal class is equivalent (in the restricted sense) to one of the at most \( {2}^{t} \) ideal classes \[ {\wp }_{1}^{{a}_{1}}\cdots {\wp }_{t}^{{a}_{t}},\;{a}_{i} = 0,1. \] Exercise 8.3.10 With the notation as in the previous two questions, show that there is exactly one relation of the form \[ {\wp }_{1}^{{a}_{1}}\cdots {\wp }_{t}^{{a}_{t}} = \rho {\mathcal{O}}_{K},\;{N}_{K}\left( \rho \right) > 0, \] with \( {a}_{i} = 0 \) or \( 1,\mathop{\sum }\limits_{{i = 1}}^{t}{a}_{i} > 0 \) . Exercise 8.3.11 Let \( K \) be a quadratic field of discriminant \( d \) . Show that the number of ambiguous ideal classes is \( {2}^{t - 1} \) where \( t \) is the number of distinct primes dividing \( d \) . Deduce that \( {2}^{t - 1} \) divides the order of the class group. Exercise 8.3.12 If \( K \) is a quadratic field of discriminant \( d \) and class number 1, show that \( d \) is prime or \( d = 4 \) or 8 . Exercise 8.3.13 If a real quadratic field \( K \) has odd class number, show that \( K \) has a unit of norm -1 . Exercise 8.3.14 Show that \( {15} + 4\sqrt{14} \) is the fundamental unit of \( \mathbb{Q}\left( \sqrt{14}\right) \) . Exercise 8.3.15 In Chapter 6 we showed that \( \mathbb{Z}\left\lbrack \sqrt{14}\right\rbrack \) is a PID (principal ideal domain). Assume the following hypothesis: given \( \alpha ,\beta \in \mathbb{Z}\left\lbrack \sqrt{14}\right\rbrack \), such that \( \gcd \left( {\alpha ,\beta }\right) = 1 \), there is a prime \( \pi \equiv \alpha \left( {\;\operatorname{mod}\;\beta }\right) \) for which the fundamental unit \( \varepsilon = {15} + 4\sqrt{14} \) generates the coprime residue classes \( \left( {\;\operatorname{mod}\;\pi }\right) \) . Show that \( \mathbb{Z}\left\lbrack \sqrt{14}\right\rbrack \) is Euclidean. It is now known that \( \mathbb{Z}\left\lbrack \sqrt{14}\right\rbrack \) is Euclidean and is the main theorem of the doctoral thesis of Harper \( \left\lbrack \mathrm{{Ha}}\right\rbrack \) . The hypothesis of the previous exercise is still unknown however and is true if the Riemann hypothesis holds for Dedekind zeta functions of number fields (see Chapter 10). The hypothesis in the question should be viewed as a number field version of a classical conjecture of Artin on primitive roots. Previously the classification of Euclidean rings of algebraic integers relied on some number field generalization of the Artin primitive root conjecture. But recently, Harper and Murty [HM] have found new techniques which circumvent the need of such a hypothesis in such questions. No doubt, these techniques will have further applications. Exercise 8.3.16 Let \( d = {a}^{2} + 1 \) . Show that if \( \left| {{u}^{2} - d{v}^{2}}\right| \neq 0,1 \) for integers \( u, v \) , then \[ \left| {{u}^{2} - d{v}^{2}}\right| > \sqrt{d} \] Exercise 8.3.17 Suppose that \( n \) is odd, \( n \geq 5 \), and that \( {n}^{2g} + 1 = d \) is squarefree. Show that the class group of \( \mathbb{Q}\left( \sqrt{d}\right) \) has an element of order \( {2g} \) . ## Chapter 9 ## Higher Reciprocity Laws ## 9.1 Cubic Reciprocity Let \( \rho = \left( {-1 + \sqrt{-3}}\right) /2 \) be as in Chapter 2, and let \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) be the ring of Eisenstein integers. Recall that \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) is a Euclidean domain and hence a PID. We set \( N\left( {a + {b\rho }}\right) = {a}^{2} - {ab} + {b}^{2} \) which is the Euclidean norm as proved in Section 2.3. Exercise 9.1.1 If \( \pi \) is a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \), show that \( N\left( \pi \right) \) is a rational prime or the square of a rational prime. Exercise 9.1.2 If \( \pi \in \mathbb{Z}\left\lbrack \rho \right\rbrack \) is such that \( N\left( \pi \right) = p \), a rational prime, show that \( \pi \) is a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . Exercise 9.1.3 If \( p \) is a rational prime congruent to 2 (mod 3), show that \( p \) is prime in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . If \( p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \), show that \( p = \pi \bar{\pi } \) where \( \pi \) is prime in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . Exercise 9.1.4 Let \( \pi \) be a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . Show that \( {\alpha }^{N\left( \pi \right) - 1} \equiv 1\left( {\;\operatorname{mod}\;\pi }\right) \) for all \( \alpha \in \mathbb{Z}\left\lbrack \rho \right\rbrack \) which are coprime to \( \pi \) . Exercise 9.1.5 Let \( \pi \) be a prime not associated to \( \left( {1 - \rho }\right) \) . First show that \( 3 \mid N\left( \pi \right) - 1 \) . If \( \left( {\alpha ,\pi }\right) = 1 \), show that there is a unique integer \( m = 0,1 \), or 2 such that \[ {\alpha }^{\left( {N\left( \pi \right) - 1}\right) /3} \equiv {\rho }^{m}\;\left( {\;\operatorname{mod}\;\pi }\right) \] Let \( N\left( \pi \right) \neq 3 \) . We define the cubic residue character of \( \alpha \left( {\;\operatorname{mod}\;\pi }\right) \) by the symbol \( {\left( \alpha /\pi \right) }_{3} \) as follows: (i) \( {\left( \alpha /\pi \right) }_{3} = 0 \) if \( \pi \mid \alpha \) ; (ii) \( {\alpha }^{\left( {N\left( \pi \right) - 1}\right) /3} \equiv {\left( \alpha /\pi \right) }_{3}\left( {\;\operatorname{mod}\;\pi }\right) \) where \( {\left( \alpha /\pi \right) }_{3} \) is the unique cube root of unity determined by the previous exercise. Exercise 9.1.6 Show that: (a) \( {\left( \alpha /\pi \right) }_{3} = 1 \) if and only if \( {x}^{3} \equiv \alpha \left( {\;\operatorname{mod}\;\pi }\right) \) is solvable in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) ; (b) \( {\left( \alpha \beta /\pi \right) }_{3} = {\left( \alpha /\pi \right) }_{3}{\left( \beta /\pi \right) }_{3} \) ; and (c) if \( \alpha \equiv \beta \left( {\;\operatorname{mod}\;\pi }\right) \), then \( {\left( \alpha /\pi \right) }_{3} = {\left( \beta /\pi \right) }_{3} \) . Let us now define the cubic character \( {\chi }_{\pi }\left( \alpha \right) = {\left( \alpha /\pi \right) }_{3} \) . Exercise 9.1.7 Show that: (a) \( \overline{{\chi }_{\pi }\left( \alpha \right) } = {\chi }_{\pi }{\left( \alpha \right) }^{2} = {\chi }_{\pi }\left( {\alpha }^{2}\right) \) ; and (b) \( \overline{{\chi }_{\pi }\left( \alpha \right) } = {\chi }_{\bar{\pi }}\left( \alpha \right) \) . Exercise 9.1.8 If \( q \equiv 2\left( {\;\operatorname{mod}\;3}\right) \), show that \( {\chi }_{q}\left( \bar{\alpha }\right) = {\chi }_{q}\left( {\alpha }^{2}\right) \) and \( {\chi }_{q}\left( n\right) = 1 \) if \( n \) is a rational integer coprime to \( q \) . This exercise shows that any rational integer is a cubic residue mod \( q \) . If \( \pi \) is prime in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \), we say \( \pi \) is primary if \( \pi \equiv 2\left( {\;\operatorname{mod}\;3}\right) \) . Therefore if \( q \equiv 2\left( {\;\operatorname{mod}\;3}\right) \), then \( q \) is primary in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . If \( \pi = a + {b\rho } \), then this means \( a \equiv 2\left( {\;\operatorname{mod}\;3}\right) \) and \( b \equiv 0\left( {\;\operatorname{mod}\;3}\right) \) . Exercise 9.1.9 Let \( N\left( \pi \right) = p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) . Among the associates of \( \pi \), show there is a unique one which is primary. We can now state the law of cubic reciprocity: let \( {\pi }_{1},{\pi }_{2} \) be primary. Suppose \( N\left( {\pi }_{1}\right), N\left( {\pi }_{2}\right) \neq 3 \) and \( N\left( {\pi }_{1}\right) \neq N\left( {\pi }_{2}\right) \) . Then \[ {\chi }_{{\pi }_{1}}\left( {\pi }_{2}\right) = {\chi }_{{\pi }_{2}}\left( {\pi }_{1}\right) \] To prove the law of cubic reciprocity, we will introduce Jacobi sums and more general Gauss sums than the ones used in Chapter 7. Let \( {\mathbb{F}}_{p} \) denote the finite field of \( p \) elements. A multiplicative character on \( {\mathbb{F}}_{p} \) is a homomorphism \( \chi : {\mathbb{F}}_{p}^{ \times } \rightarrow {\mathbb{C}}^{ \times } \) . The Legendre symbol \( \left( {a/p}\right) \) is an example of such a character. Another example is the trivial character \( {\chi }_{0} \) defined by \( {\chi }_{0}\left( a\right) = 1 \) for all \( a \in {\mathbb{F}}_{p}^{ \times } \) . It is useful to extend the definition of \( \chi \) to all of \( {\mathbb{F}}_{p} \) . We set \( \chi \left( 0\right) = 0 \) for \( \chi \neq {\chi }_{0} \) and \( {\chi }_{0}\left( 0\right) = 1 \) . For \( a \in {\mathbb{F}}_{p}^{ \times } \), define the Gauss sum \[ {g}_{a}\left( \chi \right) = \mathop{\sum }\limits_{{t \in {\mathbb{F}}_{p}}}\chi \left( t\right) {\zeta }^{at} \] whe

Exercise 8.3.7 Let \( K \) be a quadratic field of discriminant \( d \) . Let \( {P}_{0} \) denote the group of principal fractional ideals \( \alpha {\mathcal{O}}_{K} \) with \( \alpha \in K \) satisfying \( {N}_{K}\left( \alpha \right) > 0 \) . The quotient group \( {H}_{0} \) of all nonzero fractional ideals modulo \( {P}_{0} \) is called the restricted class group of \( K \) . Show that \( {H}_{0} \) is a subgroup of the ideal class group \( H \) of \( K \) and \( \left\lbrack {H : {H}_{0}}\right\rbrack \leq 2 \) .

To show that \( {H}_{0} \) is a subgroup of the ideal class group \( H \) of \( K \), we need to verify that \( {H}_{0} \) satisfies the subgroup criteria within \( H \). 1. **Identity Element**: The identity element in \( H \) is the class of the unit ideal \( {\mathcal{O}}_{K} \). Since any principal ideal \( \alpha {\mathcal{O}}_{K} \) with \( \alpha \in K \) and \( {N}_{K}\left( \alpha \right) > 0 \) is in \( {P}_{0} \), the identity element in \( H \) is also in \( {H}_{0} \). 2. **Closure**: Let \( [I] \) and \( [J] \) be elements of \( {H}_{0} \), where \( I \) and \( J \) are nonzero fractional ideals of \( K \). The product of ideals \( IJ \) is also a nonzero fractional ideal, and since multiplication of ideals is commutative and associative, the class of \( IJ \) in \( H \) is well-defined. We need to show that if both \( [I] \) and \( [J] \) are in \( {H}_{0} \), then so is their product. This follows because if both ideals are equivalent modulo principal ideals with positive norm, their product will also be equivalent modulo such principal ideals. 3. **Inverses**: For any ideal class \( [I] \in {H}_{0} \), there exists an inverse class, which corresponds to the inverse ideal (or more precisely, the inverse in the group of fractional ideals). The inverse ideal also satisfies the condition for being in the restricted class group because taking inverses preserves the property of being equivalent modulo principal ideals with positive norm. Thus, we have shown that every element in the restricted class group has an inverse within this group, ensuring that it is indeed a subgroup of the full ideal class group. Next, we need to show that the index \(\left\lbrack {H : {H}_{0}}\right\rbrack \leq 2\). This can be seen by considering the structure of the ideal class group and how principal ideals with negative norms fit into it. Since every ideal class in \( H \) can be represented by an ideal whose norm can be made positive by multiplying by a suitable unit (which does not change its class in the full class group but might change its membership in the restricted class group), it follows that each class in \( H \) corresponds either to a class already in \( {H}_{0} \) or to one obtained by multiplying by a principal ideal generated by an element with negative norm. Since there are only two such possibilities (positive or negative norm condition), it follows that \(\left\lbrack {H : {H}_{0}}\right\rbrack \leq 2\).

Proposition 9.34 Define a domain \( \operatorname{Dom}\left( \Delta \right) \) as follows: \[ \operatorname{Dom}\left( \Delta \right) = \left\{ {\psi \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \left| {\;{\left| \mathbf{k}\right| }^{2}\widehat{\psi }\left( \mathbf{k}\right) \in {L}^{2}\left( {\mathbb{R}}^{n}\right) }\right. }\right\} . \] Define \( \Delta \) on this domain by the expression \[ {\Delta \psi } = - {\mathcal{F}}^{-1}\left( {{\left| \mathbf{k}\right| }^{2}\widehat{\psi }\left( \mathbf{k}\right) }\right) \] (9.18) where \( \widehat{\psi } \) is the Fourier transform of \( \psi \) and \( {\mathcal{F}}^{-1} \) is the inverse Fourier. Then \( \Delta \) is self-adjoint on \( \operatorname{Dom}\left( \Delta \right) \) . The domain \( \operatorname{Dom}\left( \Delta \right) \) may also be described as the set of all \( \psi \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \) such that \( {\Delta \psi } \), computed in the distribution sense, belongs to \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) . If \( \psi \in \operatorname{Dom}\left( \Delta \right) \), then \( {\Delta \psi } \) as defined by (9.18) agrees with \( {\Delta \psi } \) computed in the distribution sense. The proof of Proposition 9.34 is extremely similar to that of Proposition 9.32 and is omitted. Of course, the kinetic energy operator \( - {\hslash }^{2}\Delta /\left( {2m}\right) \) is also self-adjoint on the same domain as \( \Delta \) . It is easy to see from (9.18) and the unitarity of the Fourier transform that \( - {\hslash }^{2}\Delta /\left( {2m}\right) \) is non-negative, that is, that \[ \left\langle {\psi , - \frac{{\hslash }^{2}}{2m}{\Delta \psi }}\right\rangle \geq 0 \] for all \( \psi \in \operatorname{Dom}\left( \Delta \right) \) . Using the same reasoning as in Sects. 9.6 and 9.7, it is not hard to show that the operators \( {P}_{j} \) and \( \Delta \) are essentially self-adjoint on \( {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . See Exercise 16. Care must be exercised in applying Proposition 9.34. Although the function \[ \psi \left( \mathbf{x}\right) \mathrel{\text{:=}} \frac{1}{\left| \mathbf{x}\right| } \] is harmonic on \( {\mathbb{R}}^{3} \smallsetminus \{ 0\} \), the Laplacian over \( {\mathbb{R}}^{3} \) of \( \psi \) in the distribution sense is not zero (Exercise 13). (It can be shown, by carefully analyzing the calculation in the proof of Proposition 9.35, that \( {\Delta \psi } \) is a nonzero multiple of a \( \delta \) -function.) This example shows that if a function \( \psi \) has a singularity, calculating the Laplacian of \( \psi \) away from the singularity may not give the correct distributional Laplacian of \( \psi \) . For example, the function \( \phi \) in \( {L}^{2}\left( {\mathbb{R}}^{3}\right) \) given by \[ \phi \left( \mathbf{x}\right) \mathrel{\text{:=}} \frac{{e}^{-{\left| \mathbf{x}\right| }^{2}}}{\left| \mathbf{x}\right| } \] (9.19) is not in \( \operatorname{Dom}\left( \Delta \right) \), even though both \( \phi \) and \( {\Delta \phi } \) are (by direct computation) square-integrable over \( {\mathbb{R}}^{3} \smallsetminus \{ 0\} \) . Indeed, when \( n \leq 3 \), every element of \( \operatorname{Dom}\left( \Delta \right) \) is continuous (Exercise 14). Proposition 9.35 Suppose \( \psi \left( \mathbf{x}\right) = g\left( \mathbf{x}\right) f\left( \left| \mathbf{x}\right| \right) \), where \( g \) is a smooth function on \( {\mathbb{R}}^{n} \) and \( f \) is a smooth function on \( \left( {0,\infty }\right) \) . Suppose also that \( f \) satisfies \[ \mathop{\lim }\limits_{{r \rightarrow {0}^{ + }}}{r}^{n - 1}f\left( r\right) = 0 \] \[ \mathop{\lim }\limits_{{r \rightarrow {0}^{ + }}}{r}^{n - 1}{f}^{\prime }\left( r\right) = 0. \] If both \( \psi \) and \( {\Delta \psi } \) are square-integrable over \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \), then \( \psi \) belongs to \( \operatorname{Dom}\left( \Delta \right) \) . Note that the second condition in the proposition fails if \( n = 3 \) and \( f\left( r\right) = 1/r \) . We will make use of this result in Chap. 18. Proof. To apply Proposition 9.34, we need to compute \( \langle \psi ,{\Delta \chi }\rangle \), for each \( \chi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . We choose a large cube \( C \), centered at the origin and such that the support of \( \chi \) is contained in the interior of \( C \) . Then we consider the integral of \( \bar{\psi }\left( {{\partial }^{2}\chi /\partial {x}_{j}^{2}}\right) \) over \( C \smallsetminus {C}_{\varepsilon } \), where \( {C}_{\varepsilon } \) is a cube centered at the origin and having side-length \( \varepsilon \) . We evaluate the \( {x}_{j} \) -integral first and we integrate by parts twice. For "good" values of the remaining variables, \( {x}_{j} \) ranges over all of \( C \), in which case there are no boundary terms to worry about. For "bad" values of the remaining variables, we get two kinds of boundary terms, one involving \( \bar{\psi }\left( {\partial \chi /\partial {x}_{j}}\right) \) and one involving \( \left( {\partial \bar{\psi }/\partial {x}_{j}}\right) \chi \) , in both cases integrated over two opposite faces of \( {C}_{\varepsilon } \) . Now, \[ \frac{\partial \psi }{\partial {x}_{j}} = \frac{\partial g}{\partial {x}_{j}}f\left( \left| \mathbf{x}\right| \right) + g\left( \mathbf{x}\right) \frac{df}{dr}\frac{{x}_{j}}{r}. \] Since the area of the faces of the cube is \( {\varepsilon }^{n - 1} \), the assumption on \( f \) will cause the boundary terms to disappear in the limit as \( \varepsilon \) tends to zero. Furthermore, both \( \psi \) and \( {\Delta \psi } \) are in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) and thus in \( {L}^{1}\left( C\right) \), where in the case of \( {\Delta \psi } \), we simply leave the value at the origin (which is a set of measure zero) undefined. Thus, integrals of \( \bar{\psi }{\Delta \chi } \) and \( {\left( \Delta \bar{\psi }\right) }_{\chi } \) over \( C \smallsetminus {C}_{\varepsilon } \) will converge to integrals over \( C \) . Since the boundary terms vanish in the limit, we are left with \[ \langle \psi ,{\Delta \chi }\rangle = \langle {\Delta \psi },\chi \rangle \] Thus, the distributional Laplacian of \( \psi \) is simply integration against the "pointwise" Laplacian, ignoring the origin. Proposition 9.34 then tells us that \( \psi \in \operatorname{Dom}\left( \Delta \right) \) . ∎ ## 9.9 Sums of Self-Adjoint Operators In the previous section, we have succeeded in defining the Laplacian \( \Delta \) , and hence also the kinetic energy operator \( - {\hslash }^{2}\Delta /\left( {2m}\right) \), as a self-adjoint operator on a natural dense domain in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) . We have also defined the potential energy operator \( V\left( \mathbf{X}\right) \) as a self-adjoint operator on a different dense domain, for any measurable function \( V : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) . To obtain the Schrödinger operator \( - {\hslash }^{2}\Delta /\left( {2m}\right) + V\left( \mathbf{X}\right) \), we "merely" have to make sense of the sum of two unbounded self-adjoint operators. This task, however, turns out to be more difficult than might be expected. In particular, if \( V \) is a highly singular function, then \( - {\hslash }^{2}\Delta /\left( {2m}\right) + V\left( \mathbf{X}\right) \) may fail to be self-adjoint or essentially self-adjoint on any natural domain. Definition 9.36 If \( A \) and \( B \) are unbounded operators on \( \mathbf{H} \), then \( A + B \) is the operator with domain \[ \operatorname{Dom}\left( {A + B}\right) \mathrel{\text{:=}} \operatorname{Dom}\left( A\right) \cap \operatorname{Dom}\left( B\right) \] and given by \( \left( {A + B}\right) \psi = {A\psi } + {B\psi } \) . The sum of two unbounded self-adjoint operators \( A \) and \( B \) may fail to be self-adjoint or even essentially self-adjoint. [If, however, \( B \) is bounded with \( \operatorname{Dom}\left( B\right) = \mathbf{H} \), then Proposition 9.13 shows that \( A + B \) is self-adjoint on \( \operatorname{Dom}\left( A\right) \cap \operatorname{Dom}\left( B\right) = \operatorname{Dom}\left( A\right) \) .] For one thing, if \( A \) and \( B \) are unbounded, then \( \operatorname{Dom}\left( A\right) \cap \operatorname{Dom}\left( B\right) \) may fail to be dense in \( \mathbf{H} \) . But even if \( \operatorname{Dom}\left( A\right) \cap \) \( \operatorname{Dom}\left( B\right) \) is dense in \( \mathbf{H} \), it can easily happen that \( A + B \) is not essentially self-adjoint on this domain. (See, for example, Sect. 9.10.) Many things that are simple for bounded self-adjoint operators becomes complicated when dealing with unbounded self-adjoint operators! In this section, we examine criteria on a function \( V \) under which the Schrödinger operator \[ \widehat{H} = - \frac{{\hslash }^{2}}{2m}\Delta + V \] is self-adjoint or essentially self-adjoint on some natural domain inside \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) . Theorem 9.37 (Kato-Rellich Theorem) Suppose that \( A \) and \( B \) are unbounded self-adjoint operators on \( \mathbf{H} \) . Suppose that \( \operatorname{Dom}\left( A\right) \subset \operatorname{Dom}\left( B\right) \) and that there exist positive constants \( a \) and \( b \) with \( a < 1 \) such that \[ \parallel {B\psi }\parallel \leq a\parallel {A\psi }\parallel + b\parallel \psi \parallel \] (9.20) for all \( \psi \in \operatorname{Dom}\left( A\right) \) . Then \( A + B \) is self-adjoint on \( \operatorname{Dom}\left( A\right) \) and essentially self-adjoint on any subspace of \( \operatorname{Dom}\left( A\right) \) on which \( A \) is essentially selfadjoint. Furthermore, if \( A \) is non-negative, then the spectrum of \( A + B \) is bounded below by \( - b/\left( {1 - a}\right) \) . Note that since we assume \( \operatorname{Dom}\left( B\right) \supset \operatorname{Dom}\left( A\right) \), the natural domain for \( A + B \) is \( \oper

Proposition 9.34 Define a domain \( \operatorname{Dom}\left( \Delta \right) \) as follows: \[ \operatorname{Dom}\left( \Delta \right) = \left\{ {\psi \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \left| {\;{\left| \mathbf{k}\right| }^{2}\widehat{\psi }\left( \mathbf{k}\right) \in {L}^{2}\left( {\mathbb{R}}^{n}\right) }\right. }\right\} . \] Define \( \Delta \) on this domain by the expression \[ {\Delta \psi } = - {\mathcal{F}}^{-1}\left( {{\left| \mathbf{k}\right| }^{2}\widehat{\psi }\left( \mathbf{k}\right) }\right) \] where \( \widehat{\psi } \) is the Fourier transform of \( \psi \) and \( {\mathcal{F}}^{-1} \) is the inverse Fourier. Then \( \Delta \) is self-adjoint on \( \operatorname{Dom}\left( \Delta \right) \) .

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Exercise 9.2.10 Consider the element \[ \alpha = {\left( x + y\right) }^{\ell - 2}\left( {x + {\zeta y}}\right) \] Show that: (a) the ideal \( \left( \alpha \right) \) is a perfect \( \ell \) th power. (b) \( \alpha \equiv 1 - {u\lambda }\left( {\;\operatorname{mod}\;{\lambda }^{2}}\right) \) where \( u = {\left( x + y\right) }^{\ell - 2}y \) . Exercise 9.2.11 Show that \( {\zeta }^{-u}\alpha \) is primary. Exercise 9.2.12 Use Eisenstein reciprocity to show that if \( {x}^{\ell } + {y}^{\ell } + {z}^{\ell } = 0 \) has a solution in integers, \( \ell \nmid {xyz} \), then for any \( p \mid y,{\left( \zeta /p\right) }_{\ell }^{-u} = 1 \) . (Hint: Evaluate \( \left( {p/{\zeta }^{-u}\alpha }\right) \) ,) Exercise 9.2.13 Show that if \[ {x}^{\ell } + {y}^{\ell } + {z}^{\ell } = 0 \] has a solution in integers, \( l \nmid {xyz} \), then for any \( p \mid {xyz},{\left( \zeta /p\right) }_{\ell }^{-u} = 1 \) . Exercise 9.2.14 Show that \( {\left( \zeta /p\right) }_{\ell }^{-u} = 1 \) implies that \( {p}^{\ell - 1} \equiv 1\left( {\;\operatorname{mod}\;{\ell }^{2}}\right) \) . Exercise 9.2.15 If \( \ell \) is an odd prime and \[ {x}^{\ell } + {y}^{\ell } + {z}^{\ell } = 0 \] for some integers \( x, y, z \) coprime to \( \ell \), then show that \( {p}^{\ell - 1} \equiv 1\left( {\;\operatorname{mod}\;{\ell }^{2}}\right) \) for every \( p \mid {xyz} \) . Deduce that \( {2}^{\ell - 1} \equiv 1\left( {\;\operatorname{mod}\;{\ell }^{2}}\right) \) . The congruence \( {2}^{\ell - 1} \equiv 1\left( {\;\operatorname{mod}\;{\ell }^{2}}\right) \) was first established by Wieferich in 1909 as a necessary condition in the first case of Fermat's Last Theorem. The only primes less than \( 3 \times {10}^{9} \) satisfying this congruence are 1093 and 3511 as a quick computer calculation shows. It is not known if there are infinitely many such primes. (See also Exercise 1.3.4.) ## 9.3 Supplementary Problems Exercise 9.3.1 Show that there are infinitely many primes \( p \) such that \( \left( {2/p}\right) = \) \( - 1 \) . Exercise 9.3.2 Let \( a \) be a nonsquare integer greater than 1. Show that there are infinitely many primes \( p \) such that \( \left( {a/p}\right) = - 1 \) . Exercise 9.3.3 Suppose that \( {x}^{2} \equiv a\left( {\;\operatorname{mod}\;p}\right) \) has a solution for all but finitely many primes. Show that \( a \) is a perfect square. Exercise 9.3.4 Let \( K \) be a quadratic extension of \( \mathbb{Q} \) . Show that there are infinitely many primes which do not split completely in \( K \) . Exercise 9.3.5 Suppose that \( a \) is an integer coprime to the odd prime \( q \) . If \( {x}^{q} \equiv a\left( {\;\operatorname{mod}\;p}\right) \) has a solution for all but finitely many primes, show that \( a \) is a perfect \( q \) th power. (This generalizes the penultimate exercise.) Exercise 9.3.6 Let \( p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) . Show that there are integers \( A \) and \( B \) such that \[ {4p} = {A}^{2} + {27}{B}^{2} \] \( A \) and \( B \) are unique up to sign. Exercise 9.3.7 Let \( p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) . Show that \( {x}^{3} \equiv 2\left( {\;\operatorname{mod}\;p}\right) \) has a solution if and only if \( p = {C}^{2} + {27}{D}^{2} \) for some integers \( C, D \) . Exercise 9.3.8 Show that the equation \[ {x}^{3} - 2{y}^{3} = {23}{z}^{m} \] has no integer solutions with \( \gcd \left( {x, y, z}\right) = 1 \) . ## Chapter 10 ## Analytic Methods ## 10.1 The Riemann and Dedekind Zeta Functions The Riemann zeta function \( \zeta \left( s\right) \) is defined initially for \( \operatorname{Re}\left( s\right) > 1 \) as the infinite series \[ \zeta \left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{1}{{n}^{s}} \] Exercise 10.1.1 Show that for \( \operatorname{Re}\left( s\right) > 1 \) , \[ \zeta \left( s\right) = \mathop{\prod }\limits_{p}{\left( 1 - \frac{1}{{p}^{s}}\right) }^{-1} \] where the product is over prime numbers \( p \) . Exercise 10.1.2 Let \( K \) be an algebraic number field and \( {\mathcal{O}}_{K} \) its ring of integers. The Dedekind zeta function \( {\zeta }_{K}\left( s\right) \) is defined for \( \operatorname{Re}\left( s\right) > 1 \) as the infinite series \[ {\zeta }_{K}\left( s\right) = \mathop{\sum }\limits_{\mathfrak{a}}\frac{1}{{\left( N\mathfrak{a}\right) }^{s}} \] where the sum is over all ideals of \( {\mathcal{O}}_{K} \) . Show that the infinite series is absolutely convergent for \( \operatorname{Re}\left( s\right) > 1 \) . Exercise 10.1.3 Prove that for \( \operatorname{Re}\left( s\right) > 1 \) , \[ {\zeta }_{K}\left( s\right) = \mathop{\prod }\limits_{\wp }{\left( 1 - \frac{1}{{\left( N\wp \right) }^{s}}\right) }^{-1}. \] Theorem 10.1.4 Let \( {\left\{ {a}_{m}\right\} }_{m = 1}^{\infty } \) be a sequence of complex numbers, and let \( A\left( x\right) = \mathop{\sum }\limits_{{m \leq x}}{a}_{m} = O\left( {x}^{\delta }\right) \), for some \( \delta \geq 0 \) . Then \[ \mathop{\sum }\limits_{{m = 1}}^{\infty }\frac{{a}_{m}}{{m}^{s}} \] converges for \( \operatorname{Re}\left( s\right) > \delta \) and in this half-plane we have \[ \mathop{\sum }\limits_{{m = 1}}^{\infty }\frac{{a}_{m}}{{m}^{s}} = s{\int }_{1}^{\infty }\frac{A\left( x\right) {dx}}{{x}^{s + 1}} \] for \( \operatorname{Re}\left( s\right) > 1 \) . Proof. We write \[ \mathop{\sum }\limits_{{m = 1}}^{M}\frac{{a}_{m}}{{m}^{s}} = \mathop{\sum }\limits_{{m = 1}}^{M}\left( {A\left( m\right) - A\left( {m - 1}\right) }\right) {m}^{-s} \] \[ = A\left( M\right) {M}^{-s} + \mathop{\sum }\limits_{{m = 1}}^{{M - 1}}A\left( m\right) \left\{ {{m}^{-s} - {\left( m + 1\right) }^{-s}}\right\} . \] Since \[ {m}^{-s} - {\left( m + 1\right) }^{-s} = s{\int }_{m}^{m + 1}\frac{dx}{{x}^{s + 1}} \] we get \[ \mathop{\sum }\limits_{{m = 1}}^{M}\frac{{a}_{m}}{{m}^{s}} = \frac{A\left( M\right) }{{M}^{s}} + s{\int }_{1}^{M}\frac{A\left( x\right) {dx}}{{x}^{s + 1}}. \] For \( \operatorname{Re}\left( s\right) > \delta \), we find \[ \mathop{\lim }\limits_{{M \rightarrow \infty }}\frac{A\left( M\right) }{{M}^{s}} = 0 \] since \( A\left( x\right) = O\left( {x}^{\delta }\right) \) . Hence, the partial sums converge for \( \operatorname{Re}\left( s\right) > \delta \) and we have \[ \mathop{\sum }\limits_{{m = 1}}^{\infty }\frac{{a}_{m}}{{m}^{s}} = s{\int }_{1}^{\infty }\frac{A\left( x\right) {dx}}{{x}^{s + 1}} \] in this half-plane. Exercise 10.1.5 Show that \( \left( {s - 1}\right) \zeta \left( s\right) \) can be extended analytically for \( \operatorname{Re}\left( s\right) > \) 0. Exercise 10.1.6 Evaluate \[ \mathop{\lim }\limits_{{s \rightarrow 1}}\left( {s - 1}\right) \zeta \left( s\right) \] Example 10.1.7 Let \( K = \mathbb{Q}\left( i\right) \) . Show that \( \left( {s - 1}\right) {\zeta }_{K}\left( s\right) \) extends to an analytic function for \( \operatorname{Re}\left( s\right) > \frac{1}{2} \) . Solution. Since every ideal \( \mathfrak{a} \) of \( {\mathcal{O}}_{K} \) is principal, we can write \( \mathfrak{a} = \left( {a + {ib}}\right) \) for some integers \( a, b \) . Moreover, since \[ \mathfrak{a} = \left( {a + {ib}}\right) = \left( {-a - {ib}}\right) = \left( {-a + {ib}}\right) = \left( {a - {ib}}\right) \] we can choose \( a, b \) to be both positive. In this way, we can associate with each ideal \( \mathfrak{a} \) a unique lattice point \( \left( {a, b}\right), a \geq 0, b \geq 0 \) . Conversely, to each such lattice point \( \left( {a, b}\right) \) we can associate the ideal \( \mathfrak{a} = \left( {a + {ib}}\right) \) . Moreover, \( N\mathfrak{a} = {a}^{2} + {b}^{2} \) . Thus, if we write \[ {\zeta }_{K}\left( s\right) = \mathop{\sum }\limits_{\mathfrak{a}}\frac{1}{N{\mathfrak{a}}^{s}} = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{{n}^{s}} \] we find that \[ A\left( x\right) = \mathop{\sum }\limits_{{n \leq x}}{a}_{n} \] is equal to the number of lattice points lying in the positive quadrant defined by the circle \( {a}^{2} + {b}^{2} \leq x \) . We will call such a lattice point \( \left( {a, b}\right) \) internal if \( {\left( a + 1\right) }^{2} + {\left( b + 1\right) }^{2} \leq x \) . Otherwise, we will call it a boundary lattice point. Let \( I \) be the number of internal lattice points, and \( B \) the number of boundary lattice points. Then \[ I \leq \frac{\pi }{4}x \leq I + B \] Any boundary point \( \left( {a, b}\right) \) is contained in the annulus \[ {\left( \sqrt{x} - \sqrt{2}\right) }^{2} \leq {a}^{2} + {b}^{2} \leq {\left( \sqrt{x} + \sqrt{2}\right) }^{2} \] and an upper bound for \( B \) is provided by the area of the annulus. This is easily seen to be \[ \pi {\left( \sqrt{x} + \sqrt{2}\right) }^{2} - \pi {\left( \sqrt{x} - \sqrt{2}\right) }^{2} = O\left( \sqrt{x}\right) . \] Thus \( A\left( x\right) = {\pi x}/4 + O\left( \sqrt{x}\right) \) . By Theorem 10.1.4, we deduce that \[ {\zeta }_{K}\left( s\right) = \frac{\pi }{4}s{\int }_{1}^{\infty }\frac{dx}{{x}^{s}} + s{\int }_{1}^{\infty }\frac{E\left( x\right) }{{x}^{s + 1}}{dx} \] where \( E\left( x\right) = O\left( \sqrt{x}\right) \), so that the latter integral converges for \( \operatorname{Re}\left( s\right) > \frac{1}{2} \) . Thus \[ \left( {s - 1}\right) {\zeta }_{K}\left( s\right) = \frac{\pi }{4}s + s\left( {s - 1}\right) {\int }_{1}^{\infty }\frac{E\left( x\right) }{{x}^{s + 1}}{dx} \] is analytic for \( \operatorname{Re}\left( s\right) > \frac{1}{2} \) . Exercise 10.1.8 For \( K = \mathbb{Q}\left( i\right) \), evaluate \[ \mathop{\lim }\limits_{{s \rightarrow {1}^{ + }}}\left( {s - 1}\right) {\zeta }_{K}\left( s\right) \] Exercise 10.1.9 Show that the number of integers \( \left( {a, b}\right) \) with \( a > 0 \) satisfying \( {a}^{2} + D{b}^{2} \leq x \) is \[ \frac{\pi x}{2\sqrt{D}} + O\left( \sqrt{x}\right) \] Exercise 10.1.10 Suppose \( K = \mathbb{Q}\left( \sqrt{-D}\right) \) where \( D > 0 \) and \( - D ≢ 1\left( {\;\operatorname{mod}\;4}\right) \) and \( {\mathcal{O}}_{K} \) has class number

(Theorem/Proposition/Example Problem/Lemma/Corollary Content...) Exercise 9.2.10 Consider the element \[ \alpha = {\left( x + y\right) }^{\ell - 2}\left( {x + {\zeta y}}\right) \] Show that: (a) the ideal \( \left( \alpha \right) \) is a perfect \( \ell \) th power. (b) \( \alpha \equiv 1 - {u\lambda }\left( {\;\operatorname{mod}\;{\lambda }^{2}}\right) \) where \( u = {\left( x + y\right) }^{\ell - 2}y \) .

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Proposition 3.2. In a chart \( U \times \mathbf{E} \) for \( {TX} \), let \( f : U \times \mathbf{E} \rightarrow \mathbf{E} \times \mathbf{E} \) represent \( F \), with \( f = \left( {{f}_{1},{f}_{2}}\right) \) . Then \( f \) represents a spray if and only if, for all \( s \in \mathbf{R} \) we have \[ {f}_{2}\left( {x,{sv}}\right) = {s}^{2}{f}_{2}\left( {x, v}\right) \] Proof. The proof follows at once from the definitions and the formula giving the chart representation of \( s{\left( {s}_{TX}\right) }_{ * } \) . Thus we see that the condition SPR 1 (in addition to being a second-order vector field), simply means that \( {f}_{2} \) is homogeneous of degree 2 in the variable \( v \) . By the remark in Chapter I,§3, it follows that \( {f}_{2} \) is a quadratic map in its second variable, and specifically, this quadratic map is given by \[ {f}_{2}\left( {x, v}\right) = \frac{1}{2}{D}_{2}^{2}{f}_{2}\left( {x,0}\right) \left( {v, v}\right) . \] Thus the spray is induced by a symmetric bilinear map given at each point \( x \) in a chart by (2) \[ B\left( x\right) = \frac{1}{2}{D}_{2}^{2}{f}_{2}\left( {x,0}\right) . \] Conversely, suppose given a morphism \[ U \rightarrow {L}_{\text{sym }}^{2}\left( {\mathbf{E},\mathbf{E}}\right) \;\text{ given by }\;x \mapsto B\left( x\right) \] from \( U \) into the space of symmetric bilinear maps \( \mathbf{E} \times \mathbf{E} \rightarrow \mathbf{E} \) . Thus for each \( v, w \in \mathbf{E} \) the value of \( B\left( x\right) \) at \( \left( {v, w}\right) \) is denoted by \( B\left( {x;v, w}\right) \) or \( B\left( x\right) \left( {v, w}\right) \) . Define \( {f}_{2}\left( {x, v}\right) = B\left( {x;v, v}\right) \) . Then \( {f}_{2} \) is quadratic in its second variable, and the map \( f \) defined by \[ f\left( {x, v}\right) = \left( {v, B\left( {x;v, v}\right) }\right) = \left( {v,{f}_{2}\left( {x, v}\right) }\right) \] represents a spray over \( U \) . We call \( B \) the symmetric bilinear map associated with the spray. From the local representations in (1) and (2), we conclude that a curve \( \alpha \) is a geodesic if and only if \( \alpha \) satisfies the differential equation (3) \[ {\alpha }^{\prime \prime }\left( t\right) = {B}_{\alpha \left( t\right) }\left( {{\alpha }^{\prime }\left( t\right) ,{\alpha }^{\prime }\left( t\right) }\right) \;\text{ for all }t. \] We recall the trivial fact from linear algebra that the bilinear map \( B \) is determined purely algebraically from the quadratic map, by the formula \[ B\left( {v, w}\right) = \frac{1}{2}\left\lbrack {{f}_{2}\left( {v + w}\right) - {f}_{2}\left( v\right) - {f}_{2}\left( w\right) }\right\rbrack . \] We have suppressed the \( x \) from the notation to focus on the relevant second variable \( v \) . Thus the quadratic map and the symmetric bilinear map determine each other uniquely. The above discussion has been local, over an open set \( U \) in a Banach space. In Proposition 3.4 and the subsequent discussion of connections, we show how to globalize the bilinear map \( B \) intrinsically on the manifold. Examples. As a trivial special case, we can always take \( {f}_{2}\left( {x, v}\right) = \left( {v,0}\right) \) to represent the second component of a spray in the chart. In the chapter on Riemannian metrics, we shall see how to construct a spray in a natural fashion, depending on the metric. In the chapter on covariant derivatives we show how a spray gives rise to such derivatives. Next, let us give the transformation rule for a spray under a change of charts, i.e. an isomorphism \( h : U \rightarrow V \) . On \( {TU} \), the map \( {Th} \) is represented by a morphism (its vector component) \[ H : U \times \mathbf{E} \rightarrow \mathbf{E} \times \mathbf{E}\;\text{ given by }\;H\left( {x, v}\right) = \left( {h\left( x\right) ,{h}^{\prime }\left( x\right) v}\right) . \] We then have one further lift to the double tangent bundle \( {TTU} \), and we may represent the diagram of maps symbolically as follows:  Then the derivative \( {H}^{\prime }\left( {x, v}\right) \) is given by the Jacobian matrix operating on column vectors \( {}^{t}\left( {u, w}\right) \) with \( u, w \in \mathbf{E} \), namely \[ {H}^{\prime }\left( {x, v}\right) = \left( \begin{matrix} {h}^{\prime }\left( x\right) & 0 \\ {h}^{\prime \prime }\left( x\right) v & {h}^{\prime }\left( x\right) \end{matrix}\right) \text{ so }{H}^{\prime }\left( {x, v}\right) \left( \begin{matrix} u \\ w \end{matrix}\right) = \left( \begin{matrix} {h}^{\prime }\left( x\right) & 0 \\ {h}^{\prime \prime }\left( x\right) v & {h}^{\prime }\left( x\right) \end{matrix}\right) \left( \begin{matrix} u \\ w \end{matrix}\right) . \] Thus the top map on elements in the diagram is given by \[ \left( {H,{H}^{\prime }}\right) : \left( {x, v, u, w}\right) \mapsto \left( {h\left( x\right) ,{h}^{\prime }\left( x\right) v,{h}^{\prime }\left( x\right) u,{h}^{\prime \prime }\left( x\right) \left( {u, v}\right) + {h}^{\prime }\left( x\right) w}\right) . \] For the application, we put \( u = v \) because \( {f}_{1}\left( {x, v}\right) = v \), and \( w = {f}_{U,2}\left( {x, v}\right) \) , where \( {f}_{U} \) and \( {f}_{V} \) denote the representations of the spray over \( U \) and \( V \) respectively. It follows that \( {f}_{U} \) and \( {f}_{V} \) are related by the formula \[ {f}_{V}\left( {h\left( x\right) ,{h}^{\prime }\left( x\right) v}\right) = \left( {{h}^{\prime }\left( x\right) v,{h}^{\prime \prime }\left( x\right) \left( {v, v}\right) + {h}^{\prime }\left( x\right) {f}_{U,2}\left( {x, v}\right) }\right) . \] Therefore we obtain: Proposition 3.3. Change of variable formula for the quadratic part of a spray: \[ {f}_{V.2}\left( {h\left( x\right) ,{h}^{\prime }\left( x\right) v}\right) = {h}^{\prime \prime }\left( x\right) \left( {v, v}\right) + {h}^{\prime }\left( x\right) {f}_{U.2}\left( {x, v}\right) , \] \[ {B}_{V}\left( {h\left( x\right) ;{h}^{\prime }\left( x\right) v,{h}^{\prime }\left( x\right) w}\right) = {h}^{\prime \prime }\left( x\right) \left( {v, w}\right) + {h}^{\prime }\left( x\right) {B}_{U}\left( {x;v, w}\right) . \] Proposition 3.3 admits a converse: Proposition 3.4. Suppose we are given a covering of the manifold \( X \) by open sets corresponding to charts \( U, V,\ldots \), and for each \( U \) we are given a morphism \[ {B}_{U} : U \rightarrow {L}_{\mathrm{{sym}}}^{2}\left( {\mathbf{E},\mathbf{E}}\right) \] which transforms according to the formula of Proposition 3.3 under an isomorphism \( h : U \rightarrow V \) . Then there exists a unique spray whose associated bilinear map in the chart \( U \) is given by \( {B}_{U} \) . Proof. We leave the verification to the reader. Remarks. Note that \( {B}_{U}\left( {x;v, w}\right) \) does not transform like a tensor of type \( {L}_{\mathrm{{sym}}}^{2}\left( {\mathbf{E},\mathbf{E}}\right) \), i.e. a section of the bundle \( {L}_{\mathrm{{sym}}}^{2}\left( {{TX},{TX}}\right) \) . There are several ways of defining the bilinear map \( B \) intrinsically. One of them is via second order bundles, or bundles of second order jets, and to extend the terminology we have established previously to such bundles, and even higher order jet bundles involving higher derivatives, as in [Po 62]. Another way will be done below, via connections. For our immediate purposes, it suffices to have the above discussion on second-order differential equations together with Proposition 3.3 and 3.4. Sprays were introduced by Ambrose, Palais, and Singer [APS 60], and I used them (as recommended by Palais) in the earliest version [La 62]. In [Lo 69] the bilinear map \( {B}_{U} \) is expressed in terms of second order jets. The basics of differential topology and geometry were being established in the early sixties. Cf. the bibliographical notes from [Lo 69] at the end of his first chapter. ## Connections We now show how to define the bilinear map \( B \) intrinsically and directly. Matters will be clearer if we start with an arbitrary vector bundle \[ p : E \rightarrow X \] over a manifold \( X \) . As it happens we also need the notion of a fiber bundle when the fibers are not necessarily vector spaces, so don't have a linear structure. Let \( f : Y \rightarrow X \) be a morphism. We say that \( f \) (or \( Y \) over \( X \) ) is a fiber bundle if \( f \) is surjective, and if each point \( x \) of \( X \) has an open neighborhood \( U \), and there is some manifold \( Z \) and an isomorphism \( h : {f}^{-1}\left( U\right) \rightarrow U \times Z \) such that the following diagram is commutative:  Thus locally, \( f : Y \rightarrow X \) looks like the projection from a product space. The reason why we need a fiber bundle is that the tangent bundle \[ {\pi }_{E} : {TE} \rightarrow E \] is a vector bundle over \( E \), but the composite \( f = p \circ {\pi }_{E} : {TE} \rightarrow X \) is only a fiber bundle over \( X \), a fact which is obvious by picking trivializations in charts. Indeed, if \( U \) is a chart in \( X \), and if \( U \times \mathbf{F} \rightarrow U \) is a vector bundle chart for \( E \), with fiber \( \mathbf{F} \), and \( Y = {TE} \), then we have a natural isomorphism of fiber bundles over \( U \) :  Note that \( U \) being a chart in \( X \) implies that \( U \times \mathbf{E} \rightarrow U \) is a vector bundle chart for the tangent bundle \( {TU} \) over \( U \) . The tangent bundle \( {TE} \) has two natural maps making it a vector bundle: \[ {\pi }_{E} : {TE} \rightarrow E\text{is a vector bundle over}E\text{;} \] \[ T\left( p\right) : {TE} \rightarrow {TX}\text{is a vector bundle over}{TX}\text{.} \] Therefore we have a natural morphism of fiber bundle (n

Proposition 3.2. In a chart \( U \times \mathbf{E} \) for \( {TX} \), let \( f : U \times \mathbf{E} \rightarrow \mathbf{E} \times \mathbf{E} \) represent \( F \), with \( f = \left( {{f}_{1},{f}_{2}}\right) \) . Then \( f \) represents a spray if and only if, for all \( s \in \mathbf{R} \) we have \[ {f}_{2}\left( {x,{sv}}\right) = {s}^{2}{f}_{2}\left( {x, v}\right) \]

The proof follows at once from the definitions and the formula giving the chart representation of \( s{\left( {s}_{TX}\right) }_{ * } \) . Thus we see that the condition SPR 1 (in addition to being a second-order vector field), simply means that \( {f}_{2} \) is homogeneous of degree 2 in the variable \( v \) . By the remark in Chapter I,§3, it follows that \( {f}_{2} \) is a quadratic map in its second variable, and specifically, this quadratic map is given by \[ {f}_{2}\left( {x, v}\right) = \frac{1}{2}{D}_{2}^{2}{f}_{2}\left( {x,0}\right) \left( {v, v}\right) . \] Thus the spray is induced by a symmetric bilinear map given at each point \( x \) in a chart by \[ B\left( x\right) = \frac{1}{2}{D}_{2}^{2}{f}_{2}\left( {x,0}\right) . \] Conversely, suppose given a morphism \[ U \rightarrow {L}_{\text{sym }}^{2}\left( {\mathbf{E},\mathbf{E}}\right) \;\text{ given by }\;x \mapsto B\left( x\right) \] from \( U \) into the space of symmetric bilinear maps \( \mathbf{E} \times \mathbf{E} \rightarrow \mathbf{E} \) . Thus for each \( v, w \in \mathbf{E} \) the value of \( B\left( x\right) \) at \( \left( {v, w}\right) \) is denoted by \( B\left( {x;v, w}\right) \) or \( B\left( x\right) \left( {v, w}\right) \) . Define \( {f}_{2}\left( {x, v}\right) = B\left( {x;v, v}\right) \) . Then \( {f}_{2} \) is quadratic in its second variable, and the map \( f \) defined by \[ f\left( {x, v}\right) = \left( {v, B\left( {x;v, v}\right) }\right) = \left( {v,{f}_{2}\left( {x, v}\right) }\right) \] represents a spray over \( U \) . We call \( B \) the symmetric bilinear map associated with the spray. From the local representations in (1) and (2), we conclude that a curve \( \alpha \) is a geodesic if and only if \( \alpha \) satisfies the differential equation \[ {\alpha }^{\prime \prime }\left( t\right) = {B}_{\alpha \left( t\right) }\left( {{\alpha }^{\prime }\left( t\right) ,{\alpha }^{\prime }\left( t\right) }\right) \;\text{ for all }t. \]

Theorem 2.2.8. If \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| = 1 \), a.e., then \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) if and only if there is a constant \( c \) of modulus 1 such that \( {\phi }_{1} = c{\phi }_{2} \) . Proof. Clearly \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = c{\phi }_{1}{\widetilde{\mathbf{H}}}^{2} \) when \( \left| c\right| = 1 \) . Conversely, suppose that \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) with \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| = 1 \), a.e. Then there exist functions \( {f}_{1} \) and \( {f}_{2} \) in \( {\widetilde{\mathbf{H}}}^{2} \) such that \[ {\phi }_{1} = {\phi }_{2}{f}_{2}\;\text{ and }\;{\phi }_{2} = {\phi }_{1}{f}_{1} \] Since \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = 1 = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| \) a.e., it follows that \[ {\phi }_{1}\overline{{\phi }_{2}} = {f}_{2}\;\text{ and }\;{\phi }_{2}\overline{{\phi }_{1}} = {f}_{1} \] i.e., \( {f}_{1} = \overline{{f}_{2}} \) . But since \( {f}_{1} \) and \( {f}_{2} \) are in \( {\widetilde{\mathbf{H}}}^{2},{f}_{1} = \overline{{f}_{2}} \) implies that \( {f}_{1} \) has Fourier coefficients equal to 0 for all positive and for all negative indices. Since the only nonzero coefficient is in the zeroth place, \( {f}_{1} \) and \( {f}_{2} \) are constants, obviously having moduli equal to 1 . Since the unilateral shift is a restriction of the bilateral shift to an invariant subspace, invariant subspaces of the unilateral shift are determined by Theorem 2.2.7: they are the invariant subspaces of the bilateral shift that are contained in \( {\widetilde{\mathbf{H}}}^{2} \) . In this case, the functions generating the invariant subspaces are certain analytic functions whose structure is important. Definition 2.2.9. A function \( \phi \in {\mathbf{H}}^{\infty } \) satisfying \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e. is an inner function. Theorem 2.2.10. If \( \phi \) is a nonconstant inner function, then \( \left| {\phi \left( z\right) }\right| < 1 \) for all \( z \in \mathbb{D} \) . Proof. This follows immediately from Corollary 1.1.24 and Theorem 1.1.17. The definition of inner functions requires that the functions be in \( {\mathbf{H}}^{\infty } \) . It is often useful to know that this follows if a function is in \( {\mathbf{H}}^{2} \) and has boundary values of modulus 1 a.e. Theorem 2.2.11. Let \( \phi \in {\mathbf{H}}^{2} \) . If \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e., then \( \phi \) is an inner function. Proof. It only needs to be shown that \( \phi \in {\mathbf{H}}^{\infty } \) ; this follows from Corollary 1.1.24. Corollary 2.2.12 (Beurling's Theorem). Every invariant subspace of the unilateral shift other than \( \{ 0\} \) has the form \( \phi {\mathbf{H}}^{2} \), where \( \phi \) is an inner function. Proof. The unilateral shift is the restriction of multiplication by \( {e}^{i\theta } \) to \( {\widetilde{\mathbf{H}}}^{2} \), so if \( \mathcal{M} \) is an invariant subspace of the unilateral shift, it is an invariant subspace of the bilateral shift contained in \( {\widetilde{\mathbf{H}}}^{2} \) . Thus, by Theorem 2.2.7, \( \mathcal{M} = \phi {\widetilde{\mathbf{H}}}^{2} \) for some measurable function satisfying \( \left| {\phi \left( {e}^{i\theta }\right) }\right| = 1 \) a.e. (Note that \( \{ 0\} \) is the only reducing subspace of the bilateral shift that is contained in \( {\widetilde{\mathbf{H}}}^{2} \) .) Since \( 1 \in {\widetilde{\mathbf{H}}}^{2},\phi \in {\widetilde{\mathbf{H}}}^{2} \) . Translating this situation back to \( {\mathbf{H}}^{2} \) on the disk gives \( \mathcal{M} = \phi {\mathbf{H}}^{2} \) with \( \phi \) inner, by Theorem 2.2.11. Corollary 2.2.13. Every invariant subspace of the unilateral shift is cyclic. (See Definition 1.2.17.) Proof. If \( \mathcal{M} \) is an invariant subspace of the unilateral shift, it has the form \( \phi {\mathbf{H}}^{2} \) by Beurling’s theorem (Corollary 2.2.12). For each \( n,{U}^{n}\phi = {z}^{n}\phi \), so \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}\phi }\right\} \) contains all functions of the form \( \phi \left( z\right) p\left( z\right) \), where \( p \) is a polynomial. Since the polynomials are dense in \( {\mathbf{H}}^{2} \) (as the finite sequences are dense in \( \left. {\ell }^{2}\right) \), it follows that \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}\phi }\right\} = \phi {\mathbf{H}}^{2} \) . ## 2.3 Inner and Outer Functions We shall see that every function in \( {\mathbf{H}}^{2} \), other than the constant function 0, can be written as a product of an inner function and a cyclic vector for the unilateral shift. Such cyclic vectors will be shown to have a special form. Definition 2.3.1. The function \( F \in {\mathbf{H}}^{2} \) is an outer function if \( F \) is a cyclic vector for the unilateral shift. That is, \( F \) is an outer function if \[ \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{U}^{k}F}\right\} = {\mathbf{H}}^{2} \] Theorem 2.3.2. If \( F \) is an outer function, then \( F \) has no zeros in \( \mathbb{D} \) . Proof. If \( F\left( {z}_{0}\right) = 0 \), then \( \left( {{U}^{n}F}\right) \left( {z}_{0}\right) = {z}_{0}^{n}F\left( {z}_{0}\right) = 0 \) for all \( n \) . Since the limit of a sequence of functions in \( {\mathbf{H}}^{2} \) that all vanish at \( {z}_{0} \) must also vanish at \( {z}_{0} \) (Theorem 1.1.9), \[ \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{U}^{k}F}\right\} \] cannot be all of \( {\mathbf{H}}^{2} \) . Hence there is no \( {z}_{0} \in \mathbb{D} \) with \( F\left( {z}_{0}\right) = 0 \) . Recall that a function analytic on \( \mathbb{D} \) is identically zero if it vanishes on a set that has a limit point in \( \mathbb{D} \) . The next theorem is an analogous result for boundary values of functions in \( {\mathbf{H}}^{2} \) . Theorem 2.3.3 (The F. and M. Riesz Theorem). If \( f \in {\mathbf{H}}^{2} \) and the set \[ \left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\} \] has positive measure, then \( f \) is identically 0 on \( \mathbb{D} \) . Proof. Let \( E = \left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\} \) and let \[ \mathcal{M} = \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{U}^{k}\widetilde{f}}\right\} = \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{e}^{ik\theta }\widetilde{f}}\right\} \] Then every function \( \widetilde{g} \in \mathcal{M} \) vanishes on \( E \), since all functions \( {e}^{ik\theta }\widetilde{f} \) do. If \( \widetilde{f} \) is not identically zero, it follows from Beurling's theorem (Theorem 2.2.12) that \( \mathcal{M} = \widetilde{\phi }{\widetilde{\mathbf{H}}}^{2} \) for some inner function \( \phi \) . In particular, this implies that \( \widetilde{\phi } \in \mathcal{M} \) , so \( \widetilde{\phi } \) vanishes on \( E \) . But \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e. This contradicts the hypothesis that \( E \) has positive measure, thus \( \widetilde{f} \), and hence \( f \), must be identically zero. Another beautiful result that follows from Beurling's theorem is the following factorization of functions in \( {\mathbf{H}}^{2} \) . Theorem 2.3.4. If \( f \) is a function in \( {\mathbf{H}}^{2} \) that is not identically zero, then \( f = {\phi F} \), where \( \phi \) is an inner function and \( F \) is an outer function. This factorization is unique up to constant factors. Proof. Let \( f \in {\mathbf{H}}^{2} \) and consider \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} \) . If this span is \( {\mathbf{H}}^{2} \), then \( f \) is outer by definition, and we can take \( \phi \) to be the constant function 1 and \( F = f \) to obtain the desired conclusion. If \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} \neq {\mathbf{H}}^{2} \), then, by Beurling’s theorem (Corollary 2.2.12), there must exist a nonconstant inner function \( \phi \) with \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} = \phi {\mathbf{H}}^{2} \) . Since \( f \) is in \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} = \phi {\mathbf{H}}^{2} \), there exists a function \( F \) in \( {\mathbf{H}}^{2} \) with \( f = {\phi F} \) . We shall show that \( F \) is outer. The invariant subspace \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}F}\right\} \) equals \( \psi {\mathbf{H}}^{2} \) for some inner function \( \psi \) . Then, since \( f = {\phi F} \), it follows that \( {U}^{n}f = {U}^{n}\left( {\phi F}\right) = \phi {U}^{n}F \) for every positive integer \( n \), from which we can conclude, by taking linear spans, that \( \phi {\mathbf{H}}^{2} = {\phi \psi }{\mathbf{H}}^{2} \) . Theorem 2.2.8 now implies that \( \phi \) and \( {\phi \psi } \) are constant multiples of each other. Hence \( \psi \) must be a constant function. Therefore \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}F}\right\} = {\mathbf{H}}^{2} \), so \( F \) is an outer function. Note that if \( f = {\phi F} \) with \( \phi \) inner and \( F \) outer, then \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} = \phi {\mathbf{H}}^{2} \) . Thus uniqueness of the factorization follows from the corresponding assertion in Theorem 2.2.8. Definition 2.3.5. For \( f \in {\mathbf{H}}^{2} \), if \( f = {\phi F} \) with \( \phi \) inner and \( F \) outer, we call \( \phi \) the inner part of \( f \) and \( F \) the outer part of \( f \) . Theorem 2.3.6. The zeros of an \( {\mathbf{H}}^{2} \) function are precisely the zeros of its inn

Theorem 2.2.8. If \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| = 1 \), a.e., then \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) if and only if there is a constant \( c \) of modulus 1 such that \( {\phi }_{1} = c{\phi }_{2} \).

Proof. Clearly \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = c{\phi }_{1}{\widetilde{\mathbf{H}}}^{2} \) when \( \left| c\right| = 1 \). Conversely, suppose that \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) with \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| = 1 \), a.e. Then there exist functions \( {f}_{1} \) and \( {f}_{2} \) in \( {\widetilde{\mathbf{H}}}^{2} \) such that \[ {\phi }_{1} = {\phi }_{2}{f}_{2}\;\text{ and }\;{\phi }_{2} = {\phi }_{1}{f}_{1} \] Since \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = 1 = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| \) a.e., it follows that \[ {\phi }_{1}\overline{{\phi }_{2}} = {f}_{2}\;\text{ and }\;{\phi }_{2}\overline{{\phi }_{1}} = {f}_{1} \] i.e., \( {f}_{1} = \overline{{f}_{2}} \). But since \( {f}_{1} \) and \( {f}_{2} \) are in \( {\widetilde{\mathbf{H}}}^{2},{f}_{1} = \overline{{f}_{2}} \) implies that \( {f}_{1} \) has Fourier coefficients equal to 0 for all positive and for all negative indices. Since the only nonzero coefficient is in the zeroth place, \( {f}_{1} \) and \( {f}_{2} \) are constants, obviously having moduli equal to 1.

Theorem 3.1. (Eisenstein’s Criterion). Let \( A \) be a factorial ring. Let \( K \) be its quotient field. Let \( f\left( X\right) = {a}_{n}{X}^{n} + \cdots + {a}_{0} \) be a polynomial of degree \( n \geqq 1 \) in \( A\left\lbrack X\right\rbrack \) . Let \( p \) be a prime of \( A \), and assume: \[ {a}_{n} ≢ 0\;\left( {\;\operatorname{mod}\;p}\right) ,\;{a}_{i} \equiv 0\;\left( {\;\operatorname{mod}\;p}\right) \;\text{ for all }\;i < n, \] \[ {a}_{0} ≢ 0\;\left( {\;\operatorname{mod}\;{p}^{2}}\right) \] Then \( f\left( X\right) \) is irreducible in \( K\left\lbrack X\right\rbrack \) . Proof. Extracting a g.c.d. for the coefficients of \( f \), we may assume without loss of generality that the content of \( f \) is 1 . If there exists a factorization into factors of degree \( \geqq 1 \) in \( K\left\lbrack X\right\rbrack \), then by the corollary of Gauss’ lemma there exists a factorization in \( A\left\lbrack X\right\rbrack \), say \( f\left( X\right) = g\left( X\right) h\left( X\right) \) , \[ g\left( X\right) = {b}_{d}{X}^{d} + \cdots + {b}_{0} \] \[ h\left( X\right) = {c}_{m}{X}^{m} + \cdots + {c}_{0} \] with \( d, m \geqq 1 \) and \( {b}_{d}{c}_{m} \neq 0 \) . Since \( {b}_{0}{c}_{0} = {a}_{0} \) is divisible by \( p \) but not \( {p}^{2} \), it follows that one of \( {b}_{0},{c}_{0} \) is not divisible by \( p \), say \( {b}_{0} \) . Then \( p \mid {c}_{0} \) . Since \( {c}_{m}{b}_{d} = {a}_{n} \) is not divisible by \( p \), it follows that \( p \) does not divide \( {c}_{m} \) . Let \( {c}_{r} \) be the coefficient of \( h \) furthest to the right such that \( {c}_{r} ≢ 0\left( {\;\operatorname{mod}\;p}\right) \) . Then \[ {a}_{r} = {b}_{0}{c}_{r} + {b}_{1}{c}_{r - 1} + \cdots . \] Since \( p \nmid {b}_{0}{c}_{r} \) but \( p \) divides every other term in this sum, we conclude that \( p \nmid {a}_{r} \), a contradiction which proves our theorem. Example. Let \( a \) be a non-zero square-free integer \( \neq \pm 1 \) . Then for any integer \( n \geqq 1 \), the polynomial \( {X}^{n} - a \) is irreducible over \( \mathbf{Q} \) . The polynomials \( 3{X}^{5} - {15} \) and \( 2{X}^{10} - {21} \) are irreducible over \( \mathbf{Q} \) . There are some cases in which a polynomial does not satisfy Eisenstein's criterion, but a simple transform of it does. Example. Let \( p \) be a prime number. Then the polynomial \[ f\left( X\right) = {X}^{p - 1} + \cdots + 1 \] is irreducible over \( \mathbf{Q} \) . Proof. It will suffice to prove that the polynomial \( f\left( {X + 1}\right) \) is irreducible over \( \mathbf{Q} \) . We note that the binomial coefficients \[ \left( \begin{array}{l} p \\ v \end{array}\right) = \frac{p!}{v!\left( {p - v}\right) !},\;1 \leqq v \leqq p - 1, \] are divisible by \( p \) (because the numerator is divisible by \( p \) and the denominator is not, and the coefficient is an integer). We have \[ f\left( {X + 1}\right) = \frac{{\left( X + 1\right) }^{p} - 1}{\left( {X + 1}\right) - 1} = \frac{{X}^{p} + p{X}^{p - 1} + \cdots + {pX}}{X} \] from which one sees that \( f\left( {X + 1}\right) \) satisfies Eisenstein’s criterion. Example. Let \( E \) be a field and \( t \) an element of some field containing \( E \) such that \( t \) is transcendental over \( E \) . Let \( K \) be the quotient field of \( E\left\lbrack t\right\rbrack \) . For any integer \( n \geqq 1 \) the polynomial \( {X}^{n} - t \) is irreducible in \( K\left\lbrack X\right\rbrack \) . This comes from the fact that the ring \( A = E\left\lbrack t\right\rbrack \) is factorial and that \( t \) is a prime in it. Theorem 3.2. (Reduction Criterion). Let \( A, B \) be entire rings, and let \[ \varphi : A \rightarrow B \] be a homomorphism. Let \( K, L \) be the quotient fields of \( A \) and \( B \) respectively. Let \( f \in A\left\lbrack X\right\rbrack \) be such that \( {\varphi f} \neq 0 \) and \( \deg {\varphi f} = \deg f \) . If \( {\varphi f} \) is irreducible in \( L\left\lbrack X\right\rbrack \), then \( f \) does not have a factorization \( f\left( X\right) = g\left( X\right) h\left( X\right) \) with \[ g, h \in A\left\lbrack X\right\rbrack \;\text{ and }\;\deg g,\deg h \geqq 1. \] Proof. Suppose \( f \) has such a factorization. Then \( {\varphi f} = \left( {\varphi g}\right) \left( {\varphi h}\right) \) . Since \( \deg {\varphi g} \leqq \deg g \) and \( \deg {\varphi h} \leqq \deg h \), our hypothesis implies that we must have equality in these degree relations. Hence from the irreducibility in \( L\left\lbrack X\right\rbrack \) we conclude that \( g \) or \( h \) is an element of \( A \), as desired. In the preceding criterion, suppose that \( A \) is a local ring, i.e. a ring having a unique maximal ideal \( \mathfrak{p} \), and that \( \mathfrak{p} \) is the kernel of \( \varphi \) . Then from the irreducibility of \( {\varphi f} \) in \( L\left\lbrack X\right\rbrack \) we conclude the irreducibility of \( f \) in \( A\left\lbrack X\right\rbrack \) . Indeed, any element of \( A \) which does not lie in \( \mathfrak{p} \) must be a unit in \( A \), so our last conclusion in the proof can be strengthened to the statement that \( g \) or \( h \) is a unit in \( A \) . One can also apply the criterion when \( A \) is factorial, and in that case deduce the irreducibility of \( f \) in \( K\left\lbrack X\right\rbrack \) . Example. Let \( p \) be a prime number. It will be shown later that \( {X}^{p} - X - 1 \) is irreducible over the field \( \mathbf{Z}/p\mathbf{Z} \) . Hence \( {X}^{p} - X - 1 \) is irreducible over \( \mathbf{Q} \) . Similarly, \[ {X}^{5} - 5{X}^{4} - {6X} - 1 \] is irreducible over \( \mathbf{Q} \) . There is also a routine elementary school test whether a polynomial has a root or not. Proposition 3.3. (Integral Root Test). Let \( A \) be a factorial ring and \( K \) its quotient field. Let \[ f\left( X\right) = {a}_{n}{X}^{n} + \cdots + {a}_{0} \in A\left\lbrack X\right\rbrack . \] Let \( \alpha \in K \) be a root of \( f \), with \( \alpha = b/d \) expressed with \( b, d \in A \) and \( b, d \) relatively prime. Then \( b \mid {a}_{0} \) and \( d \mid {a}_{n} \) . In particular, if the leading coefficient \( {a}_{n} \) is 1, then a root \( \alpha \) must lie in \( A \) and divides \( {a}_{0} \) . We leave the proof to the reader, who should be used to this one from way back. As an irreducibility test, the test is useful especially for a polynomial of degree 2 or 3 , when reducibility is equivalent with the existence of a root in the given field. ## §4. HILBERT'S THEOREM This section proves a basic theorem of Hilbert concerning the ideals of a polynomial ring. We define a commutative ring \( A \) to be Noetherian if every ideal is finitely generated. Theorem 4.1. Let \( A \) be a commutative Noetherian ring. Then the polynomial ring \( A\left\lbrack X\right\rbrack \) is also Noetherian. Proof. Let \( \mathfrak{A} \) be an ideal of \( A\left\lbrack X\right\rbrack \) . Let \( {\mathfrak{a}}_{i} \) consist of 0 and the set of elements \( a \in A \) appearing as leading coefficient in some polynomial \[ {a}_{0} + {a}_{1}X + \cdots + a{X}^{i} \] lying in \( \mathfrak{A} \) . Then it is clear that \( {\mathfrak{a}}_{i} \) is an ideal. (If \( a, b \) are in \( {\mathfrak{a}}_{i} \), then \( a \pm b \) is in \( {a}_{i} \) as one sees by taking the sum and difference of the corresponding polynomials. If \( x \in A \), then \( {xa} \in {\mathfrak{a}}_{i} \) as one sees by multiplying the corresponding polynomial by \( x \) .) Furthermore we have \[ {\mathfrak{a}}_{0} \subset {\mathfrak{a}}_{1} \subset {\mathfrak{a}}_{2} \subset \cdots , \] in other words, our sequence of ideals \( \left\{ {a}_{i}\right\} \) is increasing. Indeed, to see this multiply the above polynomial by \( X \) to see that \( a \in {\mathfrak{a}}_{i + 1} \) . By criterion (2) of Chapter X, \( §1 \), the sequence of ideals \( \left\{ {a}_{i}\right\} \) stops, say at \( {a}_{r} \) : \[ {\mathfrak{a}}_{0} \subset {\mathfrak{a}}_{1} \subset {\mathfrak{a}}_{2} \subset \cdots \subset {\mathfrak{a}}_{r} = {\mathfrak{a}}_{r + 1} = \cdots . \] Let \[ {a}_{01},\ldots ,{a}_{0{n}_{0}}\text{be generators for}{\mathfrak{a}}_{0}\text{,} \] .............................. \( {a}_{r1},\ldots ,{a}_{r{n}_{r}} \) be generators for \( {\mathfrak{a}}_{r} \) . For each \( i = 0,\ldots, r \) and \( j = 1,\ldots ,{n}_{i} \) let \( {f}_{ij} \) be a polynomial in \( \mathfrak{A} \), of degree \( i \), with leading coefficient \( {a}_{ij} \) . We contend that the polynomials \( {f}_{ij} \) are a set of generators for \( \mathfrak{A} \) . Let \( f \) be a polynomial of degree \( d \) in \( \mathfrak{A} \) . We shall prove that \( f \) is in the ideal generated by the \( {f}_{ij} \), by induction on \( d \) . Say \( d \geqq 0 \) . If \( d > r \), then we note that the leading coefficients of \[ {X}^{d - r}{f}_{r1},\ldots ,{X}^{d - r}{f}_{r{n}_{r}} \] generate \( {\mathfrak{a}}_{d} \) . Hence there exist elements \( {c}_{1},\ldots ,{c}_{{n}_{r}} \in A \) such that the polynomial \[ f - {c}_{1}{X}^{d - r}{f}_{r1} - \cdots - {c}_{{n}_{r}}{X}^{d - r}{f}_{r{n}_{r}} \] has degree \( < d \), and this polynomial also lies in \( \mathfrak{A} \) . If \( d \leqq r \), we can subtract a linear combination \[ f - {c}_{1}{f}_{d1} - \cdots - {c}_{{n}_{d}}{f}_{d{n}_{d}} \] to get a polynomial of degree \( < d \), also lying in \( \mathfrak{A} \) . We note that the polynomial we have subtracted from \( f \) lies in the ideal generated by the \( {f}_{ij} \) . By induction, we can subtract a polynomial \( g \) in the ideal generated by the \( {f}_{ij} \) such that \( f - g = 0 \), thereby proving our theorem. We note that if \( \varphi : A \rightarrow B \) is a surjective homomorphism of commutative rings and \( A \) is Noetherian, so is \( B \) . Indeed, let \( \mathfrak{b} \) be an ideal of \( B \), so \( {\varphi }^{-1}\left( \mathfrak{b}\right) \) is an ideal of \( A \) . Then there is a finite number of generators \( \left(

(Theorem 3.1. (Eisenstein’s Criterion). Let \( A \) be a factorial ring. Let \( K \) be its quotient field. Let \( f\left( X\right) = {a}_{n}{X}^{n} + \cdots + {a}_{0} \) be a polynomial of degree \( n \geqq 1 \) in \( A\left\lbrack X\right\rbrack \) . Let \( p \) be a prime of \( A \), and assume: \[ {a}_{n} ≢ 0\;\left( {\;\operatorname{mod}\;p}\right) ,\;{a}_{i} \equiv 0\;\left( {\;\operatorname{mod}\;p}\right) \;\text{ for all }\;i < n, \] \[ {a}_{0} ≢ 0\;\left( {\;\operatorname{mod}\;{p}^{2}}\right) \] Then \( f\left( X\right) \) is irreducible in \( K\left\lbrack X\right\rbrack \) .)

(Proof. Extracting a g.c.d. for the coefficients of \( f \), we may assume without loss of generality that the content of \( f \) is 1 . If there exists a factorization into factors of degree \( \geqq 1 \) in \( K\left\lbrack X\right\rbrack \), then by the corollary of Gauss’ lemma there exists a factorization in \( A\left\lbrack X\right\rbrack \), say \( f\left( X\right) = g\left( X\right) h\left( X\right) \) , \[ g\left( X\right) = {b}_{d}{X}^{d} + \cdots + {b}_{0} \] \[ h\left( X\right) = {c}_{m}{X}^{m} + \cdots + {c}_{0} \] with \( d, m \geqq 1 \) and \( {b}_{d}{c}_{m} \neq 0 \) . Since \( {b}_{0}{c}_{0} = {a}_{0} \) is divisible by \( p \) but not \( {p}^{2} \), it follows that one of \( {b}_{0},{c}_{0} \) is not divisible by \( p \), say \( {b}_{0} \) . Then \( p \mid {c}_{0} \) . Since \( {c}_{m}{b}_{d} = {a}_{n} \) is not divisible by \( p \), it follows that \( p \) does not divide \( {c}_{m} \) . Let \( {c}_{r} \) be the coefficient of \( h \) furthest to the right such that \( {c}_{r} ≢ 0\left( {\;\operatorname{mod}\;p}\right) \) . Then \[ {a}_{r} = {b}_{0}{c}_{r} + {b}_{1}{c}_{r - 1} + \cdots . \] Since \( p \nmid {b}_{0}{c}_{r} \) but \( p \) divides every other term in this sum, we conclude that \( p \nmid {a}_{r} \), a contradiction which proves our theorem.)

Proposition 5.46. Suppose \( M \) is a smooth manifold without boundary and \( D \subseteq M \) is a regular domain. The topological interior and boundary of \( D \) are equal to its manifold interior and boundary, respectively. Proof. Suppose \( p \in D \) is arbitrary. If \( p \) is in the manifold boundary of \( D \), Theorem 4.15 shows that there exist a smooth boundary chart \( \left( {U,\varphi }\right) \) for \( D \) centered at \( p \) and a smooth chart \( \left( {V,\psi }\right) \) for \( M \) centered at \( p \) in which \( F \) has the coordinate representation \( F\left( {{x}^{1},\ldots ,{x}^{n}}\right) = \left( {{x}^{1},\ldots ,{x}^{n}}\right) \), where \( n = \dim M = \dim D \) . Since \( D \) has the subspace topology, \( U = D \cap W \) for some open subset \( W \subseteq M \), so \( {V}_{0} = V \cap W \) is a neighborhood of \( p \) in \( M \) such that \( {V}_{0} \cap D \) consists of all the points in \( {V}_{0} \) whose \( {x}^{m} \) coordinate is nonnegative. Thus every neighborhood of \( p \) intersects both \( D \) and \( M \smallsetminus D \), so \( p \) is in the topological boundary of \( D \) . On the other hand, suppose \( p \) is in the manifold interior of \( D \) . The manifold interior is a smooth embedded codimension- 0 submanifold without boundary in \( M \) , so it is an open subset by Proposition 5.1. Thus \( p \) is in the topological interior of \( D \) . Conversely, if \( p \) is in the topological interior of \( D \), then it is not in the topological boundary, so the preceding argument shows that it is not in the manifold boundary and hence must be in the manifold interior. Similarly, if \( p \) is in the topological boundary, it is also in the manifold boundary. Here are some ways in which regular domains often arise. Proposition 5.47. Suppose \( M \) is a smooth manifold and \( f \in {C}^{\infty }\left( M\right) \) . (a) For each regular value \( b \) of \( f \), the sublevel set \( {f}^{-1}(\left( {-\infty, b\rbrack }\right) \) is a regular domain in \( M \) . (b) If \( a \) and \( b \) are two regular values of \( f \) with \( a < b \), then \( {f}^{-1}\left( \left\lbrack {a, b}\right\rbrack \right) \) is a regular domain in \( M \) . Proof. Problem 5-21. A set of the form \( {f}^{-1}(\left( {-\infty, b\rbrack }\right) \) for \( b \) a regular value of \( f \) is called a regular sublevel set of \( f \) . Part (a) of the preceding theorem shows that every regular sublevel set of a smooth real-valued function is a regular domain. If \( D \subseteq M \) is a regular domain and \( f \in {C}^{\infty }\left( M\right) \) is a smooth function such that \( D \) is a regular sublevel set of \( f \), then \( f \) is called a defining function for \( \mathbf{D} \) . Theorem 5.48. If \( M \) is a smooth manifold and \( D \subseteq M \) is a regular domain, then there exists a defining function for \( D \) . If \( D \) is compact, then \( f \) can be taken to be a smooth exhaustion function for \( M \) . Proof. Problem 5-22. Many (though not all) of the earlier results in this chapter have analogues for submanifolds with boundary. Since we will have little reason to consider nonem-bedded submanifolds with boundary, we focus primarily on the embedded case. The statements in the following proposition can be proved in the same way as their submanifold counterparts. Proposition 5.49 (Properties of Submanifolds with Boundary). Suppose \( M \) is a smooth manifold with or without boundary. (a) Every open subset of \( M \) is an embedded codimension- 0 submanifold with (possibly empty) boundary. (b) If \( N \) is a smooth manifold with boundary and \( F : N \rightarrow M \) is a smooth embedding, then with the subspace topology \( F\left( N\right) \) is a topological manifold with boundary, and it has a smooth structure making it into an embedded submani-fold with boundary in \( M \) . (c) An embedded submanifold with boundary in \( M \) is properly embedded if and only if it is closed. (d) If \( S \subseteq M \) is an immersed submanifold with boundary, then for each \( p \in S \) there exists a neighborhood \( U \) of \( p \) in \( S \) that is embedded in \( M \) . - Exercise 5.50. Prove the preceding proposition. In order to adapt the results that depended on the existence of local slice charts, we have to generalize the local \( k \) -slice condition as follows. Suppose \( M \) is a smooth manifold (without boundary). If \( \left( {U,\left( {x}^{i}\right) }\right) \) is a chart for \( M \), a \( k \) -dimensional half-slice of \( \mathbf{U} \) is a subset of the following form for some constants \( {c}^{k + 1},\ldots ,{c}^{n} \) : \[ \left\{ {\left( {{x}^{1},\ldots ,{x}^{n}}\right) \in U : {x}^{k + 1} = {c}^{k + 1},\ldots ,{x}^{n} = {c}^{n},\text{ and }{x}^{k} \geq 0}\right\} . \] We say that a subset \( S \subseteq M \) satisfies the local \( k \) -slice condition for submani-folds with boundary if each point of \( S \) is contained in the domain of a smooth chart \( \left( {U,\left( {x}^{i}\right) }\right) \) such that \( S \cap U \) is either an ordinary \( k \) -dimensional slice or a \( k \) -dimensional half-slice. In the former case, the chart is called an interior slice chart for \( S \) in \( M \), and in the latter, it is a boundary slice chart for \( S \) in \( M \) . Theorem 5.51. Let \( M \) be a smooth \( n \) -manifold without boundary. If \( S \subseteq M \) is an embedded \( k \) -dimensional submanifold with boundary, then \( S \) satisfies the local \( k \) -slice condition for submanifolds with boundary. Conversely, if \( S \subseteq M \) is a subset that satisfies the local \( k \) -slice condition for submanifolds with boundary, then with the subspace topology, \( S \) is a topological \( k \) -manifold with boundary, and it has a smooth structure making it into an embedded submanifold with boundary in \( M \) . - Exercise 5.52. Prove the preceding theorem. Using the preceding theorem in place of Theorem 5.8, one can readily prove the following theorem. Theorem 5.53 (Restricting Maps to Submanifolds with Boundary). Suppose \( M \) and \( N \) are smooth manifolds with boundary and \( S \subseteq M \) is an embedded submani-fold with boundary. (a) RESTRICTING THE DOMAIN: If \( F : M \rightarrow N \) is a smooth map, then \( {\left. F\right| }_{S} : S \rightarrow \) \( N \) is smooth. (b) RESTRICTING THE CODOMAIN: If \( \partial M = \varnothing \) and \( F : N \rightarrow M \) is a smooth map whose image is contained in \( S \), then \( F \) is smooth as a map from \( N \) to \( S \) . Remark. The requirement that \( \partial M = \varnothing \) can be removed in part (b) just as for Theorem 5.29; see Problem 9-13. - Exercise 5.54. Prove Theorem 5.53. ## Problems 5-1. Consider the map \( \Phi : {\mathbb{R}}^{4} \rightarrow {\mathbb{R}}^{2} \) defined by \[ \Phi \left( {x, y, s, t}\right) = \left( {{x}^{2} + y,{x}^{2} + {y}^{2} + {s}^{2} + {t}^{2} + y}\right) . \] Show that \( \left( {0,1}\right) \) is a regular value of \( \Phi \), and that the level set \( {\Phi }^{-1}\left( {0,1}\right) \) is diffeomorphic to \( {\mathbb{S}}^{2} \) . 5-2. Prove Theorem 5.11 (the boundary of a manifold with boundary is an embedded submanifold). 5-3. Prove Proposition 5.21 (sufficient conditions for immersed submanifolds to be embedded). 5-4. Show that the image of the curve \( \beta : \left( {-\pi ,\pi }\right) \rightarrow {\mathbb{R}}^{2} \) of Example 4.19 is not an embedded submanifold of \( {\mathbb{R}}^{2} \) . [Be careful: this is not the same as showing that \( \beta \) is not an embedding.] 5-5. Let \( \gamma : \mathbb{R} \rightarrow {\mathbb{T}}^{2} \) be the curve of Example 4.20. Show that \( \gamma \left( \mathbb{R}\right) \) is not an embedded submanifold of the torus. [Remark: the warning in Problem 5-4 applies in this case as well.] 5-6. Suppose \( M \subseteq {\mathbb{R}}^{n} \) is an embedded \( m \) -dimensional submanifold, and let \( {UM} \subseteq T{\mathbb{R}}^{n} \) be the set of all unit tangent vectors to \( M \) : \[ {UM} = \left\{ {\left( {x, v}\right) \in T{\mathbb{R}}^{n} : x \in M, v \in {T}_{x}M,\left| v\right| = 1}\right\} . \] It is called the unit tangent bundle of \( \mathbf{M} \) . Prove that \( {UM} \) is an embedded \( \left( {{2m} - 1}\right) \) -dimensional submanifold of \( T{\mathbb{R}}^{n} \approx {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) . (Used on p. 147.) 5-7. Let \( F : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be defined by \( F\left( {x, y}\right) = {x}^{3} + {xy} + {y}^{3} \) . Which level sets of \( F \) are embedded submanifolds of \( {\mathbb{R}}^{2} \) ? For each level set, prove either that it is or that it is not an embedded submanifold. 5-8. Suppose \( M \) is a smooth \( n \) -manifold and \( B \subseteq M \) is a regular coordinate ball. Show that \( M \smallsetminus B \) is a smooth manifold with boundary, whose boundary is diffeomorphic to \( {\mathbb{S}}^{n - 1} \) . (Used on p. 225.) 5-9. Let \( S \subseteq {\mathbb{R}}^{2} \) be the boundary of the square of side 2 centered at the origin (see Problem 3-5). Show that \( S \) does not have a topology and smooth structure in which it is an immersed submanifold of \( {\mathbb{R}}^{2} \) . 5-10. For each \( a \in \mathbb{R} \), let \( {M}_{a} \) be the subset of \( {\mathbb{R}}^{2} \) defined by \[ {M}_{a} = \left\{ {\left( {x, y}\right) : {y}^{2} = x\left( {x - 1}\right) \left( {x - a}\right) }\right\} . \] For which values of \( a \) is \( {M}_{a} \) an embedded submanifold of \( {\mathbb{R}}^{2} \) ? For which values can \( {M}_{a} \) be given a topology and smooth structure making it into an immersed submanifold? 5-11. Let \( \Phi : {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be defined by \( \Phi \left( {x, y}\right) = {x}^{2} - {y}^{2} \) . (a) Show that \( {\Phi }^{-1}\left( 0\right) \) is not an embedded submanifold of \( {\mathbb{R}}^{2} \) . (b) Can \( {\Phi }^{-1}\left( 0\right) \) be given a topology and

Proposition 5.46. Suppose \( M \) is a smooth manifold without boundary and \( D \subseteq M \) is a regular domain. The topological interior and boundary of \( D \) are equal to its manifold interior and boundary, respectively.

Suppose \( p \in D \) is arbitrary. If \( p \) is in the manifold boundary of \( D \), Theorem 4.15 shows that there exist a smooth boundary chart \( \left( {U,\varphi }\right) \) for \( D \) centered at \( p \) and a smooth chart \( \left( {V,\psi }\right) \) for \( M \) centered at \( p \) in which \( F \) has the coordinate representation \( F\left( {{x}^{1},\ldots ,{x}^{n}}\right) = \left( {{x}^{1},\ldots ,{x}^{n}}\right) \), where \( n = \dim M = \dim D \). Since \( D \) has the subspace topology, \( U = D \cap W \) for some open subset \( W \subseteq M \), so \( {V}_{0} = V \cap W \) is a neighborhood of \( p \) in \( M \) such that \( {V}_{0} \cap D \) consists of all the points in \( {V}_{0} \) whose \( {x}^{m} \) coordinate is nonnegative. Thus every neighborhood of \( p \) intersects both \( D \) and \( M \smallsetminus D \), so \( p \) is in the topological boundary of \( D \). On the other hand, suppose \( p \) is in the manifold interior of \( D \). The manifold interior is a smooth embedded codimension-0 submanifold without boundary in \( M \), so it is an open subset by Proposition 5.1. Thus \( p \) is in the topological interior of \( D \). Conversely, if \( p \) is in the topological interior of \( D \), then it is not in the topological boundary, so the preceding argument shows that it is not in the manifold boundary and hence must be in the manifold interior. Similarly, if \( p \) is in the topological boundary, it is also in the manifold boundary.

Lemma 3.7. Let \( f\left( z\right) \in {H}^{1} \) . Then the Fourier transform \[ \widehat{f}\left( s\right) = {\int }_{-\infty }^{\infty }f\left( t\right) {e}^{-{2\pi ist}}{dt} = 0 \] for all \( s \leq 0 \) . Proof. By the continuity of \( f \rightarrow \widehat{f} \), we may suppose \( \int \in {\mathfrak{A}}_{N} \) . Then for \( s \leq 0, F\left( z\right) = f\left( z\right) {e}^{-{2\pi isz}} \) is also in \( {\mathfrak{A}}_{N} \) . The result now follows from Cauchy’s theorem because \[ {\int }_{0}^{\pi }\left| {F\left( {R{e}^{i\theta }}\right) }\right| {Rd\theta } \rightarrow 0\;\left( {R \rightarrow \infty }\right) . \] Notice that (3.6) \[ {P}_{z}\left( t\right) = \frac{1}{2\pi i}\left( {\frac{1}{t - z} - \frac{1}{t - \bar{z}}}\right) . \] Also notice that for \( f \in {H}^{1} \), Lemma 3.7 applied to \( {\left( t - \bar{z}\right) }^{-1}f\left( t\right) \) yields \[ \int \frac{f\left( t\right) }{t - \bar{z}}{dt} = 0,\;\operatorname{Im}z > 0. \] Theorem 3.8. Let \( {d\mu }\left( t\right) \) be a finite complex measure on \( \mathbb{R} \) such that either (a) \[ \int \frac{{d\mu }\left( t\right) }{t - z} = 0\;\text{ on }\;\operatorname{Im}z < 0 \] or (b) \[ \widehat{\mu }\left( s\right) = \int {e}^{-{2\pi ist}}{d\mu }\left( t\right) = 0\;\text{ on }\;s < 0. \] Then \( {d\mu } \) is absolutely continuous and \( {d\mu } = f\left( t\right) {dt} \), where \( f\left( t\right) \in {H}^{1} \) . Proof. If (a) holds, then by (3.6) \( f\left( z\right) = {P}_{y} * \mu \left( x\right) \) is analytic and the result follows from Theorem 3.6. Assume (b) holds. The Poisson kernel \( {P}_{y}\left( t\right) \) has Fourier transform \[ \int {e}^{-{2\pi ist}}{P}_{y}\left( t\right) {dt} = {e}^{-{2\pi }\left| s\right| y} \] because \( {\widehat{P}}_{y}\left( {-s}\right) = \overline{{\widehat{P}}_{y}\left( s\right) } \) since \( {P}_{y} \) is real, and because if \( s \leq 0,{e}^{-{2\pi isz}} \) is the bounded harmonic function with boundary values \( {e}^{-{2\pi ist}} \) . Let \( {f}_{y}\left( x\right) = {P}_{y} * \) \( \mu \left( x\right) \) . By Fubini’s theorem, \( {f}_{y} \) has Fourier transform \[ {\widehat{f}}_{y}\left( s\right) = \left\{ \begin{array}{ll} {e}^{-{2\pi xy}}\widehat{\mu }\left( s\right) , & s \geq 0, \\ 0, & s < 0. \end{array}\right. \] Since \( {\widehat{f}}_{y} \in {L}^{1} \), Fourier inversion implies \[ {f}_{y}\left( x\right) = \int {e}^{2\pi ixs}{\widehat{f}}_{y}\left( s\right) {ds} = {\int }_{0}^{\infty }{e}^{{2\pi i}\left( {x + {iy}}\right) s}\widehat{\mu }\left( s\right) {ds}. \] Differentiating under the integral sign then shows that \( f\left( z\right) = {f}_{y}\left( x\right) \) is analytic, and Theorem 3.6 now implies that \( f\left( z\right) \in {H}^{1} \) . The disc version of Theorem 3.6, or equivalently Theorem 3.8, is one half of the famous F. and M. Riesz theorem. The other half asserts that if \( f\left( z\right) \in \) \( {H}^{1}, f ≢ 0 \), then \( \left| {f\left( t\right) }\right| > 0 \) almost everywhere. This is a consequence of a stronger result proved in the next section. The theorem on Carleson measures, Theorem I.5.6, also extends to the \( {H}^{p} \) spaces, \( 0 < p \leq 1 \), because the key estimate in its proof was the maximal theorem. \[ \left( {1/\pi }\right) \int \left( {\log \left| {f\left( t\right) }\right| /\left( {1 + {t}^{2}}\right) }\right) {dt} > - \infty \] Theorem 3.9 (Carleson). Let \( \sigma \) be a positive measure in the upper half plane. Then the following are equivalent: (a) \( \sigma \) is a Carleson measure: for some constant \( N\left( \sigma \right) \) , \[ \sigma \left( Q\right) \leq N\left( \sigma \right) h \] for all squares \[ Q = \left\{ {{x}_{0} < x < {x}_{0} + h,0 < y < h}\right\} . \] (b) For \( 0 < p < \infty \) , \[ \int {\left| f\right| }^{p}{d\sigma } \leq A\parallel f{\parallel }_{{H}^{p}}^{p},\;f \in {H}^{p}. \] (c) For some \( p,0 < p < \infty, f \in {L}^{p}\left( \sigma \right) \) for all \( f \in {H}^{p} \) . Proof. That (a) implies (b) follows from (3.1) and Lemma I.5.5 just as in the proof of Theorem I.5.6. Trivially, (b) implies (c). On the other hand, if (c) holds for some fixed \( p < \infty \), then (b) holds for the same value \( p \) . This follows from the closed graph theorem, which is valid here even when \( p < 1 \) (see Dunford and Schwartz [1958, p. 57]). One can also see directly that if there are \( \left\{ {f}_{n}\right\} \) in \( {H}^{p} \) with \( {\begin{Vmatrix}{f}_{n}\end{Vmatrix}}_{p} = 1 \) but \( \int {\left| {f}_{n}\right| }^{p}{d\sigma } \rightarrow \infty \), then the sum \( \sum {\alpha }_{n}{f}_{n} \), when the \( {\alpha }_{n} \), are chosen adroitly, will give an \( {H}^{p} \) function for which (c) fails. Now suppose (b) holds for some \( p > 0 \) . Let \( Q \) be the square \( \left\{ {{x}_{0} < x < }\right. \) \( \left. {{x}_{0} + {y}_{0},0 < y < {y}_{0}}\right\} \) and let \[ f\left( z\right) = {\left( \frac{1}{\pi }\frac{{y}_{0}}{{\left( z - {\bar{z}}_{0}\right) }^{2}}\right) }^{1/p}, \] where \( {z}_{0} = {x}_{0} + i{y}_{0} \) . Then \( f \in {H}^{p} \) and \( \parallel f{\parallel }_{p}^{p} = \int {P}_{{z}_{0}}\left( t\right) {dt} = 1 \) . Since \( {\left| f\left( z\right) \right| }^{p} \geq {\left( 5\pi {y}_{0}\right) }^{-1}, z \in Q \), we have \[ \sigma \left( Q\right) \leq \sigma \left( \left\{ {z : \left| {f\left( z\right) }\right| > {\left( 5\pi {y}_{0}\right) }^{-1/p}}\right\} \right) \leq {5\pi A}{y}_{0}, \] so that (a) holds. ## \(\text{4.}\left( {1/\pi }\right) \int \left( {\log \left| {f\left( t\right) }\right| /\left( {1 + {t}^{2}}\right) }\right) {dt} > - \infty \) A fundamental result of \( {H}^{p} \) theory is that the condition of this section’s title characterizes the moduli of \( {H}^{p} \) functions \( \left| {f\left( t\right) }\right| \) among the positive \( {L}^{p} \) functions. In the disc, this result is due to Szegö for \( p = 2 \) and to F. Riesz for the other \( p \) . For functions analytic across \( \partial D \), the inequality (4.1) below was first noticed by Jensen [1899] and for this reason the inequality is sometimes called Jensen's inequality. We prefer to use that name for the inequality about averages and convex functions given in Theorem I.6.1. In this section the important thing about an \( {H}^{p} \) function will be the fact that the subharmonic function \[ \log \left| {f\left( {r{e}^{i\theta }}\right) }\right| \leq \left( {1/p}\right) {\left| f\left( r{e}^{i\theta }\right) \right| }^{p} \] is majorized by a positive \( {L}^{1} \) function of \( \theta \) . It will be simpler to work at first on the disc. Theorem 4.1. If \( 0 < p \leq \infty \) and if \( f\left( z\right) \in {H}^{p}\left( D\right), f ≢ 0 \), then \[ \frac{1}{2\pi }\int \log \left| {f\left( {e}^{i\theta }\right) }\right| {d\theta } > - \infty \] If \( f\left( 0\right) ≢ 0 \), then (4.1) \[ \log \left| {f\left( 0\right) }\right| \leq \frac{1}{2\pi }\int \log \left| {f\left( {e}^{i\theta }\right) }\right| {d\theta } \] and more generally, if \( f\left( {z}_{0}\right) ≢ 0 \) (4.2) \[ \log \left| {f\left( {z}_{0}\right) }\right| \leq \frac{1}{2\pi }\int \log \left| {f\left( {e}^{i\theta }\right) }\right| {P}_{{z}_{0}}\left( \theta \right) {d\theta }. \] Proof. By Theorem I.6.7 and by the subharmonicity of \( \log \left| f\right| \) , \[ \log \left| {f\left( z\right) }\right| \leq \mathop{\lim }\limits_{{r \rightarrow 1}}\frac{1}{2\pi }\int \log \left| {f\left( {r{e}^{i\theta }}\right) }\right| {P}_{z}\left( \theta \right) {d\theta }. \] Since \( \log \left| {f\left( {r{e}^{i\theta }}\right) }\right| \rightarrow \log \mid f\left( {e}^{i\theta }\right) \) almost everywhere, and since these functions are bounded above by the integrable function \( \left( {1/p}\right) {\left| {f}^{ * }\left( \theta \right) \right| }^{p} \), where \( {f}^{ * } \) is the maximal function, we have \[ \int {\log }^{ + }\left| {f\left( {r{e}^{i\theta }}\right) }\right| {P}_{{z}_{0}}\left( \theta \right) {d\theta } \rightarrow \int {\log }^{ + }\left| {f\left( {e}^{i\theta }\right) }\right| {P}_{{z}_{0}}\left( \theta \right) {d\theta }. \] Fatou's lemma can now be applied to the negative parts to give us \[ \mathop{\lim }\limits_{{r \rightarrow 1}}\frac{1}{2\pi }\int \log \left| {f\left( {r{e}^{i\theta }}\right) }\right| {P}_{{z}_{0}}\left( \theta \right) {d\theta } \leq \frac{1}{2\pi }\int \log \left| {f\left( {e}^{i\theta }\right) }\right| {P}_{{z}_{0}}\left( \theta \right) {d\theta }. \] This proves (4.2) and (4.1). The remaining inequality follows by removing any zero at the origin. Note that the same result in the upper half plane (4.3) \[ \log \left| {f\left( {z}_{0}\right) }\right| \leq \int \log \left| {f\left( t\right) }\right| {P}_{{z}_{0}}\left( t\right) {dt},\;f \in {H}^{p}, \] follows from Theorem 4.1 and from Lemma 1.1 upon a change of variables. Corollary 4.2. If \( f\left( z\right) \in {H}^{p} \) and if \( f\left( t\right) = 0 \) on a set of positive measure, then \( f = 0 \) . Corollary 4.2 gives the other half of the F. and M. Riesz theorem. If \( {d\mu }\left( t\right) \) is a finite measure such that \( {p}_{y} * \mu \left( x\right) \) is analytic, then not only is \( {d\mu } \) absolutely continuous to \( {dt} \), but also \( {dt} \) is absolutely continuous to \( {d\mu } \) . Corollary 4.3. Let \( 0 < p, r \leq \infty \) . If \( f\left( z\right) \in {H}^{p} \) and if the boundary function \( f\left( t\right) \in {L}^{r} \), then \( f\left( z\right) \in {H}^{r} \) . \[ \left( {1/\pi }\right) \int \left( {\log \left| {f\left( t\right) }\right| /\left( {1 + {t}^{2}}\right) }\right) {dt} > - \infty \] 63 This corollary is often written \[ {H}^{p} \cap {L}^{r} \subset {H}^{r} \] Proof. Applying Jensen’s inequality, with the convex function \( \varphi \left( s\right) = \) \( \exp \left( {rs}\right) \) and with the probability measure \( {P}_{y}\left( {x - t}\right) {dt} \), to (4.3) gives \[ {\left| f\left( z\right) \right| }^{r}

Lemma 3.7. Let \( f\left( z\right) \in {H}^{1} \) . Then the Fourier transform \[ \widehat{f}\left( s\right) = {\int }_{-\infty }^{\infty }f\left( t\right) {e}^{-{2\pi ist}}{dt} = 0 \] for all \( s \leq 0 \) .

Proof. By the continuity of \( f \rightarrow \widehat{f} \), we may suppose \( \int \in {\mathfrak{A}}_{N} \) . Then for \( s \leq 0, F\left( z\right) = f\left( z\right) {e}^{-{2\pi isz}} \) is also in \( {\mathfrak{A}}_{N} \) . The result now follows from Cauchy’s theorem because \[ {\int }_{0}^{\pi }\left| {F\left( {R{e}^{i\theta }}\right) }\right| {Rd\theta } \rightarrow 0\;\left( {R \rightarrow \infty }\right) . \]

Example 2.3.14. Let \( u = {\delta }_{{x}_{0}} \) and \( f \in \mathcal{S} \) . Then \( f * {\delta }_{{x}_{0}} \) is the function \( x \mapsto f\left( {x - {x}_{0}}\right) \) , for when \( h \in \mathcal{S} \), we have \[ \left\langle {f * {\delta }_{{x}_{0}}, h}\right\rangle = \left\langle {{\delta }_{{x}_{0}},\widetilde{f} * h}\right\rangle = \left( {\widetilde{f} * h}\right) \left( {x}_{0}\right) = {\int }_{{\mathbf{R}}^{n}}f\left( {x - {x}_{0}}\right) h\left( x\right) {dx}. \] It follows that convolution with \( {\delta }_{0} \) is the identity operator. We now define the product of a function and a distribution. Definition 2.3.15. Let \( u \in {\mathcal{S}}^{\prime } \) and let \( h \) be a \( {\mathcal{C}}^{\infty } \) function that has at most polynomial growth at infinity and the same is true for all of its derivatives. This means that for all \( \alpha \) it satisfies \( \left| {\left( {{\partial }^{\alpha }h}\right) \left( x\right) }\right| \leq {C}_{\alpha }{\left( 1 + \left| x\right| \right) }^{{k}_{\alpha }} \) for some \( {C}_{\alpha },{k}_{\alpha } > 0 \) . Then define the product \( {hu} \) of \( h \) and \( u \) by \[ \langle {hu}, f\rangle = \langle u,{hf}\rangle ,\;f \in \mathcal{S}. \] (2.3.16) Note that \( {hf} \) is in \( \mathcal{S} \) and thus (2.3.16) is well defined. The product of an arbitrary \( {\mathcal{C}}^{\infty } \) function with a tempered distribution is not defined. We observe that if a function \( g \) is supported in a set \( K \), then for all \( f \in {\mathcal{C}}_{0}^{\infty }\left( {K}^{c}\right) \) we have \[ {\int }_{{\mathbf{R}}^{n}}f\left( x\right) g\left( x\right) {dx} = 0. \] (2.3.17) Moreover, the support of \( g \) is the intersection of all closed sets \( K \) with the property (2.3.17) for all \( f \) in \( {\mathcal{C}}_{0}^{\infty }\left( {K}^{c}\right) \) . Motivated by the preceding observation we give the following: Definition 2.3.16. Let \( u \) be in \( {\mathcal{D}}^{\prime }\left( {\mathbf{R}}^{n}\right) \) . The support of \( u\left( {\operatorname{supp}u}\right) \) is the intersection of all closed sets \( K \) with the property \[ \varphi \in {\mathcal{C}}_{0}^{\infty }\left( {\mathbf{R}}^{n}\right) ,\;\operatorname{supp}\varphi \subseteq {\mathbf{R}}^{n} \smallsetminus K \Rightarrow \langle u,\varphi \rangle = 0. \] (2.3.18) Distributions with compact support are exactly those whose support (as defined in the previous definition) is a compact set. To prove this assertion, we start with a distribution \( u \) with compact support as defined in Definition 2.3.3. Then there exist \( C, N, m > 0 \) such that (2.3.4) holds. For a \( {\mathcal{C}}^{\infty } \) function \( f \) whose support is contained in \( B{\left( 0, N\right) }^{c} \), the expression on the right in (2.3.4) vanishes and we must therefore have \( \langle u, f\rangle = 0 \) . This shows that the support of \( u \) is contained in \( \overline{B\left( {0, N}\right) } \) hence it is bounded, and since it is already closed (as an intersection of closed sets), it must be compact. Conversely, if the support of \( u \) as defined in Definition 2.3.16 is a compact set, then there exists an \( N > 0 \) such that \( \operatorname{supp}u \) is contained in \( B\left( {0, N}\right) \) . We take a smooth function \( \eta \) that is equal to 1 on \( B\left( {0, N}\right) \) and vanishes off \( B\left( {0, N + 1}\right) \) . Then for \( h \in {\mathcal{C}}_{0}^{\infty } \) the support of \( h\left( {1 - \eta }\right) \) does not meet the support of \( u \), and we must have \[ \langle u, h\rangle = \langle u,{h\eta }\rangle + \langle u, h\left( {1 - \eta }\right) \rangle = \langle u,{h\eta }\rangle . \] The distribution \( u \) can be thought of as an element of \( {\mathcal{E}}^{\prime } \) by defining for \( f \in {\mathcal{C}}^{\infty }\left( {\mathbf{R}}^{n}\right) \) \[ \langle u, f\rangle = \langle u,{f\eta }\rangle \] Taking \( m \) to be the integer that corresponds to the compact set \( K = \overline{B\left( {0, N + 1}\right) } \) in (2.3.2), and using that the \( {L}^{\infty } \) norm of \( {\partial }^{\alpha }\left( {f\eta }\right) \) is controlled by a finite sum of seminorms \( {\widetilde{\rho }}_{\alpha, N + 1}\left( f\right) \) with \( \left| \alpha \right| \leq m \), we obtain the validity of (2.3.4) for \( f \in {\mathcal{C}}^{\infty } \) . Example 2.3.17. The support of the Dirac mass at \( {x}_{0} \) is the set \( \left\{ {x}_{0}\right\} \) . Along the same lines, we give the following definition: Definition 2.3.18. We say that a distribution \( u \) in \( {\mathcal{D}}^{\prime }\left( {\mathbf{R}}^{n}\right) \) coincides with the function \( h \) on an open set \( \Omega \) if \[ \langle u, f\rangle = {\int }_{{\mathbf{R}}^{n}}f\left( x\right) h\left( x\right) {dx}\;\text{ for all }f\text{ in }{\mathcal{C}}_{0}^{\infty }\left( \Omega \right) . \] (2.3.19) When (2.3.19) occurs we often say that \( u \) agrees with \( h \) away from \( {\Omega }^{c} \) . This definition implies that the support of the distribution \( u - h \) is contained in the set \( {\Omega }^{c} \) . Example 2.3.19. The distribution \( {\left| x\right| }^{2} + {\delta }_{{a}_{1}} + {\delta }_{{a}_{2}} \), where \( {a}_{1},{a}_{2} \) are in \( {\mathbf{R}}^{n} \), coincides with the function \( {\left| x\right| }^{2} \) on any open set not containing the points \( {a}_{1} \) and \( {a}_{2} \) . Also, the distribution in Example 2.3.5 (8) coincides with the function \( {x}^{-1}{\chi }_{\left| x\right| \leq 1} \) away from the origin in the real line. Having ended the streak of definitions regarding operations with distributions, we now discuss properties of convolutions and Fourier transforms. Theorem 2.3.20. If \( u \in {\mathcal{S}}^{\prime } \) and \( \varphi \in \mathcal{S} \), then \( \varphi * u \) is a \( {\mathcal{C}}^{\infty } \) function and \[ \left( {\varphi * u}\right) \left( x\right) = \left\langle {u,{\tau }^{x}\widetilde{\varphi }}\right\rangle \] for all \( x \in {\mathbf{R}}^{n} \) . Moreover, for all multi-indices \( \alpha \) there exist constants \( {C}_{\alpha },{k}_{\alpha } > 0 \) such that \[ \left| {{\partial }^{\alpha }\left( {\varphi * u}\right) \left( x\right) }\right| \leq {C}_{\alpha }{\left( 1 + \left| x\right| \right) }^{{k}_{\alpha }}. \] Furthermore, if \( u \) has compact support, then \( \varphi * u \) is a Schwartz function. Proof. Let \( \psi \) be in \( \mathcal{S}\left( {\mathbf{R}}^{n}\right) \) . We have \[ \langle \varphi * u,\psi \rangle = \langle u,\widetilde{\varphi } * \psi \rangle \] \[ = u\left( {{\int }_{{\mathbf{R}}^{n}}\widetilde{\varphi }\left( {\cdot - y}\right) \psi \left( y\right) {dy}}\right) \] \[ = u\left( {{\int }_{{\mathbf{R}}^{n}}\left( {{\tau }^{y}\widetilde{\varphi }}\right) \left( \cdot \right) \psi \left( y\right) {dy}}\right) \] (2.3.20) \[ = {\int }_{{\mathbf{R}}^{n}}\left\langle {u,{\tau }^{y}\widetilde{\varphi }}\right\rangle \psi \left( y\right) {dy} \] where the last step is justified by the continuity of \( u \) and by the fact that the Riemann sums of the inner integral in (2.3.20) converge to that integral in the topology of \( \mathcal{S} \) , a fact that will be justified later. This calculation identifies the function \( \varphi * u \) as \[ \left( {\varphi * u}\right) \left( x\right) = \left\langle {u,{\tau }^{x}\widetilde{\varphi }}\right\rangle \] (2.3.21) We now show that \( \left( {\varphi * u}\right) \left( x\right) \) is a \( {\mathcal{C}}^{\infty } \) function. Let \( {e}_{j} = \left( {0,\ldots ,1,\ldots ,0}\right) \) with 1 in the \( j \) th entry and zero elsewhere. Then \[ \frac{{\tau }^{-h{e}_{j}}\left( {\varphi * u}\right) \left( x\right) - \left( {\varphi * u}\right) \left( x\right) }{h} = u\left( \frac{{\tau }^{-h{e}_{j}}\left( {{\tau }^{x}\widetilde{\varphi }}\right) - {\tau }^{x}\widetilde{\varphi }}{h}\right) \rightarrow \left\langle {u,{\tau }^{x}\left( {{\partial }_{j}\widetilde{\varphi }}\right) }\right\rangle \] by the continuity of \( u \) and the fact that \( \left( {{\tau }^{-h{e}_{j}}\left( {{\tau }^{x}\widetilde{\varphi }}\right) - {\tau }^{x}\widetilde{\varphi }}\right) /h \) tends to \( {\partial }_{j}{\tau }^{x}\widetilde{\varphi } = \) \( {\tau }^{x}\left( {{\partial }_{j}\widetilde{\varphi }}\right) \) in \( \mathcal{S} \) as \( h \rightarrow 0 \) ; see Exercise 2.3.5 (a). The same calculation for higher-order derivatives shows that \( \varphi * u \in {\mathcal{C}}^{\infty } \) and that \( {\partial }^{\gamma }\left( {\varphi * u}\right) = \left( {{\partial }^{\gamma }\varphi }\right) * u \) for all multi-indices \( \gamma \) . It follows from (2.3.3) that for some \( C, m \), and \( k \) we have \[ \left| {{\partial }^{\alpha }\left( {\varphi * u}\right) \left( x\right) }\right| \leq C\mathop{\sum }\limits_{\substack{{\left| \gamma \right| \leq m} \\ {\left| \beta \right| \leq k} }}\mathop{\sup }\limits_{{y \in {\mathbf{R}}^{n}}}\left| {{y}^{\gamma }{\tau }^{x}\left( {{\partial }^{\alpha + \beta }\widetilde{\varphi }}\right) \left( y\right) }\right| \] \[ = C\mathop{\sum }\limits_{\substack{{\left| \gamma \right| \leq m} \\ {\left| \beta \right| \leq k} }}\mathop{\sup }\limits_{{y \in {\mathbf{R}}^{n}}}\left| {{\left( x + y\right) }^{\gamma }\left( {{\partial }^{\alpha + \beta }\widetilde{\varphi }}\right) \left( y\right) }\right| \] (2.3.22) \[ \leq {C}_{m}\mathop{\sum }\limits_{{\left| \beta \right| \leq k}}\mathop{\sup }\limits_{{y \in {\mathbf{R}}^{n}}}\left( {1 + {\left| x\right| }^{m} + {\left| y\right| }^{m}}\right) \left| {\left( {{\partial }^{\alpha + \beta }\widetilde{\varphi }}\right) \left( y\right) }\right| , \] and this clearly implies that \( {\partial }^{\alpha }\left( {\varphi * u}\right) \) grows at most polynomially at infinity. We now indicate why \( \varphi * u \) is Schwartz whenever \( u \) has compact support. Applying estimate (2.3.4) to the function \( y \mapsto \varphi \left( {x - y}\right) \) yields that \[ \left

Example 2.3.14. Let \( u = {\delta }_{{x}_{0}} \) and \( f \in \mathcal{S} \) . Then \( f * {\delta }_{{x}_{0}} \) is the function \( x \mapsto f\left( {x - {x}_{0}}\right) \) , for when \( h \in \mathcal{S} \), we have \[ \left\langle {f * {\delta }_{{x}_{0}}, h}\right\rangle = \left\langle {{\delta }_{{x}_{0}},\widetilde{f} * h}\right\rangle = \left( {\widetilde{f} * h}\right) \left( {x}_{0}\right) = {\int }_{{\mathbf{R}}^{n}}f\left( {x - {x}_{0}}\right) h\left( x\right) {dx}. \]

The proof provided in the text is as follows: \[ \left\langle {f * {\delta }_{{x}_{0}}, h}\right\rangle = \left\langle {{\delta }_{{x}_{0}},\widetilde{f} * h}\right\rangle = \left( {\widetilde{f} * h}\right) \left( {x}_{0}\right) = {\int }_{{\mathbf{R}}^{n}}f\left( {x - {x}_{0}}\right) h\left( x\right) {dx}. \] This shows that the convolution \( f * {\delta }_{{x}_{0}} \) is indeed the function \( x \mapsto f(x - {x}_{0}) \).

Exercise 1.4.13 Use Exercises 1.2.7 and 1.2.8 to show that there are infinitely many primes \( \equiv 1\left( {\;\operatorname{mod}\;{2}^{r}}\right) \) for any given \( r \) . Exercise 1.4.14 Suppose \( p \) is an odd prime such that \( {2p} + 1 = q \) is also prime. Show that the equation \[ {x}^{p} + 2{y}^{p} + 5{z}^{p} = 0 \] has no solutions in integers. Exercise 1.4.15 If \( x \) and \( y \) are coprime integers, show that if \[ \left( {x + y}\right) \text{ and }\frac{{x}^{p} + {y}^{p}}{x + y} \] have a common prime factor, it must be \( p \) . Exercise 1.4.16 (Sophie Germain’s Trick) Let \( p \) be a prime such that \( {2p} + \) \( 1 = q > 3 \) is also prime. Show that \[ {x}^{p} + {y}^{p} + {z}^{p} = 0 \] has no integral solutions with \( p \nmid {xyz} \) . Exercise 1.4.17 Assuming \( {ABC} \), show that there are only finitely many consecutive cubefull numbers. Exercise 1.4.18 Show that \[ \mathop{\sum }\limits_{p}\frac{1}{p} = + \infty \] where the summation is over prime numbers. Exercise 1.4.19 (Bertrand’s Postulate) (a) If \( {a}_{0} \geq {a}_{1} \geq {a}_{2} \geq \cdots \) is a de- creasing sequence of real numbers tending to 0 , show that \[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}{a}_{n} \leq {a}_{0} - {a}_{1} + {a}_{2} \] (b) Let \( T\left( x\right) = \mathop{\sum }\limits_{{n < x}}\psi \left( {x/n}\right) \), where \( \psi \left( x\right) \) is defined as in Exercise 1.1.25. Show that \[ T\left( x\right) = x\log x - x + O\left( {\log x}\right) . \] (c) Show that \[ T\left( x\right) - {2T}\left( \frac{x}{2}\right) = \mathop{\sum }\limits_{{n \leq x}}{\left( -1\right) }^{n - 1}\psi \left( \frac{x}{n}\right) = \left( {\log 2}\right) x + O\left( {\log x}\right) . \] Deduce that \[ \psi \left( x\right) - \psi \left( \frac{x}{2}\right) \geq \frac{1}{3}\left( {\log 2}\right) x + O\left( {\log x}\right) . \] ## Chapter 2 ## Euclidean Rings ## 2.1 Preliminaries We can discuss the concept of divisibility for any commutative ring \( R \) with identity. Indeed, if \( a, b \in R \), we will write \( a \mid b \) ( \( a \) divides \( b \) ) if there exists some \( c \in R \) such that \( {ac} = b \) . Any divisor of 1 is called a unit. We will say that \( a \) and \( b \) are associates and write \( a \sim b \) if there exists a unit \( u \in R \) such that \( a = {bu} \) . It is easy to verify that \( \sim \) is an equivalence relation. Further, if \( R \) is an integral domain and we have \( a, b \neq 0 \) with \( a \mid b \) and \( b \mid a \), then \( a \) and \( b \) must be associates, for then \( \exists c, d \in R \) such that \( {ac} = b \) and \( {bd} = a \), which implies that \( {bdc} = b \) . Since we are in an integral domain, \( {dc} = 1 \), and \( d, c \) are units. We will say that \( a \in R \) is irreducible if for any factorization \( a = {bc} \), one of \( b \) or \( c \) is a unit. Example 2.1.1 Let \( R \) be an integral domain. Suppose there is a map \( n : R \rightarrow \mathbb{N} \) such that: (i) \( n\left( {ab}\right) = n\left( a\right) n\left( b\right) \forall a, b \in R \) ; and (ii) \( n\left( a\right) = 1 \) if and only if \( a \) is a unit. We call such a map a norm map, with \( n\left( a\right) \) the norm of \( a \) . Show that every element of \( R \) can be written as a product of irreducible elements. Solution. Suppose \( b \) is an element of \( R \) . We proceed by induction on the norm of \( b \) . If \( b \) is irreducible, then we have nothing to prove, so assume that \( b \) is an element of \( R \) which is not irreducible. Then we can write \( b = {ac} \) where neither \( a \) nor \( c \) is a unit. By condition (i), \[ n\left( b\right) = n\left( {ac}\right) = n\left( a\right) n\left( c\right) \] and since \( a, c \) are not units, then by condition (ii), \( n\left( a\right) < n\left( b\right) \) and \( n\left( c\right) < \) \( n\left( b\right) \) . If \( a, c \) are irreducible, then we are finished. If not, their norms are smaller than the norm of \( b \), and so by induction we can write them as products of irreducibles, thus finding an irreducible decomposition of \( b \) . Exercise 2.1.2 Let \( D \) be squarefree. Consider \( R = \mathbb{Z}\left\lbrack \sqrt{D}\right\rbrack \) . Show that every element of \( R \) can be written as a product of irreducible elements. Exercise 2.1.3 Let \( R = \mathbb{Z}\left\lbrack \sqrt{-5}\right\rbrack \) . Show that \( 2,3,1 + \sqrt{-5} \), and \( 1 - \sqrt{-5} \) are irreducible in \( R \), and that they are not associates. We now observe that \( 6 = 2 \cdot 3 = \left( {1 + \sqrt{-5}}\right) \left( {1 - \sqrt{-5}}\right) \), so that \( R \) does not have unique factorization into irreducibles. We will say that \( R \), an integral domain, is a unique factorization domain if: (i) every element of \( R \) can be written as a product of irreducibles; and (ii) this factorization is essentially unique in the sense that if \( a = {\pi }_{1}\cdots {\pi }_{r} \) . and \( a = {\tau }_{1}\cdots {\tau }_{s} \), then \( r = s \) and after a suitable permutation, \( {\pi }_{i} \sim {\tau }_{i} \) . Exercise 2.1.4 Let \( R \) be a domain satisfying (i) above. Show that (ii) is equivalent to \( \left( {\mathrm{{ii}}}^{ \star }\right) \) : if \( \pi \) is irreducible and \( \pi \) divides \( {ab} \), then \( \pi \mid a \) or \( \pi \mid b \) . An ideal \( I \subseteq R \) is called principal if it can be generated by a single element of \( R \) . A domain \( R \) is then called a principal ideal domain if every ideal of \( R \) is principal. Exercise 2.1.5 Show that if \( \pi \) is an irreducible element of a principal ideal domain, then \( \left( \pi \right) \) is a maximal ideal,(where \( \left( x\right) \) denotes the ideal generated by the element \( x \) ). Theorem 2.1.6 If \( R \) is a principal ideal domain, then \( R \) is a unique factorization domain. Proof. Let \( S \) be the set of elements of \( R \) that cannot be written as a product of irreducibles. If \( S \) is nonempty, take \( {a}_{1} \in S \) . Then \( {a}_{1} \) is not irreducible, so we can write \( {a}_{1} = {a}_{2}{b}_{2} \) where \( {a}_{2},{b}_{2} \) are not units. Then \( \left( {a}_{1}\right) \subsetneqq \left( {a}_{2}\right) \) and \( \left( {a}_{1}\right) \subsetneqq \left( {b}_{2}\right) \) . If both \( {a}_{2},{b}_{2} \notin S \), then we can write \( {a}_{1} \) as a product of irreducibles, so we assume that \( {a}_{2} \in S \) . We can inductively proceed until we arrive at an infinite chain of ideals, \[ \left( {a}_{1}\right) \subsetneqq \left( {a}_{2}\right) \subsetneqq \left( {a}_{3}\right) \subsetneqq \cdots \subsetneqq \left( {a}_{n}\right) \subsetneqq \cdots . \] Now consider \( I = \mathop{\bigcup }\limits_{{i = 1}}^{\infty }\left( {a}_{i}\right) \) . This is an ideal of \( R \), and because \( R \) is a principal ideal domain, \( I = \left( \alpha \right) \) for some \( \alpha \in R \) . Since \( \alpha \in I,\alpha \in \left( {a}_{n}\right) \) for some \( n \), but then \( \left( {a}_{n}\right) = \left( {a}_{n + 1}\right) \) . From this contradiction, we conclude that the set \( S \) must be empty, so we know that if \( R \) is a principal ideal domain, every element of \( R \) satisfies the first condition for a unique factorization domain. Next we would like to show that if we have an irreducible element \( \pi \) , and \( \pi \mid {ab} \) for \( a, b \in R \), then \( \pi \mid a \) or \( \pi \mid b \) . If \( \pi \nmid a \), then the ideal \( \left( {a,\pi }\right) = R \) , so \( \exists x, y \) such that \[ {ax} + {\pi y} = 1 \] \[ \Rightarrow \;{abx} + {\pi by} = b\text{.} \] Since \( \pi \mid {abx} \) and \( \pi \mid {\pi by} \) then \( \pi \mid b \), as desired. By Exercise 2.1.4, we have shown that \( R \) is a unique factorization domain. The following theorem describes an important class of principal ideal domains: Theorem 2.1.7 If \( R \) is a domain with a map \( \phi : R \rightarrow \mathbb{N} \), and given \( a, b \in R,\exists q, r \in R \) such that \( a = {bq} + r \) with \( r = 0 \) or \( \phi \left( r\right) < \phi \left( b\right) \), we call \( R \) a Euclidean domain. If a ring \( R \) is Euclidean, it is a principal ideal domain. Proof. Given an ideal \( I \subseteq R \), take an element \( a \) of \( I \) such that \( \phi \left( a\right) \) is minimal among elements of \( I \) . Then given \( b \in I \), we can find \( q, r \in R \) such that \( b = {qa} + r \) where \( r = 0 \) or \( \phi \left( r\right) < \phi \left( a\right) \) . But then \( r = b - {qa} \), and so \( r \in I \), and \( \phi \left( a\right) \) is minimal among the norms of elements of \( I \) . So \( r = 0 \) , and given any element \( b \) of \( I, b = {qa} \) for some \( q \in R \) . Therefore \( a \) is a generator for \( I \), and \( R \) is a principal ideal domain. Exercise 2.1.8 If \( F \) is a field, prove that \( F\left\lbrack x\right\rbrack \), the ring of polynomials in \( x \) with coefficients in \( F \), is Euclidean. The following result, called Gauss' lemma, allows us to relate factorization of polynomials in \( \mathbb{Z}\left\lbrack x\right\rbrack \) with the factorization in \( \mathbb{Q}\left\lbrack x\right\rbrack \) . More generally, if \( R \) is a unique factorization domain and \( K \) is its field of fractions, we will relate factorization of polynomials in \( R\left\lbrack x\right\rbrack \) with that in \( K\left\lbrack x\right\rbrack \) . Theorem 2.1.9 If \( R \) is a unique factorization domain, and \( f\left( x\right) \in R\left\lbrack x\right\rbrack \) , define the content of \( f \) to be the gcd of the coefficients of \( f \), denoted by \( \mathcal{C}\left( f\right) \) . For \( f\left( x\right), g\left( x\right) \in R\left\lbrack x\right\rbrack ,\mathcal{C}\left( {fg}\right) = \mathcal{C}\left( f\right) \mathcal{C}\left( g\right) \) . Proof. Consider two polynomials \( f, g \in R\left\lbrack x\right\rbrack \), with \( \mathcal{C}\left

Exercise 1.4.13 Use Exercises 1.2.7 and 1.2.8 to show that there are infinitely many primes \( \equiv 1\left( {\;\operatorname{mod}\;{2}^{r}}\right) \) for any given \( r \) .

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Theorem 11.5 For each \( h > 0 \) the difference equations (11.10)-(11.11) have a unique solution. Proof. The tridiagonal matrix \( A \) is irreducible and weakly row-diagonally dominant. Hence, by Theorem 4.7, the matrix \( A \) is invertible, and the Jacobi iterations converge. Recall that for speeding up the convergence of the Jacobi iterations we can use relaxation methods or multigrid methods as discussed in Sections 4.2 and 4.3. The error and convergence analysis is initiated by first establishing the following two lemmas. Lemma 11.6 Denote by \( A \) the matrix of the finite difference method for \( q \geq 0 \) and by \( {A}_{0} \) the corresponding matrix for \( q = 0 \) . Then \[ 0 \leq {A}^{-1} \leq {A}_{0}^{-1} \] i.e., all components of \( {A}^{-1} \) are nonnegative and smaller than or equal to the corresponding components of \( {A}_{0}^{-1} \) . Proof. The columns of the inverse \( {A}^{-1} = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) satisfy \( A{a}_{j} = {e}_{j} \) for \( j = 1,\ldots, n \) with the canonical unit vectors \( {e}_{1},\ldots ,{e}_{n} \) in \( {\mathbb{R}}^{n} \) . The Jacobi iterations for the solution of \( {Az} = {e}_{j} \) starting with \( {z}_{0} = 0 \) are given by \[ {z}_{\nu + 1} = - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) {z}_{\nu } + {D}^{-1}{e}_{j},\;\nu = 0,1,\ldots , \] with the usual splitting \( A = D + {A}_{L} + {A}_{R} \) of \( A \) into its diagonal, lower, and upper triangular parts. Since the entries of \( {D}^{-1} \) and of \( - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) \) are all nonnegative, it follows that \( {A}^{-1} \geq 0 \) . Analogously, the iterations \[ {z}_{\nu + 1} = - {D}_{0}^{-1}\left( {{A}_{L} + {A}_{R}}\right) {z}_{\nu } + {D}_{0}^{-1}{e}_{j},\;\nu = 0,1,\ldots , \] yield the columns of \( {A}_{0}^{-1} \) . Therefore, from \( {D}_{0}^{-1} \geq {D}^{-1} \) we conclude that \( {A}_{0}^{-1} \geq {A}^{-1} \) Lemma 11.7 Assume that \( u \in {C}^{4}\left\lbrack {a, b}\right\rbrack \) . Then \[ \left| {{u}^{\prime \prime }\left( x\right) - \frac{1}{{h}^{2}}\left\lbrack {u\left( {x + h}\right) - {2u}\left( x\right) + u\left( {x - h}\right) }\right\rbrack }\right| \leq \frac{{h}^{2}}{12}{\begin{Vmatrix}{u}^{\left( 4\right) }\end{Vmatrix}}_{\infty } \] for all \( x \in \left\lbrack {a + h, b - h}\right\rbrack \) . Proof. By Taylor's formula we have that \[ u\left( {x \pm h}\right) = u\left( x\right) \pm h{u}^{\prime }\left( x\right) + \frac{{h}^{2}}{2}{u}^{\prime \prime }\left( x\right) \pm \frac{{h}^{3}}{6}{u}^{\prime \prime \prime }\left( x\right) + \frac{{h}^{4}}{24}{u}^{\left( 4\right) }\left( {x \pm {\theta }_{ \pm }h}\right) \] for some \( {\theta }_{ \pm } \in \left( {0,1}\right) \) . Adding these two equations gives \[ u\left( {x + h}\right) - {2u}\left( x\right) + u\left( {x - h}\right) = {h}^{2}{u}^{\prime \prime }\left( x\right) + \frac{{h}^{4}}{24}{u}^{\left( 4\right) }\left( {x + {\theta }_{ + }h}\right) + \frac{{h}^{4}}{24}{u}^{\left( 4\right) }\left( {x - {\theta }_{ - }h}\right) , \] whence the statement of the lemma follows. Theorem 11.8 Assume that the solution to the boundary value problem (11.7)-(11.8) is four-times continuously differentiable. Then the error of the finite difference approximation can be estimated by \[ \left| {u\left( {x}_{j}\right) - {u}_{j}}\right| \leq \frac{{h}^{2}}{96}{\begin{Vmatrix}{u}^{\left( 4\right) }\end{Vmatrix}}_{\infty }{\left( b - a\right) }^{2},\;j = 1,\ldots, n. \] Proof. By Lemma 11.7, for \[ {z}_{j} \mathrel{\text{:=}} {u}^{\prime \prime }\left( {x}_{j}\right) - \frac{1}{{h}^{2}}\left\lbrack {u\left( {x}_{j + 1}\right) - {2u}\left( {x}_{j}\right) + u\left( {x}_{j - 1}\right) }\right\rbrack \] we have the estimate \[ \left| {z}_{j}\right| \leq \frac{{h}^{2}}{12}{\begin{Vmatrix}{u}^{\left( 4\right) }\end{Vmatrix}}_{\infty },\;j = 1,\ldots, n. \] (11.13) Since \[ - \frac{1}{{h}^{2}}\left\lbrack {u\left( {x}_{j + 1}\right) - \left( {2 + {h}^{2}{q}_{j}}\right) u\left( {x}_{j}\right) + u\left( {x}_{j - 1}\right) }\right\rbrack = - {u}^{\prime \prime }\left( {x}_{j}\right) + {q}_{j}u\left( {x}_{j}\right) + {z}_{j} = {r}_{j} + {z}_{j}, \] the vector \( \widetilde{U} = {\left( u\left( {x}_{1}\right) ,\ldots, u\left( {x}_{n}\right) \right) }^{T} \) given by the exact solution solves the linear system \[ A\widetilde{U} = R + Z \] where \( Z = {\left( {z}_{1},\ldots ,{z}_{n}\right) }^{T} \) . Therefore, \[ A\left( {\widetilde{U} - U}\right) = Z \] and from this, using Lemma 11.6 and the estimate (11.13), we obtain \[ \left| {u\left( {x}_{j}\right) - {u}_{j}}\right| \leq {\begin{Vmatrix}{A}^{-1}Z\end{Vmatrix}}_{\infty } \leq \frac{{h}^{2}}{12}{\begin{Vmatrix}{u}^{\left( 4\right) }\end{Vmatrix}}_{\infty }{\begin{Vmatrix}{A}_{0}^{-1}e\end{Vmatrix}}_{\infty },\;j = 1,\ldots, n \] (11.14) where \( e = {\left( 1,\ldots ,1\right) }^{T} \) . The boundary value problem \[ - {u}_{0}^{\prime \prime } = 1,\;{u}_{0}\left( a\right) = {u}_{0}\left( b\right) = 0, \] has the solution \[ {u}_{0}\left( x\right) = \frac{1}{2}\left( {x - a}\right) \left( {b - x}\right) \] Since \( {u}_{0}^{\left( 4\right) } = 0 \), in this case, as a consequence of (11.14) the finite difference approximation coincides with the exact solution; i.e., \( e = {A}_{0}U = {A}_{0}\widetilde{U} \) . Hence, \[ {\begin{Vmatrix}{A}_{0}^{-1}e\end{Vmatrix}}_{\infty } \leq {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{\infty } = \frac{1}{8}{\left( b - a\right) }^{2},\;j = 1,\ldots, n. \] Inserting this into (11.14) completes the proof. Theorem 11.8 confirms that as in the case of the initial value problems in Chapter 10, the order of the local discretization error is inherited by the global error. Note that the assumption in Theorem 11.8 on the differentiability of the solution is satisfied if \( q \) and \( r \) are twice continuously differentiable. The error estimate in Theorem 11.8 is not practical in general, since it requires a bound on the fourth derivative of the unknown exact solution. Therefore, in practice, analogously to (10.19) the error is estimated from the numerical results for step sizes \( h \) and \( h/2 \) . Similarly, as in (10.20), a Richardson extrapolation can be employed to obtain a fourth-order approximation. Of course, the finite difference approximation can be extended to the general linear ordinary differential equation of second order \[ - {u}^{\prime \prime } + p{u}^{\prime } + {qu} = r \] by using the approximation \[ {u}^{\prime }\left( {x}_{j}\right) \approx \frac{1}{2h}\left\lbrack {u\left( {x}_{j + 1}\right) - u\left( {x}_{j - 1}\right) }\right\rbrack \] (11.15) for the first derivative. This approximation again has an error of order \( O\left( {h}^{2}\right) \) (see Problem 11.9). Besides Richardson extrapolation, higher-order approximations can be obtained by using higher-order difference approximations for the derivatives such as \[ {u}^{\prime \prime }\left( x\right) \approx \frac{1}{{12}{h}^{2}}\left\lbrack {-u\left( {x + {2h}}\right) + {16u}\left( {x + h}\right) }\right. \] (11.16) \[ - {30u}\left( x\right) + {16u}\left( {x - h}\right) - u\left( {x - {2h}}\right) \rbrack \] which is of order \( O\left( {h}^{4}\right) \), provided that \( u \) is six-times continuously differentiable (see Problem 11.9). We wish also to indicate briefly how the finite difference approximations are applied to boundary value problems for partial differential equations. For this we consider the boundary value problem for \[ - \bigtriangleup u + {qu} = r\;\text{ in }D \] (11.17) in the unit square \( D = \left( {0,1}\right) \times \left( {0,1}\right) \) with boundary condition \[ u = 0\;\text{ on }\partial D. \] (11.18) Here \( \Delta \) denotes the Laplacian \[ {\Delta u} \mathrel{\text{:=}} \frac{{\partial }^{2}u}{\partial {x}_{1}^{2}} + \frac{{\partial }^{2}u}{\partial {x}_{2}^{2}} \] Proceeding as in the proof of Theorem 11.4, by partial integration it can be seen that under the assumption \( q \geq 0 \) this boundary value problem has at most one solution. It is more involved and beyond the scope of this book to establish that a solution exists under proper assumptions on the functions \( q \) and \( r \) . We refer to \( \left\lbrack {{24},{60}}\right\rbrack \) and also the remarks at the end of Section 11.4. As in Example 2.2, we choose an equidistant grid \[ {x}_{ij} = \left( {{ih},{jh}}\right) ,\;i, j = 0,\ldots, n + 1, \] with step size \( h = 1/\left( {n + 1}\right) \) and \( n \in \mathbb{N} \) . Then we approximate the Laplacian at the internal grid points by \[ \bigtriangleup u\left( {x}_{ij}\right) \approx \frac{1}{{h}^{2}}\left\{ {u\left( {x}_{i + 1, j}\right) + u\left( {x}_{i - 1, j}\right) + u\left( {x}_{i, j + 1}\right) + u\left( {x}_{i, j - 1}\right) - {4u}\left( {x}_{ij}\right) }\right\} \] and obtain the system of equations \[ \frac{1}{{h}^{2}}\left\lbrack {\left( {4 + {h}^{2}{q}_{ij}}\right) {u}_{ij} - {u}_{i + 1, j} - {u}_{i - 1, j} - {u}_{i, j + 1} - {u}_{i, j - 1}}\right\rbrack = {r}_{ij}, \] (11.19) \[ i, j = 1,\ldots, n \] for approximate values \( {u}_{ij} \) to the exact solution \( u\left( {x}_{ij}\right) \) . Here we have set \( {q}_{ij} \mathrel{\text{:=}} q\left( {x}_{ij}\right) \) and \( {r}_{ij} \mathrel{\text{:=}} r\left( {x}_{ij}\right) \) . This system has to be complemented by the boundary conditions \[ {u}_{0, j} = {u}_{n + 1, j} = 0,\;j = 0,\ldots, n + 1, \] (11.20) \[ {u}_{i,0} = {u}_{i, n + 1} = 0,\;i = 1,\ldots, n. \] We refrain from rewriting the system (11.19)-(11.20) in matrix notation and refer back to Example 2.2. Analogously to Theorem 11.5, it can be seen that the Jacobi iterations converge (and relaxation methods and multigrid methods are applicable). Hence we have the following theorem. Theorem 11.9 For each \( h > 0 \) the difference equations (11.19)-(11.20) have a unique solution. From the proof of Lemma 11.6 it can be seen that its statement also holds for the corresponding matrices of the system (11.19)-(11.2

Theorem 11.5 For each \( h > 0 \) the difference equations (11.10)-(11.11) have a unique solution.

The tridiagonal matrix \( A \) is irreducible and weakly row-diagonally dominant. Hence, by Theorem 4.7, the matrix \( A \) is invertible, and the Jacobi iterations converge.

Corollary 3.4.6. Let \( 0 < {p}_{0} < \infty \) . Then for any \( p \) with \( {p}_{0} \leq p < \infty \) and for all locally integrable functions \( f \) on \( {\mathbf{R}}^{n} \) with \( {M}_{d}\left( f\right) \in {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) we have \[ \parallel f{\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \leq {C}_{n}\left( p\right) {\begin{Vmatrix}{M}^{\# }\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{n}\right) }, \] (3.4.11) where \( {C}_{n}\left( p\right) \) is the constant in Theorem 3.4.5. Proof. Since for every point in \( {\mathbf{R}}^{n} \) there is a sequence of dyadic cubes shrinking to it, the Lebesgue differentiation theorem yields that for almost every point \( x \) in \( {\mathbf{R}}^{n} \) the averages of the locally integrable function \( f \) over the dyadic cubes containing \( x \) converge to \( f\left( x\right) \) . Consequently, \[ \left| f\right| \leq {M}_{d}\left( f\right) \;\text{ a.e. } \] Using this fact, the proof of (3.4.11) is immediate, since \[ \parallel f{\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \leq {\begin{Vmatrix}{M}_{d}\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{n}\right) }, \] and by Theorem 3.4.5 the latter is controlled by \( {C}_{n}\left( p\right) {\begin{Vmatrix}{M}^{\# }\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \) . Estimate (3.4.11) provides the sought converse to (3.4.1). ## 3.4.3 Interpolation Using BMO We continue this section by proving an interpolation result in which the space \( {L}^{\infty } \) is replaced by \( {BMO} \) . The sharp function plays a key role in the following theorem. Theorem 3.4.7. Let \( 1 \leq {p}_{0} < \infty \) . Let \( T \) be a linear operator that maps \( {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) to \( {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) with bound \( {A}_{0} \), and \( {L}^{\infty }\left( {\mathbf{R}}^{n}\right) \) to \( \operatorname{BMO}\left( {\mathbf{R}}^{n}\right) \) with bound \( {A}_{1} \) . Then for all \( p \) with \( {p}_{0} < p < \infty \) there is a constant \( {C}_{n, p} \) such that for all \( f \in {L}^{p} \) we have \[ \parallel T\left( f\right) {\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \leq {C}_{n, p,{p}_{0}}{A}_{0}^{\frac{{p}_{0}}{p}}{A}_{1}^{1 - \frac{{p}_{0}}{p}}\parallel f{\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) }. \] (3.4.12) Remark 3.4.8. In certain applications, the operator \( T \) may not be a priori defined on all of \( {L}^{{p}_{0}} + {L}^{\infty } \) but only on some subspace of it. In this case one may state that the hypotheses and the conclusion of the preceding theorem hold for a subspace of these spaces. Proof. We consider the operator \[ S\left( f\right) = {M}^{\# }\left( {T\left( f\right) }\right) \] defined for \( f \in {L}^{{p}_{0}} + {L}^{\infty } \) . It is easy to see that \( S \) is a sublinear operator. We prove that \( S \) maps \( {L}^{\infty } \) to itself and \( {L}^{{p}_{0}} \) to itself if \( {p}_{0} > 1 \) or \( {L}^{1} \) to \( {L}^{1,\infty } \) if \( {p}_{0} = 1 \) . For \( f \in {L}^{{p}_{0}} \) we have \[ \parallel S\left( f\right) {\parallel }_{{L}^{{p}_{0}}} = {\begin{Vmatrix}{M}^{\# }\left( T\left( f\right) \right) \end{Vmatrix}}_{{L}^{{p}_{0}}} \leq 2{\begin{Vmatrix}{M}_{c}\left( T\left( f\right) \right) \end{Vmatrix}}_{{L}^{{p}_{0}}} \] \[ \leq {C}_{n,{p}_{0}}\parallel T\left( f\right) {\parallel }_{{L}^{{p}_{0}}} \leq {C}_{n,{p}_{0}}{A}_{0}\parallel f{\parallel }_{{L}^{{p}_{0}}} \] where the three \( {L}^{{p}_{0}} \) norms on the top line should be replaced by \( {L}^{1,\infty } \) if \( {p}_{0} = 1 \) . For \( f \in {L}^{\infty } \) one has \[ \parallel S\left( f\right) {\parallel }_{{L}^{\infty }} = {\begin{Vmatrix}{M}^{\# }\left( T\left( f\right) \right) \end{Vmatrix}}_{{L}^{\infty }} = \parallel T\left( f\right) {\parallel }_{BMO} \leq {A}_{1}\parallel f{\parallel }_{{L}^{\infty }}. \] Interpolating between these estimates using Theorem 1.3.2 in [156], we deduce \[ {\begin{Vmatrix}{M}^{\# }\left( T\left( f\right) \right) \end{Vmatrix}}_{{L}^{p}} = \parallel S\left( f\right) {\parallel }_{{L}^{p}} \leq {C}_{p,{p}_{0}}{A}_{0}^{\frac{{p}_{0}}{p}}{A}_{1}^{1 - \frac{{p}_{0}}{p}}\parallel f{\parallel }_{{L}^{p}} \] for all \( f \in {L}^{p} \), where \( {p}_{0} < p < \infty \) . Consider now a function \( h \in {L}^{p} \cap {L}^{{p}_{0}} \) . In the case \( {p}_{0} > 1,{M}_{d}\left( {T\left( h\right) }\right) \in {L}^{{p}_{0}} \) ; hence Corollary 3.4.6 is applicable and gives \[ \parallel T\left( h\right) {\parallel }_{{L}^{p}} \leq {C}_{n}\left( p\right) {C}_{p,{p}_{0}}{A}_{0}^{\frac{{p}_{0}}{p}}{A}_{1}^{1 - \frac{{p}_{0}}{p}}\parallel h{\parallel }_{{L}^{p}}. \] Density yields the same estimate for all \( f \in {L}^{p}\left( {\mathbf{R}}^{n}\right) \) . If \( {p}_{0} = 1 \), one applies the same idea but needs the endpoint estimate of Exercise 3.4.6, since \( {M}_{d}\left( {T\left( h\right) }\right) \in {L}^{1,\infty } \) . ## 3.4.4 Estimates for Singular Integrals Involving the Sharp Function We use the sharp function to obtain pointwise estimates for singular integrals. These enable us to recover previously obtained estimates for singular integrals, but also to deduce a new endpoint boundedness result from \( {L}^{\infty } \) to \( {BMO} \) . We recall some facts about singular integral operators. Suppose that \( K \) is a function defined on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) that satisfies \[ \left| {K\left( x\right) }\right| \leq {A}_{1}{\left| x\right| }^{-n} \] (3.4.13) \[ \left| {K\left( {x - y}\right) - K\left( x\right) }\right| \leq {A}_{2}{\left| y\right| }^{\delta }{\left| x\right| }^{-n - \delta }\;\text{ when }\left| x\right| \geq 2\left| y\right| > 0, \] (3.4.14) \[ \mathop{\sup }\limits_{{r < R < \infty }}\left| {{\int }_{r \leq \left| x\right| \leq R}K\left( x\right) {dx}}\right| \leq {A}_{3} \] (3.4.15) Let \( W \) be a tempered distribution that coincides with \( K \) on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) and let \( T \) be the linear operator given by convolution with \( W \) . Under these assumptions we have that \( T \) is \( {L}^{2} \) bounded with norm at most a constant multiple of \( {A}_{1} + {A}_{2} + {A}_{3} \) (Theorem 5.4.1 in [156]), and hence it is also \( {L}^{p} \) bounded with a similar norm on \( {L}^{p} \) for \( 1 < p < \infty \) (Theorem 5.3.3 in [156]). Theorem 3.4.9. Let \( T \) be given by convolution with a distribution \( W \) that coincides with a function \( K \) on \( {\mathbf{R}}^{n} \smallsetminus \{ 0\} \) satisfying (3.4.14). Assume that \( T \) has an extension that is \( {L}^{2} \) bounded with a norm \( B \) . Then there is a constant \( {C}_{n} \) such that for any \( s > 1 \) the estimate \[ {M}^{\# }\left( {T\left( f\right) }\right) \left( x\right) \leq {C}_{n}\left( {{A}_{2} + B}\right) \max \left( {s,{\left( s - 1\right) }^{-1}}\right) M{\left( {\left| f\right| }^{s}\right) }^{\frac{1}{s}}\left( x\right) \] (3.4.16) is valid for all \( f \) in \( \mathop{\bigcup }\limits_{{s \leq p < \infty }}{L}^{p} \) and all \( x \in {\mathbf{R}}^{n} \) . Proof. In view of Proposition 3.4.2 (2), given any cube \( Q \), it suffices to find a constant \( {a}_{Q} \) such that \[ \frac{1}{\left| Q\right| }{\int }_{Q}\left| {T\left( f\right) \left( y\right) - {a}_{Q}}\right| {dy} \leq {C}_{n}\max \left( {s,{\left( s - 1\right) }^{-1}}\right) \left( {{A}_{2} + B}\right) M{\left( {\left| f\right| }^{s}\right) }^{\frac{1}{s}}\left( x\right) \] (3.4.17) for almost all \( x \in Q \) . To prove this estimate we employ a well-known theme. We write \( f = {f}_{Q}^{0} + {f}_{Q}^{\infty } \), where \( {f}_{Q}^{0} = f{\chi }_{6\sqrt{n}Q} \) and \( {f}_{Q}^{\infty } = f{\chi }_{{\left( 6\sqrt{n}Q\right) }^{c}} \) . Here \( 6\sqrt{n}Q \) denotes the cube that is concentric with \( Q \), has sides parallel to those of \( Q \), and has side length \( 6\sqrt{n}\ell \left( Q\right) \), where \( \ell \left( Q\right) \) is the side length of \( Q \) . We now fix an \( f \) in \( \mathop{\bigcup }\limits_{{s \leq p < \infty }}{L}^{p} \) and we select \( {a}_{Q} = T\left( {f}_{Q}^{\infty }\right) \left( x\right) \) . Then \( {a}_{Q} \) is finite (and thus well defined) for all \( x \in Q \) . Indeed, for all \( x \in Q \) ,(3.4.13) yields \[ \left| {T\left( {f}_{Q}^{\infty }\right) \left( x\right) }\right| = \left| {{\int }_{{\left( {Q}^{ * }\right) }^{c}}f\left( y\right) K\left( {x - y}\right) {dy}}\right| \leq \parallel f{\parallel }_{{L}^{p}}{\left( {\int }_{\left| {x - y}\right| \geq {c}_{n}\ell \left( Q\right) }\frac{{A}_{1}^{{p}^{\prime }}{dy}}{{\left| x - y\right| }^{n{p}^{\prime }}}\right) }^{\frac{1}{{p}^{\prime }}} < \infty , \] where \( {c}_{n} \) is a positive constant. It follows that \[ \frac{1}{\left| Q\right| }{\int }_{Q}\left| {T\left( f\right) \left( y\right) - {a}_{Q}}\right| {dy} \] \[ \leq \frac{1}{\left| Q\right| }{\int }_{Q}\left| {T\left( {f}_{Q}^{0}\right) \left( y\right) }\right| {dy} + \frac{1}{\left| Q\right| }{\int }_{Q}\left| {T\left( {f}_{Q}^{\infty }\right) \left( y\right) - T\left( {f}_{Q}^{\infty }\right) \left( x\right) }\right| {dy}. \] (3.4.18) In view of Theorem 5.3.3 in [156], \( T \) maps \( {L}^{s} \) to \( {L}^{s} \) with norm at most a dimensional constant multiple of \( \max \left( {s,{\left( s - 1\right) }^{-1}}\right) \left( {B + {A}_{2}}\right) \) . The first term in (3.4.18) is controlled by \[ {\left( \frac{1}{\left| Q\right| }{\int }_{Q}{\left| T\left( {f}_{Q}^{0}\right) \left( y\right) \right| }^{s}dy\right) }^{\frac{1}{s}} \leq {C}_{n}\max \left( {s,{\left( s - 1\right) }^{-1}}\right) \left( {B + {A}_{2}}\right) {\left( \frac{1}{\left| Q\right| }{\int }_{{\mathbf{R}}^{n}}{\left| {f}_{Q}^{0}\left( y\right) \right| }^{s}dy\right) }^{\frac{1}{s}} \] \[ \leq {C}_{n}^{\prime }\max \left( {s,{\left( s - 1\right) }^{-1}}\right) \left( {B + {A}_{2}}\right) M{\left( {\left| f\right| }^{s}\right) }^

Corollary 3.4.6. Let \( 0 < {p}_{0} < \infty \) . Then for any \( p \) with \( {p}_{0} \leq p < \infty \) and for all locally integrable functions \( f \) on \( {\mathbf{R}}^{n} \) with \( {M}_{d}\left( f\right) \in {L}^{{p}_{0}}\left( {\mathbf{R}}^{n}\right) \) we have \[ \parallel f{\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \leq {C}_{n}\left( p\right) {\begin{Vmatrix}{M}^{\# }\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{n}\right) }, \] where \( {C}_{n}\left( p\right) \) is the constant in Theorem 3.4.5.

Proof. Since for every point in \( {\mathbf{R}}^{n} \) there is a sequence of dyadic cubes shrinking to it, the Lebesgue differentiation theorem yields that for almost every point \( x \) in \( {\mathbf{R}}^{n} \) the averages of the locally integrable function \( f \) over the dyadic cubes containing \( x \) converge to \( f\left( x\right) \) . Consequently, \[ \left| f\right| \leq {M}_{d}\left( f\right) \;\text{ a.e. } \] Using this fact, the proof of (3.4.11) is immediate, since \[ \parallel f{\parallel }_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \leq {\begin{Vmatrix}{M}_{d}\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{n}\right) }, \] and by Theorem 3.4.5 the latter is controlled by \( {C}_{n}\left( p\right) {\begin{Vmatrix}{M}^{\# }\left( f\right) \end{Vmatrix}}_{{L}^{p}\left( {\mathbf{R}}^{n}\right) } \) .

Lemma 12.2.2 The set of lines spanned by the vectors of \( {D}_{n} \) is star-closed. ## 12.3 Reflections We can characterize star-closed sets of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) in terms of their symmetries. If \( h \) is a vector in \( {\mathbb{R}}^{n} \), then there is a unique hyperplane through the origin perpendicular to \( h \) . Let \( {\rho }_{h} \) denote the operation of reflection in this hyperplane. Simple calculations reveal that for all \( x \) , \[ {\rho }_{h}\left( x\right) = x - 2\frac{\langle x, h\rangle }{\langle h, h\rangle }h \] We make a few simple observations. It is easy to check that \( {\rho }_{h}\left( h\right) = - h \) . Also, \( {\rho }_{h}\left( x\right) = x \) if and only if \( \langle h, x\rangle = 0 \) . The product \( {\rho }_{a}{\rho }_{b} \) of two reflections is not in general a reflection. It can be shown that \( {\rho }_{a}{\rho }_{b} = {\rho }_{b}{\rho }_{a} \) if and only if either \( \langle a\rangle = \langle b\rangle \) or \( \langle a, b\rangle = 0 \) . Lemma 12.3.1 Let \( \mathcal{L} \) be a set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) in \( {\mathbb{R}}^{n} \) . Then \( \mathcal{L} \) is star-closed if and only if for every vector \( h \) that spans a line in \( \mathcal{L} \), the reflection \( {\rho }_{h} \) fixes \( \mathcal{L} \) . Proof. Let \( h \) be a vector of length \( \sqrt{2} \) spanning a line in \( \mathcal{L} \) . From our comments above, \( {\rho }_{h} \) fixes \( \langle h\rangle \) and all the lines orthogonal to \( \langle h\rangle \) . So suppose that \( \langle a\rangle \) is a line of \( \mathcal{L} \) at \( {60}^{ \circ } \) to \( \langle h\rangle \) . Without loss of generality we can assume that \( a \) has length \( \sqrt{2} \) and that \( \langle h, a\rangle = - 1 \) . Now, \[ {\rho }_{h}\left( a\right) = a - 2\frac{\left( -1\right) }{2}h = a + h, \] and \( \langle a + h\rangle \) forms a star with \( \langle a\rangle \) and \( \langle h\rangle \) . This implies that \( {\rho }_{h} \) fixes \( \mathcal{L} \) if and only if \( \mathcal{L} \) is star-closed. A root system is a set \( S \) of vectors in \( {\mathbb{R}}^{n} \) such that (a) if \( h \in S \), then \( \langle h\rangle \cap S = \{ h, - h\} \) ; (b) if \( h \in S \), then \( {\rho }_{h}\left( S\right) = S \) . Lemma 12.3.1 shows that if \( \mathcal{L} \) is a star-closed set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) in \( {\mathbb{R}}^{m} \), then the vectors of length \( \sqrt{2} \) that span these lines form a root system. For example, the set \( {D}_{n} \) is a root system. The group generated by the reflections \( {\rho }_{h} \), for \( h \) in \( S \), is the reflection group of the root system. The symmetry group of a set of lines or vectors in \( {\mathbb{R}}^{n} \) is the group of all orthogonal transformations that take the set to itself. The symmetry group of a root system or of a set of lines always contains multiplication by -1 . ## 12.4 Indecomposable Star-Closed Sets A set \( \mathcal{L} \) of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) is called decomposable if it can be partitioned into two subsets \( {\mathcal{L}}_{1} \) and \( {\mathcal{L}}_{2} \) such that every line in \( {\mathcal{L}}_{1} \) is orthogonal to every line in \( {\mathcal{L}}_{2} \) . If there is no such partition, then it is called indecomposable. Lemma 12.4.1 For \( n \geq 2 \), the set of lines \( \mathcal{L} \) spanned by the vectors in \( {D}_{n} \) is indecomposable. Proof. The lines \( \left\langle {{e}_{1} + {e}_{i}}\right\rangle \) for \( i \geq 2 \) have pairwise inner products equal to 1, and hence must be in the same part of any decomposition of \( \mathcal{L} \) . It is clear, however, that any other vector in \( {D}_{n} \) has nonzero inner product with at least one of these vectors, and so there are no lines orthogonal to all of this set. Theorem 12.4.2 Let \( \mathcal{L} \) be a star-closed indecomposable set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) . Then the reflection group of \( \mathcal{L} \) acts transitively on ordered pairs of nonorthogonal lines. Proof. First we observe that the reflection group acts transitively on the lines of \( \mathcal{L} \) . Suppose that \( \langle a\rangle \) and \( \langle b\rangle \) are two lines that are not orthogonal, and that \( \langle a, b\rangle = - 1 \) . Then \( c = - a - b \) spans the third line in the star with \( \langle a\rangle \) and \( \langle b\rangle \), and the reflection \( {\rho }_{c} \) swaps \( \langle a\rangle \) and \( \langle b\rangle \) . Therefore, \( \langle a\rangle \) can be mapped on to any line not orthogonal to it. Let \( {\mathcal{L}}^{\prime } \) be the orbit of \( \langle a\rangle \) under the reflection group of \( \mathcal{L} \) . Then every line in \( \mathcal{L} \smallsetminus {\mathcal{L}}^{\prime } \) is orthogonal to every line of \( {\mathcal{L}}^{\prime } \) . Since \( \mathcal{L} \) is indecomposable, this shows that \( {\mathcal{L}}^{\prime } = \mathcal{L} \) . Now, suppose that \( \left( {\langle a\rangle ,\langle b\rangle }\right) \) and \( \left( {\langle a\rangle ,\langle c\rangle }\right) \) are two ordered pairs of nonorthogonal lines. We will show that there is a reflection that fixes \( \langle a\rangle \) and exchanges \( \langle b\rangle \) and \( \langle c\rangle \) . Assume that \( a, b \), and \( c \) have length \( \sqrt{2} \) and that \( \langle a, b\rangle = \langle a, c\rangle = - 1 \) . Then the vector \( - a - b \) has length \( \sqrt{2} \) and spans a line in \( \mathcal{L} \) . Now, \[ 1 = \langle c, - a\rangle = \langle c, b\rangle + \langle c, - a - b\rangle . \] If \( c = b \) or \( c = - a - b \), then \( \langle c\rangle \) and \( \langle b\rangle \) are exchanged by the identity reflection or \( {\rho }_{a} \), respectively. Otherwise, \( c \) has inner product 1 with precisely one of the vectors in \( \{ b, - a - b\} \), and is orthogonal to the other. Exchanging the roles of \( b \) and \( - a - b \) if necessary, we can assume that \( \langle c, b\rangle = 1 \) . Then \( \langle b - c\rangle \in \mathcal{L} \), and the reflection \( {\rho }_{b - c} \) fixes \( \langle a\rangle \) and exchanges \( \langle b\rangle \) and \( \langle c\rangle \) . \( ▱ \) Now, suppose that \( X \) is a graph with minimum eigenvalue at least -2 . Then \( A\left( X\right) + {2I} \) is the Gram matrix of a set of vectors of length \( \sqrt{2} \) that span a set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) . Let \( \mathcal{L}\left( X\right) \) denote the star-closure of this set of lines. Notice that the Gram matrix determines the set of vectors up to orthogonal transformations of the underlying vector space, and therefore \( \mathcal{L}\left( X\right) \) is uniquely determined up to orthogonal transformations. Lemma 12.4.3 If \( X \) is a graph with minimum eigenvalue at least -2, then the star-closed set of lines \( \mathcal{L}\left( X\right) \) is indecomposable if and only if \( X \) is connected. Proof. First suppose that \( X \) is connected. Let \( {\mathcal{L}}^{\prime } \) be the lines spanned by the vectors whose Gram matrix is \( A\left( X\right) + {2I} \) . Lines corresponding to adjacent vertices of \( X \) are not orthogonal, and hence must be in the same part of any decomposition of \( \mathcal{L}\left( X\right) \) . Therefore, all the lines in \( {\mathcal{L}}^{\prime } \) are in the same part. Any line lying in a star with two other lines is not orthogonal to either of them, and therefore lies in the same part of any decomposition of \( \mathcal{L}\left( X\right) \) . Hence the star-closure of \( {\mathcal{L}}^{\prime } \) is all in the same part of any decomposition, which shows that \( \mathcal{L}\left( X\right) \) is indecomposable. If \( X \) is not connected, then \( {\mathcal{L}}^{\prime } \) has a decomposition into two parts. Any line orthogonal to two lines in a star is orthogonal to all three lines of the star, and so any line added to \( {\mathcal{L}}^{\prime } \) to complete a star can be assigned to one of the two parts of the decomposition, eventually yielding a decomposition of \( \mathcal{L} \) . Therefore, we see that any connected graph with minimum eigenvalue at least -2 is associated with a star-closed indecomposable set of lines. Our strategy will be to classify all such sets, and thereby classify all the graphs with minimum eigenvalue at least -2 . ## 12.5 A Generating Set We now show that any indecomposable star-closed set of lines \( \mathcal{L} \) at \( {60}^{ \circ } \) and \( {90}^{ \circ } \) is the star-closure of a special subset of those lines. Eventually, we will see that the structure of this subset is very restricted. Lemma 12.5.1 Let \( \mathcal{L} \) be an indecomposable star-closed set of lines at \( {60}^{ \circ } \) and \( {90}^{ \circ } \), and let \( \langle a\rangle ,\langle b\rangle \), and \( \langle c\rangle \) form a star in \( \mathcal{L} \) . Every other line of \( \mathcal{L} \) is orthogonal to either one or three lines in the star. Proof. Without loss of generality we may assume that \( a, b \), and \( c \) all have length \( \sqrt{2} \) and that \[ \langle a, b\rangle = \langle b, c\rangle = \langle c, a\rangle = - 1. \] It follows then that \( c = - a - b \), and so for any other line \( \langle x\rangle \) of \( \mathcal{L} \) we have \[ \langle x, a\rangle + \langle x, b\rangle + \langle x, c\rangle = 0. \] Because each of the terms is in \( \{ 0, \pm 1\} \), we see that either all three terms are zero or the three terms are \( 1,0 \), and -1 in some order. Now, fix a star \( \langle a\rangle ,\langle b\rangle \), and \( \langle c\rangle \), and as above choose \( a, b \), and \( c \) to be vectors of length \( \sqrt{2} \) with pairwise inner products -1 . Let \( D \) be the set of lines of \( \mathcal{L} \) that are orthogonal to all three lines in the star. The remaining lines of \(

Lemma 12.2.2 The set of lines spanned by the vectors of \( {D}_{n} \) is star-closed.

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Lemma 9.3.7. If \( \left( {{g}_{1},{S}_{1}}\right) \sim \left( {{g}_{2},{S}_{2}}\right) \) then for all \( i\left( {{g}_{1},{e}_{i}\left( {S}_{1}\right) }\right) \sim \left( {{g}_{2},{e}_{i}\left( {S}_{2}\right) }\right) \) . Proof. There exist \( e \) and \( T \) such that \( \left( {e\left( {S}_{i}\right), T}\right) \) is a balanced pair representing \( {g}_{i} \) for \( i = 1,2 \) . We write \( \left( {{g}_{1},{S}_{1}}\right) \underset{k}{ \sim }\left( {{g}_{2},{S}_{2}}\right) \) if the length of this \( e \) is \( \leq k \) . The lemma is proved by induction on \( k \) using 9.3.3. Lemma 9.3.8. Each \( {E}_{i} \) is an \( F \) -function; i.e., \( {E}_{i}\left( {g\left\lbrack {h, T}\right\rbrack }\right) = g{E}_{i}\left( \left\lbrack {h, T}\right\rbrack \right) \) . Define \( f : B \rightarrow \mathbb{N} \) by \( f\left( \left\lbrack {h, T}\right\rbrack \right) = \) the number of leaves of \( T \) . This is well defined. Lemma 9.3.9. Let \( n \geq i \) . The function \( {E}_{i} \) maps \( {f}^{-1}\left( n\right) \) bijectively onto \( {f}^{-1}\left( {n + 1}\right) \) . Proof. That \( {E}_{i} \) is injective is clear. The proof that \( {E}_{i} \) is surjective should be clear from Example 9.3.10, below. Let \( {B}_{n} = {f}^{-1}\left( \left\lbrack {1, n}\right\rbrack \right) \) . Then 9.3.9 says that when \( n \geq i,{E}_{i} \) maps \( B - {B}_{n - 1} \) bijectively onto \( B - {B}_{n} \) . We call \( {E}_{i} \) a simple expansion operator on \( B - {B}_{n - 1} \) and we call its inverse \( {C}_{i} : B - {B}_{n} \rightarrow B - {B}_{n - 1} \) a simple contraction operator. Example 9.3.10. Let \( T \) have more than \( i \) leaves. If the \( {i}^{\text{th }} \) and \( {\left( i + 1\right) }^{\text{th }} \) leaves form a caret (i.e., \( T = {e}_{i}\left( {T}^{\prime }\right) \) for some \( {T}^{\prime } \in \mathcal{T} \) ) then \( {C}_{i}\left( \left\lbrack {h, T}\right\rbrack \right) = \left\lbrack {h,{T}^{\prime }}\right\rbrack \) . If these leaves do not form a caret, let \( S \) be any tree with the same number of leaves as \( T \) such that \( S = {e}_{i}\left( {S}^{\prime }\right) \) for some \( {S}^{\prime } \in \mathcal{T} \) . Let \( g \in F \) be represented by \( \left( {S, T}\right) \) . Then \( \left\lbrack {g, S}\right\rbrack = \left\lbrack {1, T}\right\rbrack \), so \( \left\lbrack {{hg}, S}\right\rbrack = \left\lbrack {h, T}\right\rbrack \) and \( {C}_{i}\left( \left\lbrack {h, T}\right\rbrack \right) = \left\lbrack {{hg},{S}^{\prime }}\right\rbrack \) . We make \( B \) into an \( F \) -poset by defining \( \left\lbrack {g, S}\right\rbrack \leq \left\lbrack {h, T}\right\rbrack \) if for some \( {i}_{1},{i}_{2},\ldots ,{i}_{r} \), with \( r \geq 0,{E}_{{i}_{r}} \circ \cdots \circ {E}_{{i}_{1}}\left( \left\lbrack {g, S}\right\rbrack \right) = \left\lbrack {h, T}\right\rbrack \) . Each \( {B}_{n} \) is an \( F \) - sub-poset of \( B \) . Lemma 9.3.11. The \( F \) -action on the set \( B \) is a free action. ## D. Finiteness Properties of \( F \) : As usual we reuse the symbol \( B \) for the associated ordered abstract simplicial complex defined by the poset \( B \) . Proposition 9.3.12. The induced \( F \) -action on \( \left| B\right| \) is a free action. Proof. The stabilizers of vertices are trivial by 9.3.11. Since the action of \( F \) on \( \left\lbrack {h, S}\right\rbrack \) preserves the number of leaves of \( S \), the stabilizer of each simplex of \( S \) is also trivial. Proposition 9.3.13. The poset \( B \) is a directed set. Proof. Let \( \left\{ {{b}_{1},\ldots ,{b}_{k}}\right\} \subset B \) . Write \( {b}_{i} = \left\lbrack {{h}_{i},{S}_{i}}\right\rbrack \) . Then \( {S}_{i} \) expands to \( {S}_{i}^{\prime } \) such that \( \left( {{h}_{i},{S}_{i}^{\prime }}\right) \sim \left( {{1}_{F},{T}_{i}^{\prime }}\right) \) for some \( {T}_{i}^{\prime } \) . Write \( {b}_{i}^{\prime } = \left\lbrack {{h}_{i},{S}_{i}^{\prime }}\right\rbrack = \left\lbrack {{1}_{F},{T}_{i}^{\prime }}\right\rbrack \) . Then \( {b}_{i} \leq {b}_{i}^{\prime } \) for all \( i \) . Let \( T = \mathop{\bigcup }\limits_{{i = 1}}^{k}{T}_{i}^{\prime } \) and write \( b = \left\lbrack {{1}_{F}, T}\right\rbrack \) . Then \( {b}_{i}^{\prime } \leq b \) for all \( i \) . Proposition 9.3.14. If a poset \( P \) is a directed set then \( \left| P\right| \) is contractible. Proof. Whenever \( K \) is a finite subcomplex of \( P \) there exists \( v \in P \) such that the cone \( v * \left| K\right| \) is a subcomplex of \( \left| P\right| \) . Thus the homotopy groups of \( \left| P\right| \) are trivial, so, by the Whitehead Theorem, \( \left| P\right| \) is contractible. Corollary 9.3.15. \( \left| B\right| \) is contractible. The function \( f : B \rightarrow \mathbb{N} \) extends affinely to a Morse function \( {}^{8} \) (also denoted by) \( f : \left| B\right| \rightarrow \mathbb{R} \) . Then \( \left| {B}_{n}\right| = {f}^{-1}(\left( {-\infty, n\rbrack }\right) \) . Proposition 9.3.16. The \( {CW} \) complex \( F \smallsetminus \left| {B}_{n}\right| \) is finite. Proof. Let \( \left\lbrack {h, S}\right\rbrack \in {B}_{n} \) . Its \( F \) -orbit contains \( \left\lbrack {1, S}\right\rbrack \) and there are only finitely many finite trees having at most \( n \) leaves. This shows that the 0 -skeleton of \( F \smallsetminus \left| {B}_{n}\right| \) is finite. The rest is clear. Proposition 9.3.17. For each integer \( k \) there is an integer \( m\left( k\right) \) such that if \( b \) is a vertex of \( B \) and \( f\left( b\right) \geq m\left( k\right) \), then the downward link \( {\operatorname{lk}}_{\left| B\right| }^{ \downarrow }b \) is \( k \) - connected. We postpone the proof of 9.3.17 until the next subsection. Proposition 9.3.18. For \( n \geq m\left( k\right) ,\left| {B}_{n}\right| \) is \( k \) -connected. Hence \( \left\{ \left| {B}_{n}\right| \right\} \) is essentially \( k \) -connected for all \( k \) . Proof. By 9.3.17 and 8.3.4 we conclude that \( \left( {\left| {B}_{n}\right| ,\left| {B}_{n - 1}\right| }\right) \) is \( \left( {k + 1}\right) \) -connected if \( n \geq m\left( k\right) \) . When combined with the Whitehead Theorem and 9.3.15 this proves what is claimed. By 7.4.1 and 7.2.2 we conclude: Theorem 9.3.19. (Brown-Geoghegan Theorem) Thompson's Group \( F \) has type \( {F}_{\infty } \) . ## E. Analysis of the downward links: It remains to prove 9.3.17. We begin with two topics of general interest (9.3.20 and 9.3.21). If \( \mathcal{U} = \left\{ {X}_{\alpha }\right\} \) is a cover of the CW complex \( X \) by subcomplexes, the nerve of \( \mathcal{U} \) is the abstract simplicial complex \( N\left( \mathcal{U}\right) \) having a vertex \( {v}_{\alpha } \) for each \( {X}_{\alpha } \) , and a simplex \( \left\{ {{v}_{{\alpha }_{0}},\ldots ,{v}_{{\alpha }_{k}}}\right\} \) whenever \( \mathop{\bigcap }\limits_{{i = 0}}^{k}{X}_{{\alpha }_{i}} \neq \varnothing \) . The following property of nerves is widely used in topology. --- \( {}^{8} \) See Section 8.3. --- Proposition 9.3.20. If the cover \( \mathcal{U} \) is finite and if \( \mathop{\bigcap }\limits_{{i = 0}}^{k}{X}_{{\alpha }_{i}} \) is contractible whenever it is non-empty, then \( \left| {N\left( \mathcal{U}\right) }\right| \) and \( X \) are homotopy equivalent. Proof. There is a vertex \( {v}_{{\alpha }_{0},\ldots ,{\alpha }_{k}} \) of the first derived \( {sd}\left| {N\left( \mathcal{U}\right) }\right| \) for each simplex \( \left\{ {{v}_{{\alpha }_{0}},\ldots ,{v}_{{\alpha }_{k}}}\right\} \) of \( N\left( \mathcal{U}\right) \) . Pick a point \( {x}_{{\alpha }_{0},\ldots ,{\alpha }_{k}} \in \mathop{\bigcap }\limits_{{i = 0}}^{k}{X}_{{\alpha }_{i}} \) . Define a map \( \alpha : \operatorname{sd}\left| {N\left( \mathcal{U}\right) }\right| \rightarrow X \) taking each vertex \( {v}_{{\alpha }_{0},\ldots ,{\alpha }_{k}} \) to the point \( {x}_{{\alpha }_{0},\ldots ,{\alpha }_{k}} \) ; the contractibility hypothesis makes it possible to extend \( \alpha \) so that the simplex whose first \( {}^{9} \) vertex is \( {v}_{{\alpha }_{0},\ldots ,{\alpha }_{k}} \) is mapped into \( \mathop{\bigcap }\limits_{{i = 0}}^{k}{X}_{{\alpha }_{i}} \) . A straightforward generalization of the proof of 4.1.5 shows that \( \alpha \) is a homotopy equivalence. The particular nerve to which this will be applied is the abstract simplicial \( {\operatorname{complex}}^{10}{L}_{n} \) whose simplexes are the sets of pairwise disjoint adjacent pairs in the ordered set \( \left( {1,2,\ldots, n}\right) \) . Proposition 9.3.21. For any integer \( k \geq 0 \) there is an integer \( m\left( k\right) \) such that \( \left| {L}_{n}\right| \) is \( k \) -connected when \( n \geq m\left( k\right) \) . Proof. By induction on \( k \) we prove a sharper statement: Claim: given \( k \geq 0 \) there are integers \( m\left( k\right) \) and \( q\left( k\right) \) such that when \( n \geq \) \( m\left( k\right) \) the \( k \) -skeleton \( {\left| {L}_{n}\right| }^{k} \) is homotopically trivial by means of a homotopy \( {H}^{\left( k\right) } \) in which, for every simplex \( \sigma \) of \( {\left| {L}_{n}\right| }^{k},{H}^{\left( k\right) }\left( {\left| \sigma \right| \times I}\right) \) is supported by a subcomplex having \( \leq q\left( k\right) \) vertices. For \( k = 0 \) we can take \( m\left( 0\right) = 5 \) and \( q\left( 0\right) = 3 \) . Assume the Claim holds when \( k \) is replaced by \( k - 1 \geq 0 \) . For any \( k \) -simplex \( \sigma ,{H}^{\left( k - 1\right) }\left( {\left| \sigma \right| \times I}\right) \) is supported by a subcomplex \( J \) having at most \( r = \left( {k + 1}\right) \left( {q\left( {k - 1}\right) }\right) \) vertices. If \( n \geq {2r} + 2 \) there is a vertex \( v \) of \( {L}_{n} \) such that \( J \) lies in the link of \( v \) . Extend \( {H}^{\left( k - 1\right) } \) to \( {H}^{\left( k\right) } \) in two steps: first, \( {H}^{\left( k\right) } \) is to be the identity map on \( \left| \sigma

Lemma 9.3.7. If \( \left( {{g}_{1},{S}_{1}}\right) \sim \left( {{g}_{2},{S}_{2}}\right) \) then for all \( i\left( {{g}_{1},{e}_{i}\left( {S}_{1}\right) }\right) \sim \left( {{g}_{2},{e}_{i}\left( {S}_{2}\right) }\right) \) .

There exist \( e \) and \( T \) such that \( \left( {e\left( {S}_{i}\right), T}\right) \) is a balanced pair representing \( {g}_{i} \) for \( i = 1,2 \) . We write \( \left( {{g}_{1},{S}_{1}}\right) \underset{k}{ \sim }\left( {{g}_{2},{S}_{2}}\right) \) if the length of this \( e \) is \( \leq k \) . The lemma is proved by induction on \( k \) using 9.3.3.

Theorem 4.11. Every module can be embedded into an injective module. Noetherian rings. Our last result is due to Bass (cf. Chase [1961]). Theorem 4.12. A ring \( R \) is left Noetherian if and only if every direct sum of injective left \( R \) -modules is injective. Proof. Assume that every direct sum of injective left \( R \) -modules is injective, and let \( {L}_{1} \subseteq {L}_{2} \subseteq \cdots \subseteq {L}_{n} \subseteq \cdots \) be an ascending sequence of left ideals of \( R \) . Then \( L = \mathop{\bigcup }\limits_{{n > 0}}{L}_{n} \) is a left ideal of \( R \) . By 4.11 there is a monomorphism of \( R/{L}_{n} \) into an injective \( R \) -module \( {J}_{n} \) . By the hypothesis, \( J = {\bigoplus }_{n > 0}{J}_{n} \) is injective. Construct a module homomorphism \( \varphi : L \rightarrow J \) as follows. Let \( {\varphi }_{n} \) be the homomorphism \( R \rightarrow R/{L}_{n} \rightarrow {J}_{n} \) . If \( x \in L \), then \( x \in {L}_{n} \) for some \( n \), and then \( {\varphi }_{k}\left( x\right) = 0 \) for all \( k \geqq n \) . Let \( \varphi \left( x\right) = {\left( {\varphi }_{k}\left( x\right) \right) }_{k > 0} \in J \) . Since \( J \) is injective, \( \varphi \) extends to a module homomorphism \( \psi : {}_{R}R \rightarrow J \) . Then \( \psi \left( 1\right) = {\left( {t}_{k}\right) }_{k > 0} \in J \) for some \( {t}_{k} \in {J}_{k},{t}_{k} = 0 \) for almost all \( k \) . Hence \( \psi \left( 1\right) \in {\bigoplus }_{k < n}{J}_{k} \) for some \( n \) . Then \( \varphi \left( x\right) = \psi \left( {x1}\right) = {x\psi }\left( 1\right) \in {\bigoplus }_{k < n}{J}_{k} \) for all \( x \in L \) ; in particular, \( {\varphi }_{n}\left( x\right) = 0 \) and \( x \in {L}_{n} \), since \( R/{L}_{n} \rightarrow {J}_{n} \) is injective. Thus \( L = {L}_{n} \), and then \( {L}_{k} = {L}_{n} \) for all \( k \geqq n \) . Conversely, assume that \( R \) is left Noetherian, and let \( J = {\bigoplus }_{i \in I}{J}_{i} \) be a direct sum of injective \( R \) -modules \( {J}_{i} \) . Let \( L \) be a left ideal and let \( \varphi : L \rightarrow J \) be a module homomorphism. Since \( R \) is left Noetherian, \( L \) is finitely generated: \( L = R{r}_{1} + \cdots + R{r}_{n} \) for some \( {r}_{1},\ldots ,{r}_{n} \in L \) . Each \( \varphi \left( {r}_{k}\right) \) has finitely many nonzero components in \( {\bigoplus }_{i \in I}{J}_{i} \) and belongs to a finite direct sum \( {\bigoplus }_{i \in {S}_{k}}{J}_{i} \) . Hence there is a finite direct sum \( {\bigoplus }_{i \in S}{J}_{i}, S = {S}_{1} \cup \cdots \cup {S}_{n} \), that contains every \( \varphi \left( {r}_{k}\right) \) and contains \( \varphi \left( L\right) \) . Since \( S \) is finite, \( {\bigoplus }_{i \in S}{J}_{i} \) is injective, by 4.4; hence \( \varphi : L \rightarrow {\bigoplus }_{i \in S}{J}_{i} \) extends to a module homomorphism \( {}_{R}R \rightarrow {\bigoplus }_{i \in S}{J}_{i} \subseteq \) \( {\bigoplus }_{i \in I}{J}_{i} \) . Hence \( {\bigoplus }_{i \in I}{J}_{i} \) is injective, by 4.5. \( ▱ \) ## Exercises 1. Show that every direct summand of an injective module is injective. 2. Show that every direct sum of injective left \( R \) -modules is injective. 3. Let \( L \) be a left ideal of \( R \) and let \( \varphi : L \rightarrow M \) be a module homomorphism. Show that \( \varphi \) can be extended to \( {}_{R}R \) if and only if there exists \( m \in M \) such that \( \varphi \left( r\right) = {rm} \) for all \( r \in L \) . 4. Let \( J \) be an injective left \( R \) -module. Let \( a \in J \) and \( r \in R \) satisfy \( \operatorname{Ann}\left( r\right) \subseteq \operatorname{Ann}\left( a\right) \) (if \( t \in R \) and \( {tr} = 0 \), then \( {ta} = 0 \) ). Prove that \( a = {rx} \) for some \( x \in J \) . 5. Show that the quotient field of a Noetherian domain \( R \) is an injective \( R \) -module. 6. Find all subgroups of \( {\mathbb{Z}}_{p}\infty \) . 7. Show that \( {\mathbb{Z}}_{p}\infty \) is indecomposable. 8. Let \( U \) be the multiplicative group of all complex numbers of modulus 1 . Show that \( U\left( p\right) \cong {\mathbb{Z}}_{p}\infty \) . 9. Show that the additive group \( \mathbb{Q}/\mathbb{Z} \) is isomorphic to the direct sum \( {\bigoplus }_{p}{\mathbb{Z}}_{{p}^{\infty }} \) with one term for every prime \( p \) . *10. Can you extend Theorem 4.9 to modules over any PID? ## 5. The Injective Hull In this section we show that every module has, up to isomorphism, a smallest injective extension. We show this by comparing injective extensions to another kind of extensions, essential extensions. Definition. A submodule \( S \) of a left \( R \) -module \( M \) is essential when \( S \cap T \neq 0 \) for every submodule \( T \neq 0 \) of \( M \) . Essential submodules are also called large. Readers may prove the following: Proposition 5.1. If \( A \subseteq B \) are submodules of \( C \), then \( A \) is essential in \( C \) if and only if \( A \) is essential in \( B \) and \( B \) is essential in \( C \) . A monomorphism \( \varphi : A \rightarrow B \) is essential when \( \operatorname{Im}\varphi \) is an essential submodule of \( B \) . Proposition 5.2. If \( \mu \) is an essential monomorphism, and \( \varphi \circ \mu \) is injective, then \( \varphi \) is injective. Proof. If \( \varphi \circ \mu \) is injective, then \( \operatorname{Ker}\varphi \cap \operatorname{Im}\mu = 0 \) ; hence \( \operatorname{Ker}\varphi = 0 \) . Definition. An essential extension of a left R-module \( A \) is a left R-module \( B \) such that \( A \) is an essential submodule of \( B \) ; more generally, a left \( R \) -module \( B \) with an essential monomorphism \( A \rightarrow B \) . These two definitions are equivalent up to isomorphisms: if \( A \) is an essential submodule of \( B \), then the inclusion homomorphism \( A \rightarrow B \) is an essential monomorphism; if \( \mu : A \rightarrow B \) is an essential monomorphism, then \( A \) is isomorphic to the essential submodule \( \operatorname{Im}\mu \) of \( B \) ; moreover, using surgery, one can construct a module \( {B}^{\prime } \cong B \) in which \( A \) is an essential submodule. Proposition 5.3. If \( \mu : A \rightarrow B \) and \( v : B \rightarrow C \) are monomorphisms, then \( v \circ \mu \) is essential if and only if \( \mu \) and \( v \) are essential. This follows from Proposition 5.1; the details make nifty exercises. Proposition 5.4. A left \( R \) -module \( J \) is injective if and only if \( J \) has no proper essential extension \( J \subsetneqq M \), if and only if every essential monomorphism \( J \rightarrow M \) is an isomorphism. Proof. Let \( J \) be injective. If \( J \subseteq M \), then \( J \) is a direct summand of \( M \) , \( M = J \oplus N \) ; then \( N \cap J = 0 \) ; if \( J \) is essential in \( M \), then \( N = 0 \) and \( J = M \) . If in turn \( J \) has no proper essential extension, and \( \mu : J \rightarrow M \) is an essential monomorphism, then \( \operatorname{Im}\mu \cong J \) has no proper essential extension, hence \( M = \) \( \operatorname{Im}\mu \) and \( \mu \) is an isomorphism. Finally, assume that every essential monomorphism \( J \rightarrow A \) is an isomorphism. We show that \( J \) is a direct summand of every module \( M \supseteq J \) . By Zorn’s lemma there is a submodule \( K \) of \( M \) maximal such that \( J \cap K = 0 \) . Readers will verify that the projection \( M \rightarrow M/K \) induces an essential monomorphism \( \mu : J \rightarrow M/K \) . By the hypothesis, \( \mu \) is an isomorphism; hence \( J + K = M \) and \( M = J \oplus K \) . \( ▱ \) Proposition 5.5. Let \( \mu : M \rightarrow N \) and \( v : M \rightarrow J \) be monomorphisms. If \( \mu \) is essential and \( J \) is injective, then \( v = \kappa \circ \mu \) for some monomorphism \( \kappa : N \rightarrow J \) . Proof. Since \( J \) is injective, there exists a homomorphism \( \kappa : N \rightarrow J \) such that \( v = \kappa \circ \mu \), which is injective by 5.2. \( ▱ \) By 5.5, every essential extension of \( M \) is, up to isomorphism, contained in every injective extension of \( M \) . This leads to the main result in this section. Theorem 5.6. Every left R-module \( M \) is an essential submodule of an injective \( R \) -module, which is unique up to isomorphism. Proof. By Theorem 4.11, \( M \) is a submodule of an injective module \( K \) . Let \( \mathcal{S} \) be the set of all submodules \( M \subseteq S \subseteq K \) of \( K \) in which \( M \) is essential (for instance, \( M \) itself). If \( {\left( {S}_{i}\right) }_{i \in I} \) is a chain in \( \mathcal{S} \), then \( S = \mathop{\bigcup }\limits_{{i \in I}}{S}_{i} \in \mathcal{S} \) : if \( N \neq 0 \) is a submodule of \( S \), then \( {S}_{i} \cap N \neq 0 \) for some \( i \), and then \( M \cap N = M \cap {S}_{i} \cap N \neq 0 \) since \( M \) is essential in \( {S}_{i} \) ; thus \( M \) is essential in \( S \) . By Zorn’s lemma, \( \mathcal{S} \) has a maximal element \( J \) . If \( J \) had a proper essential extension, then by \( {5.5}\mathrm{\;J} \) would have a proper essential extension \( J \subsetneqq {J}^{\prime } \subseteq K \) and would not be maximal; therefore \( J \) is injective, by 5.4. Now, assume that \( M \) is essential in two injective modules \( J \) and \( {J}^{\prime } \) . The inclusion monomorphisms \( \mu : M \rightarrow J \) and \( v : M \rightarrow {J}^{\prime } \) are essential. By 5.5 there is a monomorphism \( \theta : J \rightarrow {J}^{\prime } \) such that \( v = \theta \circ \mu \) . Then \( \theta \) is essential, by 5.3, and is an isomorphism by 5.4. \( ▱ \) Definition. The injective hull of a left \( R \) -module \( M \) is the injective module, unique up to isomorphism, in which \( M \) is an essential submodule. The injective hull or injective envelope \( E\left( M\right) \) of \( M \) can be characterized in several ways: \( E\left( M\right) \) is

(Theorem 4.12. A ring \( R \) is left Noetherian if and only if every direct sum of injective left \( R \) -modules is injective.)

Assume that every direct sum of injective left \( R \) -modules is injective, and let \( {L}_{1} \subseteq {L}_{2} \subseteq \cdots \subseteq {L}_{n} \subseteq \cdots \) be an ascending sequence of left ideals of \( R \) . Then \( L = \mathop{\bigcup }\limits_{{n > 0}}{L}_{n} \) is a left ideal of \( R \) . By 4.11 there is a monomorphism of \( R/{L}_{n} \) into an injective \( R \) -module \( {J}_{n} \) . By the hypothesis, \( J = {\bigoplus }_{n > 0}{J}_{n} \) is injective. Construct a module homomorphism \( \varphi : L \rightarrow J \) as follows. Let \( {\varphi }_{n} \) be the homomorphism \( R \rightarrow R/{L}_{n} \rightarrow {J}_{n} \) . If \( x \in L \), then \( x \in {L}_{n} \) for some \( n \), and then \( {\varphi }_{k}\left( x\right) = 0 \) for all \( k \geqq n \) . Let \( \varphi \left( x\right) = {\left( {\varphi }_{k}\left( x\right) \right) }_{k > 0} \in J \) . Since \( J \) is injective, \( \varphi \) extends to a module homomorphism \( \psi : {}_{R}R \rightarrow J \) . Then \( \psi \left( 1\right) = {\left( {t}_{k}\right) }_{k > 0} \in J \) for some \( {t}_{k} \in {J}_{k},{t}_{k} = 0 \) for almost all \( k \) . Hence \( \psi \left( 1\right) \in {\bigoplus }_{k < n}{J}_{k} \) for some \( n \) . Then \( \varphi \left( x\right) = \psi \left( {x1}\right) = {x\psi }\left( 1\right) \in {\bigoplus }_{k < n}{J}_{k} \) for all \( x \in L \) ; in particular, \( {\varphi }_{n}\left( x\right) = 0 \) and \( x \in {L}_{n} \), since \( R/{L}_{n} \rightarrow {J}_{n} \) is injective. Thus \( L = {L}_{n} \), and then \( {L}_{k} = {L}_{n} \) for all \( k \geqq n \) . Conversely, assume that \( R \) is left Noetherian, and let \( J = {\bigoplus }_{i \in I}{J}_{i} \) be a direct sum of injective \( R \) -modules \( {J}_{i} \) . Let

Lemma 10.55. If \( \alpha \) and \( \beta \) are distinct variables, \( \alpha \) does not occur bound in \( \varphi \) , and \( \beta \) does not occur in \( \varphi \) at all, then \( \vdash \forall {\alpha \varphi } \leftrightarrow \forall \beta {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi \) . Proof \[ \vdash \forall {\alpha \varphi } \rightarrow {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \] 10.53 \[ \vdash \forall \beta \forall {\alpha \varphi } \rightarrow \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \] using \( {10.23}\left( 2\right) \) \[ \vdash \forall {\alpha \varphi } \rightarrow \forall \alpha \forall {\alpha \varphi } \] 10.23(3) \[ \vdash \forall {\alpha \varphi } \rightarrow \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \] Now we can apply the result just obtained to \( \beta ,\alpha \), Subf \( {}_{\beta }^{\alpha }\varphi \) instead of \( \alpha ,\beta ,\varphi \) and obtain \( \vdash \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \rightarrow \forall \alpha {\operatorname{Subf}}_{\beta }^{\alpha }{\operatorname{Subf}}_{\alpha }^{\beta }\varphi \) . Since \( {\operatorname{Subf}}_{\alpha }^{\beta }{\operatorname{Subf}}_{\beta }^{\alpha }\varphi = \varphi \), the desired theorem follows. Definition 10.56. By \( {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \) we mean the formula obtained from \( \varphi \) by replacing each bound occurrence of \( \alpha \) in \( \varphi \) by \( \beta \) . Again one should check that \( {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \) is really always again a formula. The formation of \( {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \) from \( \varphi \) is clearly effective. The following proposition is established analogously to 10.52 . Proposition 10.57. If \( m \) is the Gödel number of a variable \( \alpha, x \) is the Gödel number of a variable \( \beta \), and \( y \) is the Gödel number of a formula \( \varphi \), let \( h\left( {m, x, y}\right) \) be the Gödel number of \( {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \) ; otherwise, let \( h\left( {m, x, y}\right) = 0 \) . Then \( h \) is recursive. Lemma 10.58. If \( \alpha \) occurs bound in \( \varphi \), then there is a formula \( \forall {\alpha \psi } \) which occurs in \( \varphi \) such that \( \alpha \) does not occur bound in \( \psi \) . Proof. Induction on \( \varphi \) . Theorem 10.59 (Change of bound variables). If \( \beta \) does not occur in \( \varphi \), then \( \vdash \varphi \leftrightarrow {\mathrm{{Subb}}}_{\beta }^{\alpha }\varphi . \) Proof. We proceed by induction on the number \( m \) of bound occurrences of \( \alpha \) in \( \varphi \) . If \( m = 0 \), the desired conclusion is trivial. We now assume that \( m > 0 \), and that our result is known for all formulas with fewer than \( m \) bound occurrences of \( \alpha \) . By Lemma 10.58, let \( \forall {\alpha \psi } \) be a formula occurring in \( \varphi \) such that \( \alpha \) does not occur bound in \( \psi \) . Let \( \gamma \) be a variable not occurring in \( \varphi \) (hence not in \( \psi \) ) with \( \gamma \neq \alpha ,\beta \) . Then by 10.55 we have (1) \[ \vdash \forall {\alpha \psi } \leftrightarrow \forall \gamma {\operatorname{Subf}}_{\gamma }^{\alpha }\psi \] Now let \( \chi \) be obtained from \( \varphi \) by replacing an occurrence of \( \forall {\alpha \psi } \) in \( \varphi \) by \( \forall \gamma {\text{Subf }}_{\gamma }^{\alpha }\psi \) . Then by (1) and 10.54 we have (2) \[ \vdash \varphi \leftrightarrow \chi \text{.} \] Now \( \beta \) does not occur in \( \chi \), and \( \chi \) has fewer than \( m \) bound occurrences of \( \alpha \) . Hence by the induction assumption, (3) \[ { \vdash }_{\chi } \leftrightarrow {\operatorname{Subb}}_{\beta }^{\alpha }\chi \] Now clearly \( {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \) can be obtained from \( {\operatorname{Subb}}_{\beta }^{\alpha }\chi \) by replacing an occurrence of \( \forall \gamma {\operatorname{Subf}}_{\gamma }^{\alpha }\psi \) in \( {\operatorname{Subb}}_{\beta }^{\alpha }\chi \) by \( \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\psi \) . Thus by 10.54 it suffices to show (4) \[ \vdash \forall \gamma {\operatorname{Subf}}_{\gamma }^{\alpha }\psi \leftrightarrow \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\psi \] In fact, \( \gamma \neq \beta ,\gamma \) does not occur bound in \( {\operatorname{Subf}}_{\gamma }^{\alpha }\psi \), and \( \beta \) does not occur in \( {\operatorname{Subf}}_{\gamma }^{\alpha }\psi \) at all; furthermore, \( {\operatorname{Subf}}_{\beta }^{\gamma }{\operatorname{Subf}}_{\gamma }^{\alpha }\psi = {\operatorname{Subf}}_{\beta }^{\alpha }\psi \) . Thus 10.55 yields (4), and the proof is complete. The restriction on \( \beta \) in 10.59 is necessary, as one sees intuitively by the example \( \varphi = \exists \alpha \neg \alpha = \beta \) with \( \alpha \neq \beta \) . Here \( {\operatorname{Subb}}_{\beta }^{\alpha }\varphi = \exists \beta \neg \beta = \beta \), and \( \varphi \rightarrow {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \) is obviously not a valid formula, so \( {\psi }_{\varphi } \rightarrow {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \) . This can be rigorously established after we have introduced the notion of truth (see 11.50). We now turn to properties of \( {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi \) . The main result,10.61, removes unnecessary hypotheses from 10.53. Lemma 10.60. If the variable \( \alpha \) does not occur in \( \sigma \), and if no free occurrence of \( \alpha \) in \( \varphi \) is within the scope of a quantifier on a variable occurring in \( \sigma \), then \( \vdash \forall {\alpha \varphi } \rightarrow {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi . \) Proof. Let \( \beta \) be a variable not occurring in \( \varphi \), not occurring in \( \sigma \), and different from \( \alpha \) . Then by change of bound variables, \[ \vdash \varphi \leftrightarrow {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \] Hence using \( {10.23}\left( 2\right) \) we infer that (1) \[ \vdash \forall {\alpha \varphi } \leftrightarrow \forall \alpha {\operatorname{Subb}}_{\beta }^{\alpha }\varphi . \] Now \( \alpha \) does not occur bound in \( {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \), so by 10.53 we see that (2) \[ \vdash \forall \alpha {\operatorname{Subb}}_{\beta }^{\alpha }\varphi \rightarrow {\operatorname{Subf}}_{\sigma }^{\alpha }{\operatorname{Subb}}_{\beta }^{\alpha }\varphi . \] Now \( \alpha \) does not occur at all in \( {\operatorname{Subf}}_{\sigma }^{\alpha }{\operatorname{Subb}}_{\beta }^{\alpha }\varphi \), and clearly \[ {\operatorname{Subb}}_{\alpha }^{\beta }{\operatorname{Subf}}_{\sigma }^{\alpha }{\operatorname{Subb}}_{\beta }^{\alpha }\varphi = {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi , \] so by change of bound variable, (3) \[ {\mathrm{{FSubf}}}_{\sigma }^{\alpha }{\mathrm{{Subb}}}_{\beta }^{\alpha }\varphi \leftrightarrow {\mathrm{{Subf}}}_{\sigma }^{\alpha }\varphi . \] Conditions (1), (2), (3) immediately yield the desired result. Theorem 10.61 (Universal specification). If no free occurrence in \( \varphi \) of the variable \( \alpha \) is within the scope of a quantifier on a variable occurring in \( \sigma \) , then \( \vdash \forall {\alpha \varphi } \rightarrow {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi \) . Proof. Let \( \beta \) be a variable not occurring in \( \varphi \) or in \( \sigma \), and distinct from \( \alpha \) . Then \[ \vdash \forall {\alpha \varphi } \rightarrow {\mathrm{{Subf}}}_{\beta }^{\alpha }\varphi \] 10.60 (1) \[ \vdash \forall \beta \forall {\alpha \varphi } \rightarrow \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \] using \( {10.23}\left( 2\right) \) (2) \[ H\xrightarrow[]{\alpha }\varphi \rightarrow \forall \beta \;\forall {\alpha \varphi } \] 10.23(3) Now no free occurrence of \( \beta \) in \( \operatorname{Sub}{f}_{\beta }^{\alpha }\varphi \) is within the scope of a quantifier on a variable occurring in \( \sigma \) . Clearly also Subf \( {}_{\sigma }^{\beta } \) Subf \( {}_{\beta }^{\alpha }\varphi = \) Subf \( {}_{\sigma }^{\alpha }\varphi \) . Hence by 10.60 (3) \[ \vdash \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \rightarrow {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi . \] Conditions (1), (2), (3) immediately yield the desired result. Theorem 10.61 gives the most important property of \( {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi \) . This property is frequently taken as one of the axiom schemas for derivability. Again, the hypothesis on \( \alpha \) is necessary, as is seen by the example \( \varphi = \exists \beta \neg \left( {\alpha = \beta }\right) \) ; \( \forall {\alpha \varphi } \rightarrow {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \) is not logically valid. We say here that a clash of bound variables has occurred. We now give some important corollaries of 10.61. Corollary 10.62. \( \; \vdash \forall {\alpha \varphi } \rightarrow \varphi \) . Corollary 10.63. If the variable \( \alpha \) does not occur free in \( \varphi \), then \( \vdash \varphi \leftrightarrow \forall {\alpha \varphi } \) . Proof. By 10.62, (1) \[ \vdash \forall {\alpha \varphi } \rightarrow \varphi \text{.} \] 182 For the other direction, let \( \beta \) be a variable not occurring in \( \varphi \) . Then by change of bound variable, (2) \[ \vdash \varphi \leftrightarrow {\mathrm{{Subb}}}_{\beta }^{\alpha }\varphi . \] Hence using \( {10.23}\left( 2\right) \) we obtain (3) \[ \vdash \forall \alpha {\mathrm{{Subb}}}_{\beta }^{\alpha }\varphi \rightarrow \forall {\alpha \varphi }, \] and by \( {10.23}\left( 3\right) \) we obtain (4) \[ {\mathrm{{FSubb}}}_{\beta }^{\alpha }\varphi \rightarrow \forall \alpha {\mathrm{{Subb}}}_{\beta }^{\alpha }\varphi . \] From (1)-(4) the desired conclusion eas

Lemma 10.55. If \( \alpha \) and \( \beta \) are distinct variables, \( \alpha \) does not occur bound in \( \varphi \) , and \( \beta \) does not occur in \( \varphi \) at all, then \( \vdash \forall {\alpha \varphi } \leftrightarrow \forall \beta {\operatorname{Subf}}_{\sigma }^{\alpha }\varphi \) .

\[ \vdash \forall {\alpha \varphi } \rightarrow {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \] \[ \vdash \forall \beta \forall {\alpha \varphi } \rightarrow \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \] using \( {10.23}\left( 2\right) \) \[ \vdash \forall {\alpha \varphi } \rightarrow \forall \alpha \forall {\alpha \varphi } \] \[ \vdash \forall {\alpha \varphi } \rightarrow \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \] Now we can apply the result just obtained to \( \beta ,\alpha \), Subf \( {}_{\beta }^{\alpha }\varphi \) instead of \( \alpha ,\beta ,\varphi \) and obtain \( \vdash \forall \beta {\operatorname{Subf}}_{\beta }^{\alpha }\varphi \rightarrow \forall \alpha {\operatorname{Subf}}_{\beta }^{\alpha }{\operatorname{Subf}}_{\alpha }^{\beta }\varphi \) . Since \( {\operatorname{Subf}}_{\alpha }^{\beta }{\operatorname{Subf}}_{\beta }^{\alpha }\varphi = \varphi \), the desired theorem follows.

Theorem 2.6 (Homomorphism Theorem). If \( \varphi : A \rightarrow B \) is a homomorphism of left \( R \) -modules, then \[ A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi \] in fact, there is an isomorphism \( \theta : A/\operatorname{Ker}f \rightarrow \operatorname{Im}f \) unique such that \( \varphi = \iota \circ \theta \circ \pi \), where \( \iota : \operatorname{Im}f \rightarrow B \) is the inclusion homomorphism and \( \pi : A \rightarrow A/\operatorname{Ker}f \) is the canonical projection. Thus every module homomorphism is the composition of an inclusion homomorphism, an isomorphism, and a canonical projection to a quotient module:  Theorem 2.7 (First Isomorphism Theorem). If \( A \) is a left \( R \) -module and \( B \supseteq C \) are submodules of \( A \), then \[ A/B \cong \left( {A/C}\right) /\left( {B/C}\right) ; \] in fact, there is a unique isomorphism \( \theta : A/B \rightarrow \left( {A/C}\right) /\left( {B/C}\right) \) such that \( \theta \circ \rho = \tau \circ \pi \), where \( \pi : A \rightarrow A/C,\rho : A \rightarrow A/B \), and \( \tau : A/C \rightarrow \) \( \left( {A/C}\right) /\left( {B/C}\right) \) are the canonical projections:  Theorem 2.8 (Second Isomorphism Theorem). If \( A \) and \( B \) are submodules of a left \( R \) -module, then \[ \left( {A + B}\right) /B \cong A/\left( {A \cap B}\right) \] in fact, there is an isomorphism \( \theta : A/\left( {A \cap B}\right) \rightarrow \left( {A + B}\right) /B \) unique such that \( \theta \circ \rho = \pi \circ \iota \), where \( \pi : A + B \rightarrow \left( {A + B}\right) /B \) and \( \rho : A \rightarrow A/\left( {A \cap B}\right) \) are the canonical projections and \( \iota : A \rightarrow A + B \) is the inclusion homomorphism:  Proofs. In Theorem 2.6, there is by Theorem I.5.2 a unique isomorphism \( \theta \) of abelian groups such that \( \varphi = \iota \circ \theta \circ \pi \) . Then \( \theta \left( {a + \operatorname{Ker}\varphi }\right) = \varphi \left( a\right) \) for all \( a \in A \) . Therefore \( \theta \) is a module homomorphism. Theorems 2.7 and 2.8 are proved similarly. \( ▱ \) As in Section I.5, the isomorphisms theorems are often numbered so that 2.6 is the first isomorphism theorem. Then our first and second isomorphism theorems, 2.7 and 2.8, are the second and third isomorphism theorems. As another application of Theorem 2.6 we construct all cyclic modules. Proposition 2.9. A unital left \( R \) -module is cyclic if and only if it is isomorphic to \( R/L\left( { = {}_{R}R/L}\right) \) for some left ideal \( L \) of \( R \) . If \( M = {Rm} \) is cyclic, then \( M \cong R/\operatorname{Ann}\left( m\right) \), where \[ \operatorname{Ann}\left( m\right) = \{ r \in R \mid {rm} = 0\} \] is a left ideal of \( R \) . If \( R \) is commutative, then \( \operatorname{Ann}\left( {Rm}\right) = \operatorname{Ann}\left( m\right) \) . Proof. Let \( M = {Rm} \) be cyclic. Then \( \varphi : r \mapsto {rm} \) is a module homomorphism of \( {}_{R}R \) onto \( M \) . By 2.6, \( M \cong R/\operatorname{Ker}\varphi \), and we see that \( \operatorname{Ker}\varphi = \operatorname{Ann}\left( m\right) \) . In particular, \( \operatorname{Ann}\left( m\right) \) is a left ideal of \( R \) . Moreover, \( \operatorname{Ann}\left( M\right) \subseteq \operatorname{Ann}\left( m\right) \) ; if \( R \) is commutative, then, conversely, \( {sm} = 0 \) implies \( s\left( {rm}\right) = 0 \) for all \( r \in R \), and \( \operatorname{Ann}\left( m\right) = \operatorname{Ann}\left( M\right) \) . Conversely, if \( L \) is a left ideal of \( R \), then \( R/L \) is cyclic, generated by \( 1 + L \) , since \( r + L = r\left( {1 + L}\right) \) for every \( r \in R \) . Hence any \( M \cong R/L \) is cyclic. \( ▱ \) The left ideal \( \operatorname{Ann}\left( m\right) \) is the annihilator of \( m \) . In any left \( R \) -module \( M \) , Ann \( \left( m\right) \) is a left ideal of \( R \) ; moreover, \( \operatorname{Ann}\left( M\right) = \mathop{\bigcap }\limits_{{m \in M}}\operatorname{Ann}\left( m\right) \) . ## Exercises 1. Let \( \varphi : A \rightarrow B \) be a homomorphism of left \( R \) -modules. Show that \( \varphi \left( {{\varphi }^{-1}\left( C\right) }\right) = \) \( C \cap \operatorname{Im}\varphi \), for every submodule \( C \) of \( B \) . 2. Let \( \varphi : A \rightarrow B \) be a homomorphism of left \( R \) -modules. Show that \( {\varphi }^{-1}\left( {\varphi \left( C\right) }\right) = \) \( C + \operatorname{Ker}\varphi \), for every submodule \( C \) of \( A \) . 3. Let \( \varphi : A \rightarrow B \) be a homomorphism of left \( R \) -modules. Show that direct and inverse image under \( \varphi \) induce a one-to-one correspondence, which preserves inclusion, between submodules of \( A \) that contain \( \operatorname{Ker}\varphi \), and submodules of \( \operatorname{Im}\varphi \) . 4. Let \( M \) be a [unital] left \( R \) -module and let \( I \) be a two-sided ideal of \( R \) . Make \( M/{IM} \) an \( R/I \) -module. 5. Let \( R \) be a (commutative) domain. Show that all nonzero principal ideals of \( R \) are isomorphic (as \( R \) -modules). 6. Let \( A \) and \( B \) be submodules of \( M \) . Show by an example that \( A \cong B \) does not imply \( M/A \cong M/B \) . 7. Let \( R \) be a ring with an identity element. If \( x, y \in R \) and \( {xR} = {yR} \), then show that \( {Rx} \cong {Ry} \) (as left \( R \) -modules); in fact, there is an isomorphism \( {Rx} \rightarrow {Ry} \) that sends \( x \) to \( y \) . 8. Let \( \varphi : A \rightarrow B \) and \( \psi : B \rightarrow C \) be module homomorphisms. Show that \( \psi \circ \varphi = 0 \) if and only if \( \varphi \) factors through the inclusion homomorphism \( \operatorname{Ker}\psi \rightarrow B \) . 9. Let \( \varphi : A \rightarrow B \) and \( \psi : B \rightarrow C \) be module homomorphisms. Show that \( \psi \circ \varphi = 0 \) if and only if \( \psi \) factors through the projection \( B \rightarrow B/\operatorname{Im}\varphi \) . 10. If \( \varphi : A \rightarrow B \) and \( \rho : A \rightarrow C \) are module homomorphisms, \( \rho \) is surjective, and \( \operatorname{Ker}\rho \subseteq \operatorname{Ker}\varphi \), then show that \( \varphi \) factors uniquely through \( \rho \) . ## 3. Direct Sums and Products Direct sums and products construct modules from simpler modules, and their universal properties help build diagrams. The definitions and basic properties in this section are stated for left modules but apply to right modules as well. Direct products. The direct product of a family of modules is their Cartesian product, with componentwise operations: Definition. The direct product of a family \( {\left( {A}_{i}\right) }_{i \in I} \) of left \( R \) -modules is their Cartesian product \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) (the set of all families \( {\left( {x}_{i}\right) }_{i \in I} \) such that \( {x}_{i} \in {A}_{i} \) for all \( i \) ) with componentwise addition and action of \( R \) : \[ {\left( {x}_{i}\right) }_{i \in I} + {\left( {y}_{i}\right) }_{i \in I} = {\left( {x}_{i} + {y}_{i}\right) }_{i \in I}, r{\left( {x}_{i}\right) }_{i \in I} = {\left( r{x}_{i}\right) }_{i \in I}. \] It is immediate that these operations make \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) a left \( R \) -module. If \( I = \varnothing \) , then \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} = \{ 0\} \) . If \( I = \{ 1\} \), then \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} \cong {A}_{1} \) . If \( I = \{ 1,2,\ldots, n\} \) , then \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) is also denoted by \( {A}_{1} \times {A}_{2} \times \cdots \times {A}_{n} \) . The direct product \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) comes with a projection \( {\pi }_{j} : \mathop{\prod }\limits_{{i \in I}}{A}_{i} \rightarrow {A}_{j} \) for every \( j \in I \), which sends \( {\left( {x}_{i}\right) }_{i \in I} \) to its \( j \) component \( {x}_{j} \), and is a homomorphism; in fact, the left \( R \) -module structure on \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) is the only module structure such that every projection is a module homomorphism. The direct product and its projections have a universal property: Proposition 3.1. Let \( M \) and \( {\left( {A}_{i}\right) }_{i \in I} \) be left \( R \) -modules. For every family \( {\left( {\varphi }_{i}\right) }_{i \in I} \) of module homomorphisms \( {\varphi }_{i} : M \rightarrow {A}_{i} \) there exists a unique module homomorphism \( \varphi : M \rightarrow \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) such that \( {\pi }_{i} \circ \varphi = {\varphi }_{i} \) for all \( i \in I \) :  The proof is an exercise. By 3.1, every family of homomorphisms \( {\varphi }_{i} \) : \( {A}_{i} \rightarrow {B}_{i} \) induces a homomorphism \( \varphi = \mathop{\prod }\limits_{{i \in I}}{\varphi }_{i} \) unique such that every square  commutes ( \( {\rho }_{i} \circ \varphi = {\varphi }_{i} \circ {\pi }_{i} \) for all \( i \) ), where \( {\pi }_{i} \) and \( {\rho }_{i} \) are the projections; namely, \( \varphi \left( {\left( {x}_{i}\right) }_{i \in I}\right) = {\left( {\varphi }_{i}\left( {x}_{i}\right) \right) }_{i \in I} \) . This can also be shown directly. If \( I = \) \( \{ 1,2,\ldots, n\} \), then \( \mathop{\prod }\limits_{{i \in I}}{\varphi }_{i} \) is also denoted by \( {\varphi }_{1} \times {\varphi }_{2} \times \cdots \times {\va

Theorem 2.6 (Homomorphism Theorem). If \( \varphi : A \rightarrow B \) is a homomorphism of left \( R \) -modules, then \[ A/\operatorname{Ker}\varphi \cong \operatorname{Im}\varphi \] in fact, there is an isomorphism \( \theta : A/\operatorname{Ker}f \rightarrow \operatorname{Im}f \) unique such that \( \varphi = \iota \circ \theta \circ \pi \), where \( \iota : \operatorname{Im}f \rightarrow B \) is the inclusion homomorphism and \( \pi : A \rightarrow A/\operatorname{Ker}f \) is the canonical projection.

In Theorem 2.6, there is by Theorem I.5.2 a unique isomorphism \( \theta \) of abelian groups such that \( \varphi = \iota \circ \theta \circ \pi \) . Then \( \theta \left( {a + \operatorname{Ker}\varphi }\right) = \varphi \left( a\right) \) for all \( a \in A \) . Therefore \( \theta \) is a module homomorphism.

Exercise 1.3.2 Let \( p \) be an odd prime. Suppose that \( {2}^{n} \equiv 1\left( {\;\operatorname{mod}\;p}\right) \) and \( {2}^{n} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . Show that \( {2}^{d} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) where \( d \) is the order of 2 \( \left( {\;\operatorname{mod}\;p}\right) \) . Exercise 1.3.3 Assuming the \( {ABC} \) Conjecture, show that there are infinitely many primes \( p \) such that \( {2}^{p - 1} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . Exercise 1.3.4 Show that the number of primes \( p \leq x \) for which \[ {2}^{p - 1} ≢ 1\;\left( {\;\operatorname{mod}\;{p}^{2}}\right) \] is \( \gg \log x/\log \log x \), assuming the \( {ABC} \) Conjecture. In 1909, Wieferich proved that if \( p \) is a prime satisfying \[ {2}^{p - 1} ≢ 1\;\left( {\;\operatorname{mod}\;{p}^{2}}\right) \] then the equation \( {x}^{p} + {y}^{p} = {z}^{p} \) has no nontrivial integral solutions satisfying \( p \nmid {xyz} \) . It is still unknown without assuming \( {ABC} \) if there are infinitely many primes \( p \) such that \( {2}^{p - 1} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . (See also Exercise 9.2.15.) A natural number \( n \) is called squarefull (or powerfull) if for every prime \( p \mid n \) we have \( {p}^{2} \mid n \) . In 1976 Erdös [Er] conjectured that we cannot have three consecutive squarefull natural numbers. Exercise 1.3.5 Show that if the Erdös conjecture above is true, then there are infinitely many primes \( p \) such that \( {2}^{p - 1} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . Exercise 1.3.6 Assuming the \( {ABC} \) Conjecture, prove that there are only finitely many \( n \) such that \( n - 1, n, n + 1 \) are squarefull. Exercise 1.3.7 Suppose that \( a \) and \( b \) are odd positive integers satisfying \[ \operatorname{rad}\left( {{a}^{n} - 2}\right) = \operatorname{rad}\left( {{b}^{n} - 2}\right) \] for every natural number \( n \) . Assuming \( {ABC} \), prove that \( a = b \) . (This problem is due to \( \mathrm{H} \) . Kisilevsky.) ## 1.4 Supplementary Problems Exercise 1.4.1 Show that every proper ideal of \( \mathbb{Z} \) is of the form \( n\mathbb{Z} \) for some integer \( n \) . Exercise 1.4.2 An ideal \( I \) is called prime if \( {ab} \in I \) implies \( a \in I \) or \( b \in I \) . Prove that every prime ideal of \( \mathbb{Z} \) is of the form \( p\mathbb{Z} \) for some prime integer \( p \) . Exercise 1.4.3 Prove that if the number of prime Fermat numbers is finite, then the number of primes of the form \( {2}^{n} + 1 \) is finite. Exercise 1.4.4 If \( n > 1 \) and \( {a}^{n} - 1 \) is prime, prove that \( a = 2 \) and \( n \) is prime. Exercise 1.4.5 An integer is called perfect if it is the sum of its divisors. Show that if \( {2}^{n} - 1 \) is prime, then \( {2}^{n - 1}\left( {{2}^{n} - 1}\right) \) is perfect. Exercise 1.4.6 Prove that if \( p \) is an odd prime, any prime divisor of \( {2}^{p} - 1 \) is of the form \( {2kp} + 1 \), with \( k \) a positive integer. Exercise 1.4.7 Show that there are no integer solutions to the equation \( {x}^{4} - {y}^{4} = \) \( 2{z}^{2} \) . Exercise 1.4.8 Let \( p \) be an odd prime number. Show that the numerator of \[ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p - 1} \] is divisible by \( p \) . Exercise 1.4.9 Let \( p \) be an odd prime number greater than 3 . Show that the numerator of \[ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p - 1} \] is divisible by \( {p}^{2} \) . Exercise 1.4.10 (Wilson’s Theorem) Show that \( n > 1 \) is prime if and only if \( n \) divides \( \left( {n - 1}\right) ! + 1 \) . Exercise 1.4.11 For each \( n > 1 \), let \( Q \) be the product of all numbers \( a < n \) which are coprime to \( n \) . Show that \( Q \equiv \pm 1\left( {\;\operatorname{mod}\;n}\right) \) . Exercise 1.4.12 In the previous exercise, show that \( Q \equiv 1\left( {\;\operatorname{mod}\;n}\right) \) whenever \( n \) is odd and has at least two prime factors. Exercise 1.4.13 Use Exercises 1.2.7 and 1.2.8 to show that there are infinitely many primes \( \equiv 1\left( {\;\operatorname{mod}\;{2}^{r}}\right) \) for any given \( r \) . Exercise 1.4.14 Suppose \( p \) is an odd prime such that \( {2p} + 1 = q \) is also prime. Show that the equation \[ {x}^{p} + 2{y}^{p} + 5{z}^{p} = 0 \] has no solutions in integers. Exercise 1.4.15 If \( x \) and \( y \) are coprime integers, show that if \[ \left( {x + y}\right) \text{ and }\frac{{x}^{p} + {y}^{p}}{x + y} \] have a common prime factor, it must be \( p \) . Exercise 1.4.16 (Sophie Germain’s Trick) Let \( p \) be a prime such that \( {2p} + \) \( 1 = q > 3 \) is also prime. Show that \[ {x}^{p} + {y}^{p} + {z}^{p} = 0 \] has no integral solutions with \( p \nmid {xyz} \) . Exercise 1.4.17 Assuming \( {ABC} \), show that there are only finitely many consecutive cubefull numbers. Exercise 1.4.18 Show that \[ \mathop{\sum }\limits_{p}\frac{1}{p} = + \infty \] where the summation is over prime numbers. Exercise 1.4.19 (Bertrand’s Postulate) (a) If \( {a}_{0} \geq {a}_{1} \geq {a}_{2} \geq \cdots \) is a de- creasing sequence of real numbers tending to 0 , show that \[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}{a}_{n} \leq {a}_{0} - {a}_{1} + {a}_{2} \] (b) Let \( T\left( x\right) = \mathop{\sum }\limits_{{n < x}}\psi \left( {x/n}\right) \), where \( \psi \left( x\right) \) is defined as in Exercise 1.1.25. Show that \[ T\left( x\right) = x\log x - x + O\left( {\log x}\right) . \] (c) Show that \[ T\left( x\right) - {2T}\left( \frac{x}{2}\right) = \mathop{\sum }\limits_{{n \leq x}}{\left( -1\right) }^{n - 1}\psi \left( \frac{x}{n}\right) = \left( {\log 2}\right) x + O\left( {\log x}\right) . \] Deduce that \[ \psi \left( x\right) - \psi \left( \frac{x}{2}\right) \geq \frac{1}{3}\left( {\log 2}\right) x + O\left( {\log x}\right) . \] ## Chapter 2 ## Euclidean Rings ## 2.1 Preliminaries We can discuss the concept of divisibility for any commutative ring \( R \) with identity. Indeed, if \( a, b \in R \), we will write \( a \mid b \) ( \( a \) divides \( b \) ) if there exists some \( c \in R \) such that \( {ac} = b \) . Any divisor of 1 is called a unit. We will say that \( a \) and \( b \) are associates and write \( a \sim b \) if there exists a unit \( u \in R \) such that \( a = {bu} \) . It is easy to verify that \( \sim \) is an equivalence relation. Further, if \( R \) is an integral domain and we have \( a, b \neq 0 \) with \( a \mid b \) and \( b \mid a \), then \( a \) and \( b \) must be associates, for then \( \exists c, d \in R \) such that \( {ac} = b \) and \( {bd} = a \), which implies that \( {bdc} = b \) . Since we are in an integral domain, \( {dc} = 1 \), and \( d, c \) are units. We will say that \( a \in R \) is irreducible if for any factorization \( a = {bc} \), one of \( b \) or \( c \) is a unit. Example 2.1.1 Let \( R \) be an integral domain. Suppose there is a map \( n : R \rightarrow \mathbb{N} \) such that: (i) \( n\left( {ab}\right) = n\left( a\right) n\left( b\right) \forall a, b \in R \) ; and (ii) \( n\left( a\right) = 1 \) if and only if \( a \) is a unit. We call such a map a norm map, with \( n\left( a\right) \) the norm of \( a \) . Show that every element of \( R \) can be written as a product of irreducible elements. Solution. Suppose \( b \) is an element of \( R \) . We proceed by induction on the norm of \( b \) . If \( b \) is irreducible, then we have nothing to prove, so assume that \( b \) is an element of \( R \) which is not irreducible. Then we can write \( b = {ac} \) where neither \( a \) nor \( c \) is a unit. By condition (i), \[ n\left( b\right) = n\left( {ac}\right) = n\left( a\right) n\left( c\right) \] and since \( a, c \) are not units, then by condition (ii), \( n\left( a\right) < n\left( b\right) \) and \( n\left( c\right) < \) \( n\left( b\right) \) . If \( a, c \) are irreducible, then we are finished. If not, their norms are smaller than the norm of \( b \), and so by induction we can write them as products of irreducibles, thus finding an irreducible decomposition of \( b \) . Exercise 2.1.2 Let \( D \) be squarefree. Consider \( R = \mathbb{Z}\left\lbrack \sqrt{D}\right\rbrack \) . Show that every element of \( R \) can be written as a product of irreducible elements. Exercise 2.1.3 Let \( R = \mathbb{Z}\left\lbrack \sqrt{-5}\right\rbrack \) . Show that \( 2,3,1 + \sqrt{-5} \), and \( 1 - \sqrt{-5} \) are irreducible in \( R \), and that they are not associates. We now observe that \( 6 = 2 \cdot 3 = \left( {1 + \sqrt{-5}}\right) \left( {1 - \sqrt{-5}}\right) \), so that \( R \) does not have unique factorization into irreducibles. We will say that \( R \), an integral domain, is a unique factorization domain if: (i) every element of \( R \) can be written as a product of irreducibles; and (ii) this factorization is essentially unique in the sense that if \( a = {\pi }_{1}\cdots {\pi }_{r} \) . and \( a = {\tau }_{1}\cdots {\tau }_{s} \), then \( r = s \) and after a suitable permutation, \( {\pi }_{i} \sim {\tau }_{i} \) . Exercise 2.1.4 Let \( R \) be a domain satisfying (i) above. Show that (ii) is equivalent to \( \left( {\mathrm{{ii}}}^{ \star }\right) \) : if \( \pi \) is irreducible and \( \pi \) divides \( {ab} \), then \( \pi \mid a \) or \( \pi \mid b \) . An ideal \( I \subseteq R \) is called principal if it can be generated by a single element of \( R \) . A domain \( R \) is then called a principal ideal domain if every ideal of \( R \) is principal. Exercise 2.1.5 Show that if \( \pi \) is an irreducible element of a principal ideal domain, then \( \left( \pi \right) \) is a maximal ideal,(where \( \left( x\right) \) denotes the ideal generated by the element \( x \) ). Theorem 2.1.6 If \( R \) is a principal ideal domain, then \( R \) is a unique factorization domain. Proof. Let \( S \) be the set of elements of \( R

Exercise 1.3.2 Let \( p \) be an odd prime. Suppose that \( {2}^{n} \equiv 1\left( {\;\operatorname{mod}\;p}\right) \) and \( {2}^{n} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . Show that \( {2}^{d} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) where \( d \) is the order of 2 \( \left( {\;\operatorname{mod}\;p}\right) \) .

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Proposition 2.7 Suppose \( U \) is a simply connected domain in \( {\mathbb{R}}^{n} \) and \( \mathbf{F} \) is a smooth, \( {\mathbb{R}}^{n} \) -valued function on \( U \) . Then \( \mathbf{F} \) is conservative if and only if \( \mathbf{F} \) satisfies \[ \frac{\partial {F}_{j}}{\partial {x}_{k}} - \frac{\partial {F}_{k}}{\partial {x}_{j}} = 0 \] (2.8) at each point in \( U \) . When \( n = 3 \), it is easy to check that the condition (2.8) is equivalent to the curl \( \nabla \times \mathbf{F} \) of \( \mathbf{F} \) being zero on \( U \) . The hypothesis that \( U \) be simply connected cannot be omitted; see Exercise 7. Proof. If \( \mathbf{F} \) is conservative, then \[ \frac{\partial {F}_{j}}{\partial {x}_{k}} = - \frac{{\partial }^{2}V}{\partial {x}_{k}\partial {x}_{j}} = - \frac{{\partial }^{2}V}{\partial {x}_{j}\partial {x}_{k}} = \frac{\partial {F}_{k}}{\partial {x}_{j}} \] at every point in \( U \) . In the other direction, if \( \mathbf{F} \) satisfies (2.8), \( V \) can be obtained by integrating \( \mathbf{F} \) along paths and using the Stokes theorem to establish independence of choice of path. See, for example, Theorem 4.3 on p. 549 of [44] for a proof in the \( n = 3 \) case. The proof in higher dimensions is the same, provided one knows the general version of the Stokes theorem. We may also consider velocity-dependent forces. If, for example, F(x, v) \( = - \gamma \mathbf{v} + {\mathbf{F}}_{1}\left( \mathbf{x}\right) \), where \( \gamma \) is a positive constant, then we will again have energy that is decreasing with time. There is another new phenomenon, however, in dimension greater than 1 , namely the possibility of having a conserved energy even when the force depends on velocity. Proposition 2.8 Suppose a particle in \( {\mathbb{R}}^{n} \) moves in the presence of a force \( \mathbf{F} \) of the form \[ \mathbf{F}\left( {\mathbf{x},\mathbf{v}}\right) = - \nabla V\left( \mathbf{x}\right) + {\mathbf{F}}_{2}\left( {\mathbf{x},\mathbf{v}}\right) \] where \( V \) is a smooth function and where \( {\mathbf{F}}_{2} \) satisfies \[ \mathbf{v} \cdot {\mathbf{F}}_{2}\left( {\mathbf{x},\mathbf{v}}\right) = 0 \] (2.9) for all \( \mathbf{x} \) and \( \mathbf{v} \) in \( {\mathbb{R}}^{n} \) . Then the energy function \( E\left( {\mathbf{x},\mathbf{v}}\right) = \frac{1}{2}m{\left| \mathbf{v}\right| }^{2} + V\left( \mathbf{x}\right) \) is constant along each trajectory. If, for example, \( {\mathbf{F}}_{2} \) is the force exerted on a charged particle in \( {\mathbb{R}}^{3} \) by a magnetic field \( \mathbf{B}\left( \mathbf{x}\right) \), then \[ {\mathbf{F}}_{2}\left( {\mathbf{x},\mathbf{v}}\right) = q\mathbf{v} \times \mathbf{B}\left( \mathbf{x}\right) \] where \( q \) is the charge of the particle, which clearly satisfies (2.9). Proof. See Exercise 8. ## 2.3 Systems of Particles If we have a system if \( N \) particles, each moving in \( {\mathbb{R}}^{n} \), then we denote the position of the \( j \) th particle by \[ {\mathbf{x}}^{j} = \left( {{x}_{1}^{j},\ldots ,{x}_{n}^{j}}\right) \] Thus, in the expression \( {x}_{k}^{j} \), the superscript \( j \) indicates the \( j \) th particle, while the subscript \( k \) indicates the \( k \) th component. Newton’s law then takes the form \[ {m}_{j}{\ddot{\mathbf{x}}}^{j} = {\mathbf{F}}^{j}\left( {{\mathbf{x}}^{1},\ldots ,{\mathbf{x}}^{N},{\dot{\mathbf{x}}}^{1},\ldots ,{\dot{\mathbf{x}}}^{N}}\right) ,\;j = 1,2,\ldots, N, \] where \( {m}_{j} \) is the mass of the \( j \) th particle. Here, \( {\mathbf{F}}^{j} \) is the force on the \( j \) th particle, which in general will depend on the position and velocity not only of that particle, but also on the position and velocity of the other particles. ## 2.3.1 Conservation of Energy In a system of particles, we cannot expect that the energy of each individual particle will be conserved, because as the particles interact, they can exchange energy. Rather, we should expect that, under suitable assumptions on the forces \( {\mathbf{F}}^{j} \), we can define a conserved energy function for the whole system (the total energy of the system). Let us consider forces depending only on the position of the particles, and let us assume that the energy function will be of the form \[ E\left( {{\mathbf{x}}^{1},\ldots ,{\mathbf{x}}^{N},{\mathbf{v}}^{1},\ldots ,{\mathbf{v}}^{N}}\right) = \mathop{\sum }\limits_{{j = 1}}^{N}\frac{1}{2}{m}_{j}{\left| {\mathbf{v}}^{j}\right| }^{2} + V\left( {{\mathbf{x}}^{1},\ldots ,{\mathbf{x}}^{N}}\right) . \] (2.10) We will now try to see what form for \( V \) (if any) will allow \( E \) to be constant along each trajectory. Proposition 2.9 An energy function of the form (2.10) is constant along each trajectory if \[ {\nabla }^{j}V = - {\mathbf{F}}^{j} \] (2.11) for each \( j \), where \( {\nabla }^{j} \) is the gradient with respect to the variable \( {\mathbf{x}}^{j} \) . Proof. We compute that \[ \frac{dE}{dt} = \mathop{\sum }\limits_{{j = 1}}^{N}\left\lbrack {{m}_{j}{\dot{\mathbf{x}}}^{j} \cdot {\ddot{\mathbf{x}}}^{j} + {\nabla }^{j}V \cdot {\dot{\mathbf{x}}}^{j}}\right\rbrack \] \[ = \mathop{\sum }\limits_{{j = 1}}^{N}{\dot{\mathbf{x}}}^{j} \cdot \left\lbrack {{m}_{j}{\ddot{\mathbf{x}}}^{j} + {\nabla }^{j}V}\right\rbrack \] \[ = \mathop{\sum }\limits_{{j = 1}}^{N}{\dot{\mathbf{x}}}^{j} \cdot \left\lbrack {{\mathbf{F}}^{j} + {\nabla }^{j}V}\right\rbrack \] If \( {\nabla }^{j}V = - {\mathbf{F}}^{j} \), then \( E \) will be conserved. - As in the one-particle case, there is a simple condition for the existence of a potential function \( V \) satisfying (2.11). Proposition 2.10 Suppose a force function \( \mathbf{F} = \left( {{\mathbf{F}}^{1},\ldots ,{\mathbf{F}}^{N}}\right) \) is defined on a simply connected domain \( U \) in \( {\mathbb{R}}^{nN} \) . Then there exists a smooth function \( V \) on \( U \) satisfying \[ {\nabla }^{j}V = - {\mathbf{F}}^{j} \] for all \( j \) if and only if we have \[ \frac{\partial {F}_{k}^{j}}{\partial {x}_{m}^{l}} = \frac{\partial {F}_{m}^{l}}{\partial {x}_{k}^{j}} \] (2.12) for all \( j, k, l \), and \( m \) . Proof. Apply Proposition 2.7 with \( n \) replaced by \( {nN} \) and with \( j \) and \( k \) replaced by the pairs \( \left( {j, k}\right) \) and \( \left( {l, m}\right) \) . ## 2.3.2 Conservation of Momentum We now introduce the notion of the momentum of a particle. Definition 2.11 In an \( N \) -particle system, the momentum of the jth particle, denoted \( {\mathbf{p}}^{j} \), is the product of the mass and the velocity of that particle: \[ {\mathbf{p}}^{j} = {m}_{j}{\dot{\mathbf{x}}}^{j} \] The total momentum of the system, denoted \( \mathbf{p} \), is defined as \[ \mathbf{p} = \mathop{\sum }\limits_{{j = 1}}^{N}{\mathbf{p}}^{j} \] Observe that \[ \frac{d{\mathbf{p}}^{j}}{dt} = {m}_{j}{\mathbf{\ddot{x}}}^{j} = {\mathbf{F}}^{j} \] Thus, Newton's law may be reformulated as saying, "The force is the rate of change of the momentum." This is how Newton originally formulated his second law. Newton's third law says, "For every action, there is an equal and opposite reaction." This law will apply if all forces are of the "two-particle" variety and satisfy a natural symmetry property. Having two-particle forces means that the force \( {\mathbf{F}}^{j} \) on the \( j \) th particle is a sum of terms \( {\mathbf{F}}^{j, k}, j \neq k \), where \( {\mathbf{F}}^{j, k} \) depends only \( {\mathbf{x}}^{j} \) and \( {\mathbf{x}}^{k} \) . The relevant symmetry property is that \( {\mathbf{F}}^{j, k}\left( {{\mathbf{x}}^{j},{\mathbf{x}}^{k}}\right) = - {\mathbf{F}}^{k, j}\left( {{\mathbf{x}}^{k},{\mathbf{x}}^{j}}\right) \) ; that is, the force exerted by the \( j \) th particle on the \( k \) th particle is the negative (i.e.,"equal and opposite") of the force exerted by the \( k \) th particle on the \( j \) th particle. If the forces are assumed also to be conservative, then the potential energy of the system will be of the form \[ V\left( {{\mathbf{x}}^{1},{\mathbf{x}}^{2},\ldots ,{\mathbf{x}}^{N}}\right) = \mathop{\sum }\limits_{{j < k}}{V}^{j, k}\left( {{\mathbf{x}}^{j} - {\mathbf{x}}^{k}}\right) . \] (2.13) One important consequence of Newton's third law is conservation of the total momentum of the system. Proposition 2.12 Suppose that for each \( j \), the force on the \( j \) th particle is of the form \[ {\mathbf{F}}^{j}\left( {{\mathbf{x}}^{1},{\mathbf{x}}^{2},\ldots ,{\mathbf{x}}^{N}}\right) = \mathop{\sum }\limits_{{k \neq j}}{\mathbf{F}}^{j, k}\left( {{\mathbf{x}}^{j},{\mathbf{x}}^{k}}\right) , \] for certain functions \( {\mathbf{F}}^{j, k} \) . Suppose also that we have the "equal and opposite” condition \[ {\mathbf{F}}^{j, k}\left( {{\mathbf{x}}^{j},{\mathbf{x}}^{k}}\right) = - {\mathbf{F}}^{k, j}\left( {{\mathbf{x}}^{j},{\mathbf{x}}^{k}}\right) . \] Then the total momentum of the system is conserved. Note that since the rate of change of \( {\mathbf{p}}^{j} \) is \( {\mathbf{F}}^{j} \), the force on the \( j \) th particle, the momentum of each individual particle is not constant in time, except in the trivial case of a noninteracting system (one in which all forces are zero). Proof. Differentiating gives \[ \frac{d\mathbf{p}}{dt} = \mathop{\sum }\limits_{{j = 1}}^{N}\frac{d{\mathbf{p}}^{j}}{dt} = \mathop{\sum }\limits_{{j = 1}}^{N}{\mathbf{F}}^{j} = \mathop{\sum }\limits_{j}\mathop{\sum }\limits_{{k \neq j}}{\mathbf{F}}^{j, k}\left( {{\mathbf{x}}^{j},{\mathbf{x}}^{k}}\right) . \] By the equal and opposite condition, \( {\mathbf{F}}^{j, k}\left( {{\mathbf{x}}^{j},{\mathbf{x}}^{k}}\right) \) cancels with \( {\mathbf{F}}^{k, j}\left( {{\mathbf{x}}^{j},{\mathbf{x}}^{k}}\right) \) , so \( d\mathbf{p}/{dt} = 0 \) . ∎ Let us consider, now, a more general situation in which we have conservative forces, but not necessarily of the "two-particle" form. It is still possible to have conservation of momentum, as the following result shows. Proposition 2.13 If a multiparticle system has a force l

Proposition 2.7 Suppose \( U \) is a simply connected domain in \( {\mathbb{R}}^{n} \) and \( \mathbf{F} \) is a smooth, \( {\mathbb{R}}^{n} \) -valued function on \( U \) . Then \( \mathbf{F} \) is conservative if and only if \( \mathbf{F} \) satisfies \[ \frac{\partial {F}_{j}}{\partial {x}_{k}} - \frac{\partial {F}_{k}}{\partial {x}_{j}} = 0 \] at each point in \( U \) .

Proof. If \( \mathbf{F} \) is conservative, then there exists a potential function \( V \) such that \( \mathbf{F} = \nabla V \). Therefore, we have: \[ \frac{\partial {F}_{j}}{\partial {x}_{k}} = \frac{\partial}{\partial {x}_{k}} \left( \frac{\partial V}{\partial {x}_{j}} \right) = \frac{{\partial }^{2}V}{\partial {x}_{k}\partial {x}_{j}} \] and \[ \frac{\partial {F}_{k}}{\partial {x}_{j}} = \frac{\partial}{\partial {x}_{j}} \left( \frac{\partial V}{\partial {x}_{k}} \right) = \frac{{\partial }^{2}V}{\partial {x}_{j}\partial {x}_{k}} \] Since the mixed partial derivatives of \( V \) are equal, we have: \[ \frac{\partial {F}_{j}}{\partial {x}_{k}} = \frac{\partial {F}_{k}}{\partial {x}_{j}} \] Thus, the condition \(\frac{\partial {F}_{j}}{\partial {x}_{k}} - \frac{\partial {F}_{k}}{\partial {x}_{j}} = 0\) holds at every point in \( U \). In the other direction, if \( \mathbf{F} \) satisfies the condition \(\frac{\partial {F}_{j}}{\partial {x}_{k}} - \frac{\partial {F}_{k}}{\partial {x}_{j}} = 0\), then we can construct a potential function \( V \) by integrating \( \mathbf{F} \) along paths and using the Stokes theorem to establish independence of choice of path. Specifically, for any two points \( A \) and \( B \) in \( U \), the line integral of \( \mathbf{F} \) over any path connecting \( A \) to \( B \) is path-independent due to the condition \(\frac{\partial {F}_{j}}{\partial {x}_{k}} - \frac{\partial {F}_{k}}{\partial {x}_{j}} = 0\). This allows us to define \( V(B) - V(A) \) as the line integral of \( \mathbf{F} \) from \( A \) to \( B \). Since this integral is path-independent, \( V \) is well-defined and differentiable, and its gradient is precisely \( \mathbf{F} \). Therefore, \( \mathbf{F} \) is conservative.

Theorem 3.2.2. There exist finite constants \( {C}_{n} \) and \( {C}_{n}^{\prime } \) such that the following statements are valid: (a) Given \( b \in {BMO}\left( {\mathbf{R}}^{n}\right) \), the linear functional \( {L}_{b} \) lies in \( {\left( {H}^{1}\left( {\mathbf{R}}^{n}\right) \right) }^{ * } \) and has norm at most \( {C}_{n}\parallel b{\parallel }_{BMO} \) . Moreover, the mapping \( b \mapsto {L}_{b} \) from BMO to \( {\left( {H}^{1}\right) }^{ * } \) is injective. (b) For every bounded linear functional \( L \) on \( {H}^{1} \) there exists a BMO function b such that for all \( f \in {H}_{0}^{1} \) we have \( L\left( f\right) = {L}_{b}\left( f\right) \) and also \[ \parallel b{\parallel }_{BMO} \leq {C}_{n}^{\prime }{\begin{Vmatrix}{L}_{b}\end{Vmatrix}}_{{H}^{1} \rightarrow \mathbf{C}} \] Proof. We have already proved that for all \( b \in {BMO}\left( {\mathbf{R}}^{n}\right) ,{L}_{b} \) lies in \( {\left( {H}^{1}\left( {\mathbf{R}}^{n}\right) \right) }^{ * } \) and has norm at most \( {C}_{n}\parallel b{\parallel }_{BMO} \) . The embedding \( b \mapsto {L}_{b} \) is injective as a consequence of Exercise 3.2.2. It remains to prove (b). Fix a bounded linear functional \( L \) on \( {H}^{1}\left( {\mathbf{R}}^{n}\right) \) and also fix a cube \( Q \) . Consider the space \( {L}^{2}\left( Q\right) \) of all square integrable functions supported in \( Q \) with norm \[ \parallel g{\parallel }_{{L}^{2}\left( Q\right) } = {\left( {\int }_{Q}{\left| g\left( x\right) \right| }^{2}dx\right) }^{\frac{1}{2}}. \] We denote by \( {L}_{0}^{2}\left( Q\right) \) the closed subspace of \( {L}^{2}\left( Q\right) \) consisting of all functions in \( {L}^{2}\left( Q\right) \) with mean value zero. We show that every element in \( {L}_{0}^{2}\left( Q\right) \) is in \( {H}^{1}\left( {\mathbf{R}}^{n}\right) \) and we have the inequality \[ \parallel g{\parallel }_{{H}^{1}} \leq {c}_{n}{\left| Q\right| }^{\frac{1}{2}}\parallel g{\parallel }_{{L}^{2}} \] (3.2.4) To prove (3.2.4) we use the square function characterization of \( {H}^{1} \) . We fix a Schwartz function \( \Psi \) on \( {\mathbf{R}}^{n} \) whose Fourier transform is supported in the annulus \( \frac{1}{2} \leq \left| \xi \right| \leq 2 \) and that satisfies (1.3.6) for all \( \xi \neq 0 \) and we let \( {\Delta }_{j}\left( g\right) = {\Psi }_{{2}^{-j}} * g \) . To estimate the \( {L}^{1} \) norm of \( {\left( \mathop{\sum }\limits_{j}{\left| {\Delta }_{j}\left( g\right) \right| }^{2}\right) }^{1/2} \) over \( {\mathbf{R}}^{n} \), consider the part of the integral over \( 3\sqrt{n}Q \) and the integral over \( {\left( 3\sqrt{n}Q\right) }^{c} \) . First we use Hölder’s inequality and an \( {L}^{2} \) estimate to prove that \[ {\int }_{3\sqrt{n}Q}{\left( \mathop{\sum }\limits_{j}{\left| {\Delta }_{j}\left( g\right) \left( x\right) \right| }^{2}\right) }^{\frac{1}{2}}{dx} \leq {c}_{n}{\left| Q\right| }^{\frac{1}{2}}\parallel g{\parallel }_{{L}^{2}}. \] Now for \( x \notin 3\sqrt{n}Q \) we use the mean value property of \( g \) to obtain \[ \left| {{\Delta }_{j}\left( g\right) \left( x\right) }\right| \leq \frac{{c}_{n}\parallel g{\parallel }_{{L}^{2}}{2}^{{nj} + j}{\left| Q\right| }^{\frac{1}{n} + \frac{1}{2}}}{{\left( 1 + {2}^{j}\left| x - {c}_{Q}\right| \right) }^{n + 2}}, \] (3.2.5) where \( {c}_{Q} \) is the center of \( Q \) . Estimate (3.2.5) is obtained in a way similar to that we obtained the corresponding estimate for one atom; see Theorem 2.3.11 for details. Now (3.2.5) implies that \[ {\int }_{{\left( 3\sqrt{n}Q\right) }^{c}}{\left( \mathop{\sum }\limits_{j}{\left| {\Delta }_{j}\left( g\right) \left( x\right) \right| }^{2}\right) }^{\frac{1}{2}}{dx} \leq {c}_{n}{\left| Q\right| }^{\frac{1}{2}}\parallel g{\parallel }_{{L}^{2}}, \] which proves (3.2.4). Since \( {L}_{0}^{2}\left( Q\right) \) is a subspace of \( {H}^{1} \), it follows from (3.2.4) that the linear functional \( L : {H}^{1} \rightarrow \mathbf{C} \) is also a bounded linear functional on \( {L}_{0}^{2}\left( Q\right) \) with norm \[ \parallel L{\parallel }_{{L}_{0}^{2}\left( Q\right) \rightarrow \mathbf{C}} \leq {c}_{n}{\left| Q\right| }^{1/2}\parallel L{\parallel }_{{H}^{1} \rightarrow \mathbf{C}} \] (3.2.6) By the Riesz representation theorem for the Hilbert space \( {L}_{0}^{2}\left( Q\right) \), there is an element \( {F}^{Q} \) in \( {\left( {L}_{0}^{2}\left( Q\right) \right) }^{ * } = {L}^{2}\left( Q\right) /\{ \) constants \( \} \) such that \[ L\left( g\right) = {\int }_{Q}{F}^{Q}\left( x\right) g\left( x\right) {dx} \] (3.2.7) for all \( g \in {L}_{0}^{2}\left( Q\right) \), and this \( {F}^{Q} \) satisfies \[ {\begin{Vmatrix}{F}^{Q}\end{Vmatrix}}_{{L}^{2}\left( Q\right) } \leq \parallel L{\parallel }_{{L}_{0}^{2}\left( Q\right) \rightarrow \mathbf{C}} \] (3.2.8) Thus for any cube \( Q \) in \( {\mathbf{R}}^{n} \), there is square integrable function \( {F}^{Q} \) supported in \( Q \) such that (3.2.7) is satisfied. We observe that if a cube \( Q \) is contained in another cube \( {Q}^{\prime } \), then \( {F}^{Q} \) differs from \( {F}^{{Q}^{\prime }} \) by a constant on \( Q \) . Indeed, for all \( g \in {L}_{0}^{2}\left( Q\right) \) we have \[ {\int }_{Q}{F}^{{Q}^{\prime }}\left( x\right) g\left( x\right) {dx} = L\left( g\right) = {\int }_{Q}{F}^{Q}\left( x\right) g\left( x\right) {dx} \] and thus \[ {\int }_{Q}\left( {{F}^{{Q}^{\prime }}\left( x\right) - {F}^{Q}\left( x\right) }\right) g\left( x\right) {dx} = 0. \] Consequently, \[ g \rightarrow {\int }_{Q}\left( {{F}^{{Q}^{\prime }}\left( x\right) - {F}^{Q}\left( x\right) }\right) g\left( x\right) {dx} \] is the zero functional on \( {L}_{0}^{2}\left( Q\right) \) ; hence \( {F}^{{Q}^{\prime }} - {F}^{Q} \) must be the zero function in the space \( {\left( {L}_{0}^{2}\left( Q\right) \right) }^{ * } \), i.e., \( {F}^{{Q}^{\prime }} - {F}^{Q} \) is a constant on \( Q \) . Let \[ {Q}_{m} = {\left\lbrack -m/2, m/2\right\rbrack }^{n} \] for \( m = 1,2,\ldots \) . Then \( \left| {Q}_{1}\right| = 1 \) . We define a locally integrable function \( b\left( x\right) \) on \( {\mathbf{R}}^{n} \) by setting \[ b\left( x\right) = {F}^{{Q}_{m}}\left( x\right) - \frac{1}{\left| {Q}_{1}\right| }{\int }_{{Q}_{1}}{F}^{{Q}_{m}}\left( t\right) {dt} \] (3.2.9) whenever \( x \in {Q}_{m} \) . We check that this definition is unambiguous. Let \( 1 \leq \ell < m \) . Then for \( x \in {Q}_{\ell }, b\left( x\right) \) is also defined as in (3.2.9) with \( \ell \) in the place of \( m \) . The difference of these two functions is \[ {F}^{{Q}_{m}} - {F}^{{Q}_{\ell }} - \underset{{Q}_{1}}{\operatorname{Avg}\left( {{F}^{{Q}_{m}} - {F}^{{Q}_{\ell }}}\right) } = 0, \] since the function \( {F}^{{Q}_{m}} - {F}^{{Q}_{\ell }} \) is constant in the cube \( {Q}_{\ell } \) (which is contained in \( {Q}_{m} \) ), as indicated earlier. Next we claim that for any cube \( Q \) there is a constant \( {C}_{Q} \) such that \[ {F}^{Q} = b - {C}_{Q}\;\text{ on }Q. \] (3.2.10) Indeed, given a cube \( Q \) pick the smallest \( m \) such that \( Q \) is contained in \( {Q}^{m} \) and observe that \[ {F}^{Q} = \underset{\text{constant on }Q}{\underbrace{{F}^{Q} - {F}^{{Q}_{m}}}} + \underset{\begin{matrix} {Q}_{1} \\ b\left( x\right) \end{matrix}}{\underbrace{{F}^{{Q}_{m}} - \operatorname{Avg}{F}^{{Q}_{m}}}} + \underset{\begin{matrix} {Q}_{1} \\ \text{ constant on }Q \end{matrix}}{\underbrace{\operatorname{Avg}{F}^{{Q}_{m}}}} \] and let \( - {C}_{Q} \) be the sum of the two preceding constant expressions on \( Q \) . We have now found a locally integrable function \( b \) such that for all cubes \( Q \) and all \( g \in {L}_{0}^{2}\left( Q\right) \) we have \[ {\int }_{Q}b\left( x\right) g\left( x\right) {dx} = {\int }_{Q}\left( {{F}^{Q}\left( x\right) + {C}_{Q}}\right) g\left( x\right) {dx} = {\int }_{Q}{F}^{Q}\left( x\right) g\left( x\right) {dx} = L\left( g\right) , \] (3.2.11) as follows from (3.2.7) and (3.2.10). We conclude the proof by showing that \( b \) lies in \( \operatorname{BMO}\left( {\mathbf{R}}^{n}\right) \) . By (3.2.10),(3.2.8), and (3.2.6) we have \[ \mathop{\sup }\limits_{Q}\frac{1}{\left| Q\right| }{\int }_{Q}\left| {b\left( x\right) - {C}_{Q}}\right| {dx} = \mathop{\sup }\limits_{Q}\frac{1}{\left| Q\right| }{\int }_{Q}\left| {{F}^{Q}\left( x\right) }\right| {dx} \] \[ \leq \mathop{\sup }\limits_{Q}{\left| Q\right| }^{-1}{\left| Q\right| }^{\frac{1}{2}}{\begin{Vmatrix}{F}^{Q}\end{Vmatrix}}_{{L}^{2}\left( Q\right) } \] \[ \leq \mathop{\sup }\limits_{Q}{\left| Q\right| }^{-\frac{1}{2}}\parallel L{\parallel }_{{L}_{0}^{2}\left( Q\right) \rightarrow \mathbf{C}} \] \[ \leq {c}_{n}\parallel L{\parallel }_{{H}^{1} \rightarrow \mathbf{C}} < \infty . \] Using Proposition 3.1.2 (3), we deduce that \( b \in {BMO} \) and \( \parallel b{\parallel }_{BMO} \leq 2{c}_{n}\parallel L{\parallel }_{{H}^{1} \rightarrow \mathbf{C}} \) . Finally, (3.2.11) implies that \[ L\left( g\right) = {\int }_{{\mathbf{R}}^{n}}b\left( x\right) g\left( x\right) {dx} = {L}_{b}\left( g\right) \] for all \( g \in {H}_{0}^{1}\left( {\mathbf{R}}^{n}\right) \), proving that the linear functional \( L \) coincides with \( {L}_{b} \) on a dense subspace of \( {H}^{1} \) . Consequently, \( L = {L}_{b} \), and this concludes the proof of part (b). ## Exercises 3.2.1. Given \( b \) in \( {BMO} \), let \( {L}_{b} \) be as in Definition 3.2.1. Prove that for \( b \) in \( {BMO} \) we have \[ \parallel b{\parallel }_{BMO} \approx \mathop{\sup }\limits_{{\parallel f{\parallel }_{{H}^{1}} \leq 1}}\left| {{L}_{b}\left( f\right) }\right| \] and for a given \( f \) in \( {H}^{1} \) we have \[ \parallel f{\parallel }_{{H}^{1}} \approx \mathop{\sup }\limits_{{\parallel b{\parallel }_{BMO} \leq 1}}\left| {{L}_{b}\left( f\right) }\right| . \] [Hint: Use \( \parallel T{\parallel }_{{X}^{ * }} = \mathop{\sup }\limits_{\substack{{x \in X} \\ {\parallel x{\parallel }_{X} \leq 1} }}\left| {T\left( x\right) }\right| \) for all \( T \) in the dual of a Banach spa

(Theorem 3.2.2. There exist finite constants \( {C}_{n} \) and \( {C}_{n}^{\prime } \) such that the following statements are valid: (a) Given \( b \in {BMO}\left( {\mathbf{R}}^{n}\right) \), the linear functional \( {L}_{b} \) lies in \( {\left( {H}^{1}\left( {\mathbf{R}}^{n}\right) \right) }^{ * } \) and has norm at most \( {C}_{n}\parallel b{\parallel }_{BMO} \) . Moreover, the mapping \( b \mapsto {L}_{b} \) from BMO to \( {\left( {H}^{1}\right) }^{ * } \) is injective. (b) For every bounded linear functional \( L \) on \( {H}^{1} \) there exists a BMO function b such that for all \( f \in {H}_{0}^{1} \) we have \( L\left( f\right) = {L}_{b}\left( f\right) \) and also \[ \parallel b{\parallel }_{BMO} \leq {C}_{n}^{\prime }{\begin{Vmatrix}{L}_{b}\end{Vmatrix}}_{{H}^{1} \rightarrow \mathbf{C}} \] )

We have already proved that for all \( b \in {BMO}\left( {\mathbf{R}}^{n}\right) ,{L}_{b} \) lies in \( {\left( {H}^{1}\left( {\mathbf{R}}^{n}\right) \right) }^{ * } \) and has norm at most \( {C}_{n}\parallel b{\parallel }_{BMO} \) . The embedding \( b \mapsto {L}_{b} \) is injective as a consequence of Exercise 3.2.2. It remains to prove (b). Fix a bounded linear functional \( L \) on \( {H}^{1}\left( {\mathbf{R}}^{n}\right) \) and also fix a cube \( Q \) . Consider the space \( {L}^{2}\left( Q\right) \) of all square integrable functions supported in \( Q \) with norm \[ \parallel g{\parallel }_{{L}^{2}\left( Q\right) } = {\left( {\int }_{Q}{\left| g\left( x\right) \right| }^{2}dx\right) }^{\frac{1}{2}}. \] We denote by \( {L}_{0}^{2}\left( Q\right) \) the closed subspace of \( {L}^{2}\left( Q\right) \) consisting of all functions in \( {L}^{2}\left( Q\right) \) with mean value zero. We show that every element in \( {L}_{0}^{2}\left( Q\right) \) is in \( {H}^{1}\left( {\mathbf{R}}^{n}\right) \) and we have the inequality \[ \parallel g{\parallel }_{{H}^{1}} \leq {c}_{n}{\left| Q\right| }^{\frac{1}{2}}\parallel g{\parallel }_{{L}^{2}} \] To prove this, we use the square function characterization of \( {H}^{1} \). We fix a Schwartz function \( \Psi \) on \( {\mathbf{R}}^{n} \) whose Fourier transform is supported in the annulus \( \frac{1}{2} \leq \left| \xi \right| \leq 2 \) and that satisfies (1.3.6) for all \( \xi \neq 0 \) and we let \( {\Delta }_{j}\left( g\right) = {\Psi }_{{2}^{-j}} * g \) . To estimate the \( {L}^{1} \) norm of \( {\left( \mathop{\sum }\limits_{j}{\left| {\Delta }_{j}\left( g\right) \right| }^{2}\right) }^{1/2} \) over \( {\mathbf{R}}^{n} \), consider the part of the integral over \( 3\sqrt{n}Q \) and the integral over \( {\left( 3\sqrt{n}Q\right) }^{c} \) . First we use Hölder’s inequality and an \( {L}^{2} \) estimate to prove that \[ {\int }_{3\sqrt{n}Q}{\left( \mathop{\sum }\limits_{j}{\left| {\Delta }_{j}\left( g\right) \left( x\right) \right| }^{2}\right) }^{\frac{1}{2}}{dx} \leq {c}_{n}{\left| Q\right| }^{\frac{1}{2}}\parallel g{\parallel }_{{L}^{2}}. \] Now for \( x \notin 3\sqrt{n}Q \) we use the mean value property of \( g \) to obtain \[ \left| {{\Delta }_{j}\left( g\

Theorem 3.3.4. Suppose that \( X \) is a Banach space with an unconditional basis. If \( X \) is not reflexive, then either \( {c}_{0} \) is complemented in \( X \), or \( {\ell }_{1} \) is complemented in \( X \) (or both). In either case, \( {X}^{* * } \) is nonseparable. Proof. The first statement of the theorem follows immediately from Theorem 3.2.19, Theorem 3.3.1, and Theorem 3.3.2. Now, for the latter statement, if \( {c}_{0} \) were complemented in \( X \), then \( {X}^{* * } \) would contain a (complemented) copy \( {\ell }_{\infty } \) . If \( {\ell }_{1} \) were complemented in \( X \), then \( {X}^{ * } \) would be nonseparable, since it would contain a (complemented) copy of \( {\ell }_{\infty } \) . In either case, \( {X}^{* * } \) is nonseparable. ## 3.4 The James Space \( \mathcal{J} \) Continuing with the classic paper of James [126], we come to his construction of one of the most important examples in Banach space theory. This space, nowadays known as the James space, is, in fact, quite a natural space, consisting of sequences of bounded 2-variation. The James space will provide an example of a Banach space with a basis but with no unconditional basis; it also answered several other open questions at the time. For example, it was not known whether a Banach space \( X \) was necessarily reflexive if its bidual was separable. The James space \( \mathcal{J} \) is separable and has codimension one in \( {\mathcal{J}}^{* * } \), and so gives a counterexample. Later, James [127] went further and modified the definition of the norm to make \( \mathcal{J} \) isometric to \( {\mathcal{J}}^{* * } \) , thus showing that a Banach space can be isometrically isomorphic to its bidual yet fail to be reflexive! Let us define \( \widetilde{\mathcal{J}} \) to be the space of all sequences \( \xi = {\left( \xi \left( n\right) \right) }_{n = 1}^{\infty } \) of real numbers with finite square variation; that is, \( \xi \in \widetilde{\mathcal{J}} \) if and only if there is a constant \( M \) such that for every choice of integers \( {\left( {p}_{j}\right) }_{j = 0}^{n} \) with \( 1 \leq {p}_{0} < {p}_{1} < \cdots < {p}_{n} \) we have \[ \mathop{\sum }\limits_{{j = 1}}^{n}{\left( \xi \left( {p}_{j}\right) - \xi \left( {p}_{j - 1}\right) \right) }^{2} \leq {M}^{2} \] It is easy to verify that if \( \xi \in \widetilde{\mathcal{J}} \), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\xi \left( n\right) \) exists. We then define \( \mathcal{J} \) as the subspace of \( \widetilde{\mathcal{J}} \) of all \( \xi \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\xi \left( n\right) = 0 \) . Definition 3.4.1. The James space \( \mathcal{J} \) is the (real) Banach space of all sequences \( \xi = {\left( \xi \left( n\right) \right) }_{n = 1}^{\infty } \in \widetilde{\mathcal{J}} \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\xi \left( n\right) = 0 \), endowed with the norm \[ \parallel \xi {\parallel }_{\mathcal{J}} = \frac{1}{\sqrt{2}}\sup \left\{ {\left( {\left( \xi \left( {p}_{n}\right) - \xi \left( {p}_{0}\right) \right) }^{2} + \mathop{\sum }\limits_{{k = 1}}^{n}{\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2}\right) }^{1/2}\right\} , \] where the supremum is taken over all \( n \in \mathbb{N} \) and all choices of integers \( {\left( {p}_{j}\right) }_{j = 0}^{n} \) with \( 1 \leq {p}_{0} < {p}_{1} < \cdots < {p}_{n} \) The definition of the norm in the James space is not quite natural; clearly, the norm is equivalent to the alternative norm given by the formula \[ \parallel \xi {\parallel }_{0} = \sup \left\{ {\left( \mathop{\sum }\limits_{{k = 1}}^{n}{\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2}\right) }^{1/2}\right\} , \] where again, the supremum is taken over all sequences of integers \( {\left( {p}_{j}\right) }_{j = 0}^{n} \) with \( 1 \leq \) \( {p}_{0} < {p}_{1} < \cdots < {p}_{n} \) . In fact, \[ \frac{1}{\sqrt{2}}\parallel \xi {\parallel }_{0} \leq \parallel \xi {\parallel }_{\mathcal{J}} \leq \sqrt{2}\parallel \xi {\parallel }_{0},\;\xi \in \mathcal{J}. \] Notice that \( {\begin{Vmatrix}{e}_{n}\end{Vmatrix}}_{\mathcal{J}} = 1 \) for all \( n \), but \( {\begin{Vmatrix}{e}_{n}\end{Vmatrix}}_{0} = \sqrt{2} \) for \( n \geq 2 \) . We also note that \( \parallel \cdot {\parallel }_{\mathcal{J}} \) can be canonically extended to \( \widetilde{\mathcal{J}} \) by \[ \parallel \xi {\parallel }_{\mathcal{J}} = \frac{1}{\sqrt{2}}\sup \left\{ {\left( {\left( \xi \left( {p}_{n}\right) - \xi \left( {p}_{0}\right) \right) }^{2} + \mathop{\sum }\limits_{{k = 1}}^{n}{\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2}\right) }^{1/2}\right\} , \] but this defines only a seminorm on \( \widetilde{\mathcal{J}} \) vanishing on all constant sequences. Proposition 3.4.2. The sequence \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) of standard unit vectors is a monotone basis for \( \mathcal{J} \) in both norms \( \parallel \cdot {\parallel }_{\mathcal{J}} \) and \( \parallel \cdot {\parallel }_{0} \) . Proof. We will leave for the reader the verification that \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a monotone basic sequence in both norms. To prove that it is a basis, we need only consider the norm \( \parallel \cdot {\parallel }_{0} \) . Suppose \( \xi \in \mathcal{J} \) . For each \( N \) let \[ {\xi }_{N} = \xi - \mathop{\sum }\limits_{{j = 1}}^{N}\xi \left( j\right) {e}_{j} \] Given \( \epsilon > 0 \), pick \( 1 \leq {p}_{0} < {p}_{1} < \cdots < {p}_{n} \) for which \( \mathop{\sum }\limits_{{j = 1}}^{n}{\left( \xi \left( {p}_{j}\right) - \xi \left( {p}_{j - 1}\right) \right) }^{2} > \) \( \parallel \xi {\parallel }_{0}^{2} - {\epsilon }^{2} \) . In order to estimate the norm of \( {\xi }_{N} \) when \( N > {p}_{n} \), it is enough to consider positive integers \( {q}_{0} \leq {q}_{1} < {q}_{2} < \cdots < {q}_{m} \), where \( N \leq {q}_{0} \) . Then for the partition \( 1 \leq {p}_{0} < {p}_{1} < \cdots < {p}_{n} < {q}_{0} < {q}_{2} < \cdots < {q}_{m} \) we have \[ \parallel \xi {\parallel }_{0}^{2} \geq \mathop{\sum }\limits_{{j = 1}}^{n}{\left( \xi \left( {p}_{j}\right) - \xi \left( {p}_{j - 1}\right) \right) }^{2} + {\left( \xi \left( {q}_{0}\right) - \xi \left( {p}_{n}\right) \right) }^{2} + \mathop{\sum }\limits_{{j = 1}}^{m}{\left( \xi \left( {q}_{j}\right) - \xi \left( {q}_{j - 1}\right) \right) }^{2} \] \[ \geq \mathop{\sum }\limits_{{j = 1}}^{n}{\left( \xi \left( {p}_{j}\right) - \xi \left( {p}_{j - 1}\right) \right) }^{2} + \mathop{\sum }\limits_{{j = 1}}^{m}{\left( \xi \left( {q}_{j}\right) - \xi \left( {q}_{j - 1}\right) \right) }^{2}. \] Hence \[ \mathop{\sum }\limits_{{j = 1}}^{m}{\left( \xi \left( {q}_{j}\right) - \xi \left( {q}_{j - 1}\right) \right) }^{2} \leq {\epsilon }^{2} \] Thus \( {\begin{Vmatrix}{\xi }_{N}\end{Vmatrix}}_{0} < \epsilon \) for \( N > {p}_{n} \) . Proposition 3.4.3. Let \( {\left( {\eta }_{k}\right) }_{k = 1}^{\infty } \) be a normalized block basic sequence with respect to \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in \( \left( {\mathcal{J},\parallel \cdot {\parallel }_{0}}\right) \) . Then for every sequence of scalars \( {\left( {\lambda }_{k}\right) }_{k = 1}^{n} \) , \[ {\begin{Vmatrix}\mathop{\sum }\limits_{{k = 1}}^{n}{\lambda }_{k}{\eta }_{k}\end{Vmatrix}}_{0} \leq \sqrt{5}{\left( \mathop{\sum }\limits_{{k = 1}}^{n}{\lambda }_{k}^{2}\right) }^{1/2}. \] Proof. For each \( k \) let \[ {\eta }_{k} = \mathop{\sum }\limits_{{j = {q}_{k - 1} + 1}}^{{q}_{k}}{\eta }_{k}\left( j\right) {e}_{j} \] where \( 0 = {q}_{0} < {q}_{1} < \cdots \), and put \[ \xi = \mathop{\sum }\limits_{{k = 1}}^{n}{\lambda }_{k}{\eta }_{k} \] Suppose \( 1 \leq {p}_{0} < {p}_{1} < \cdots < {p}_{m} \) . Fix \( i \leq n \) . Let \( {A}_{i} \) be the set of \( k \) such that \( {q}_{i - 1} < {p}_{k - 1} < {p}_{k} \leq {q}_{i} \) . If \( k \in {A}_{i} \), then \[ \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) = {\lambda }_{i}\left( {{\eta }_{i}\left( {p}_{k}\right) - {\eta }_{i}\left( {p}_{k - 1}\right) }\right) . \] Hence \[ \mathop{\sum }\limits_{{k \in {A}_{i}}}{\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2} \leq {\lambda }_{i}^{2} \] If \( A = { \cup }_{i}{A}_{i} \), we thus have \[ \mathop{\sum }\limits_{{k \in A}}{\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2} \leq \mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}^{2} \] Let \( B \) be the set of \( 1 \leq k \leq m \) with \( k \notin A \) . For each such \( k \) there exist \( i = i\left( k\right), j = j\left( k\right) \) such that \( {q}_{i - 1} < {p}_{k - 1} \leq {q}_{i} \) and \( {q}_{j - 1} < {p}_{k} \leq {q}_{j} \) . Then, \[ {\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2} = {\left( {\lambda }_{j}{\eta }_{j}\left( {p}_{k}\right) - {\lambda }_{i}{\eta }_{i}\left( {p}_{k - 1}\right) \right) }^{2} \] \[ \leq 2\left( {{\lambda }_{j}^{2}{\eta }_{j}{\left( {p}_{k}\right) }^{2} + {\lambda }_{i}^{2}{\eta }_{j}{\left( {p}_{k - 1}\right) }^{2}}\right) \] \[ \leq 2\left( {{\lambda }_{j}^{2} + {\lambda }_{i}^{2}}\right) \] Thus, \[ \mathop{\sum }\limits_{{k = 1}}^{m}{\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2} \leq \mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}^{2} + 2\mathop{\sum }\limits_{{k \in B}}{\lambda }_{i\left( k\right) }^{2} + 2\mathop{\sum }\limits_{{k \in B}}{\lambda }_{j\left( k\right) }^{2}. \] Since the \( i\left( k\right) \) ’s and similarly the \( j\left( k\right) \) ’s are distinct for \( k \in B \), it follows that \[ \mathop{\sum }\limits_{{k = 1}}^{m}{\left( \xi \left( {p}_{k}\right) - \xi \left( {p}_{k - 1}\right) \right) }^{2} \leq 5\mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}^{2} \] and this completes the proof. Proposition 3.4.4. The sequence \( {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a shrink

Theorem 3.3.4. Suppose that \( X \) is a Banach space with an unconditional basis. If \( X \) is not reflexive, then either \( {c}_{0} \) is complemented in \( X \), or \( {\ell }_{1} \) is complemented in \( X \) (or both). In either case, \( {X}^{* * } \) is nonseparable.

The first statement of the theorem follows immediately from Theorem 3.2.19, Theorem 3.3.1, and Theorem 3.3.2. Now, for the latter statement, if \( {c}_{0} \) were complemented in \( X \), then \( {X}^{* * } \) would contain a (complemented) copy \( {\ell }_{\infty } \). If \( {\ell }_{1} \) were complemented in \( X \), then \( {X}^{ * } \) would be nonseparable, since it would contain a (complemented) copy of \( {\ell }_{\infty } \). In either case, \( {X}^{* * } \) is nonseparable.

Proposition 19. Let \( R \) be a commutative ring with 1 . (1) Prime ideals are primary. (2) The ideal \( Q \) is primary if and only if every zero divisor in \( R/Q \) is nilpotent. (3) If \( Q \) is primary then rad \( Q \) is a prime ideal, and is the unique smallest prime ideal containing \( Q \) . (4) If \( Q \) is an ideal whose radical is a maximal ideal, then \( Q \) is a primary ideal. (5) Suppose \( M \) is a maximal ideal and \( Q \) is an ideal with \( {M}^{n} \subseteq Q \subseteq M \) for some \( n \geq 1 \) . Then \( Q \) is a primary ideal with \( \operatorname{rad}Q = M \) . Proof: The first two statements are immediate from the definition of a primary ideal. For (3), suppose \( {ab} \in \operatorname{rad}Q \) . Then \( {a}^{m}{b}^{m} = {\left( ab\right) }^{m} \in Q \), and since \( Q \) is primary, either \( {a}^{m} \in Q \), in which case \( a \in \operatorname{rad}Q \), or \( {\left( {b}^{m}\right) }^{n} \in Q \) for some positive integer \( n \), in which case \( b \in \operatorname{rad}Q \) . This proves that \( \operatorname{rad}Q \) is a prime ideal, and it follows that \( \operatorname{rad}Q \) is the smallest prime ideal containing \( Q \) (Proposition 12). To prove (4) we pass to the quotient ring \( R/Q \) ; by (2), it suffices to show that every zero divisor in this quotient ring is nilpotent. We are reduced to the situation where \( Q = \left( 0\right) \) and \( M = \operatorname{rad}Q = \operatorname{rad}\left( 0\right) \), which is the nilradical, is a maximal ideal. Since the nilradical is contained in every prime ideal (Proposition 12), it follows that \( M \) is the unique prime ideal, so also the unique maximal ideal. If \( d \) were a zero divisor, then the ideal \( \left( d\right) \) would be a proper ideal, hence contained in a maximal ideal. This implies that \( d \in M \), hence every zero divisor is indeed nilpotent. Finally, suppose \( {M}^{n} \subseteq Q \subseteq M \) for some \( n \geq 1 \) where \( M \) is a maximal ideal. Then \( Q \subseteq M \) so rad \( Q \subseteq \operatorname{rad}M = M \) . Conversely, \( {M}^{n} \subseteq Q \) shows that \( M \subseteq \operatorname{rad}Q \), so \( \operatorname{rad}Q = M \) is a maximal ideal, and \( Q \) is primary by (4). Definition. If \( Q \) is a primary ideal, then the prime ideal \( P = \operatorname{rad}Q \) is called the associated prime to \( Q \), and \( Q \) is said to belong to \( P \) (or to be \( P \) -primary). It is easy to check that a finite intersection of \( P \) -primary ideals is again a \( P \) -primary ideal (cf. the exercises). ## Examples (1) The primary ideals in \( \mathbb{Z} \) are 0 and the ideals \( \left( {p}^{m}\right) \) for \( p \) a prime and \( m \geq 1 \) . (2) For any field \( k \), the ideal \( \left( x\right) \) in \( k\left\lbrack {x, y}\right\rbrack \) is primary since it is a prime ideal. For any \( n \geq 1 \), the ideal \( {\left( x, y\right) }^{n} \) is primary since it is a power of the maximal ideal \( \left( {x, y}\right) \) . (3) The ideal \( Q = \left( {{x}^{2}, y}\right) \) in the polynomial ring \( k\left\lbrack {x, y}\right\rbrack \) is primary since we have \( {\left( x, y\right) }^{2} \subseteq \left( {{x}^{2}, y}\right) \subseteq \left( {x, y}\right) \) . Similarly, \( {Q}^{\prime } = \left( {4, x}\right) \) in \( \mathbb{Z}\left\lbrack x\right\rbrack \) is a \( \left( {2, x}\right) \) -primary ideal. (4) Primary ideals need not be powers of prime ideals. For example, the primary ideal \( Q \) in the previous example is not the power of a prime ideal, as follows. If \( \left( {{x}^{2}, y}\right) = {P}^{k} \) for some prime ideal \( P \) and some \( k \geq 1 \), then \( {x}^{2}, y \in {P}^{k} \subseteq P \) so \( x, y \in P \) . Then \( P = \left( {x, y}\right) \), and since \( y \notin {\left( x, y\right) }^{2} \), it would follow that \( k = 1 \) and \( Q = \left( {x, y}\right) \) . Since \( x \notin \left( {{x}^{2}, y}\right) \), this is impossible. (5) If \( R \) is Noetherian, and \( Q \) is a primary ideal belonging to the prime ideal \( P \), then \[ {P}^{m} \subseteq Q \subseteq P \] for some \( m \geq 1 \) by Proposition 14. If \( P \) is a maximal ideal, then the last statement in Proposition 19 shows that the converse also holds. This is not necessarily true if \( P \) is a prime ideal that is not maximal. For example, consider the ideal \( I = \left( {{x}^{2},{xy}}\right) \) in \( k\left\lbrack {x, y}\right\rbrack \) . Then \( \left( {x}^{2}\right) \subset I \subset \left( x\right) \), and \( \left( x\right) \) is a prime ideal, but \( I \) is not primary: \( {xy} \in I \) and \( x \notin I \), but no positive power of \( y \) is an element of \( I \) . This example also shows that an ideal whose radical is prime (but not maximal as in (4) of the proposition) is not necessarily primary. (6) Powers of prime ideals need not be primary. For example, consider the quotient ring \( R = \mathbb{R}\left\lbrack {x, y, z}\right\rbrack /\left( {{xy} - {z}^{2}}\right) \), the coordinate ring of the cone \( {z}^{2} = {xy} \) in \( {\mathbb{R}}^{3} \), and let \( P = \left( {\bar{x},\bar{z}}\right) \) be the ideal generated by \( \bar{x} \) and \( \bar{z} \) in \( R \) . This is a prime ideal in \( R \) since the quotient is \( R/\left( {\bar{x},\bar{z}}\right) \cong \mathbb{R}\left\lbrack {x, y, z}\right\rbrack /\left( {x, z}\right) \cong \mathbb{R}\left\lbrack y\right\rbrack \) (because \( \left( {{xy} - {z}^{2}}\right) \subset \left( {x, z}\right) \) ). The ideal \[ {P}^{2} = \left( {{\bar{x}}^{2},\bar{x}\bar{z},{\bar{z}}^{2}}\right) = \left( {{\bar{x}}^{2},\bar{x}\bar{z},\bar{x}\bar{y}}\right) = \bar{x}\left( {\bar{x},\bar{y},\bar{z}}\right) , \] however, is not primary: \( \bar{x}\bar{y} = {\bar{z}}^{2} \in {P}^{2} \), but \( \bar{x} \notin {P}^{2} \), and no power of \( \bar{y} \) is in \( {P}^{2} \) . Note that \( {P}^{2} \) is another example of an ideal that is not primary whose radical is prime. (7) Suppose \( R \) is a U.F.D. If \( \pi \) is an irreducible element of \( R \) then it is easy to see that the powers \( \left( {\pi }^{n}\right) \) for \( n = 1,2,\ldots \) are \( \left( \pi \right) \) -primary ideals. Conversely, suppose \( Q \) is a \( \left( \pi \right) \) -primary ideal, and let \( n \) be the largest integer with \( Q \subseteq \left( {\pi }^{n}\right) \) (such an integer exists since, for example, \( {\pi }^{k} \in Q \) for some \( k \geq 1 \), so \( n \leq k \) ). If \( q \) is an element of \( Q \) not contained in \( \left( {\pi }^{n + 1}\right) \), then \( q = r{\pi }^{n} \) for some \( r \in R \) and \( r \notin \left( \pi \right) \) . Since \( r \notin \left( \pi \right) \) and \( Q \) is \( \left( \pi \right) \) -primary, it follows that \( {\pi }^{n} \in Q \) . This shows that \( Q = \left( {\pi }^{n}\right) \) . In the examples above, the ideal \( \left( {{x}^{2},{xy}}\right) \) in \( k\left\lbrack {x, y}\right\rbrack \) is not a primary ideal, but it can be written as the intersection of primary ideals: \( \left( {{x}^{2},{xy}}\right) = \left( x\right) \cap {\left( x, y\right) }^{2} \) . ## Definition. (1) An ideal \( I \) in \( R \) has a primary decomposition if it may be written as a finite intersection of primary ideals: \[ I = \mathop{\bigcap }\limits_{{i = 1}}^{m}{Q}_{i}\;{Q}_{i}\text{ a primary ideal. } \] (2) The primary decomposition above is minimal and the \( {Q}_{i} \) are called the primary components of \( I \) if (a) no primary ideal contains the intersection of the remaining primary ideals, i.e., \( {Q}_{i} \nsupseteq { \cap }_{j \neq i}{Q}_{j} \) for all \( i \), and (b) the associated prime ideals are all distinct: \( \operatorname{rad}{Q}_{i} \neq \operatorname{rad}{Q}_{j} \) for \( i \neq j \) . We now prove that in a Noetherian ring every proper ideal has a minimal primary decomposition. This result is often called the Lasker-Noether Decomposition Theorem, since it was first proved for polynomial rings by the chess master Emanuel Lasker and the proof was later greatly simplified and generalized by Emmy Noether. Definition. A proper ideal \( I \) in the commutative ring \( R \) is said to be irreducible if \( I \) cannot be written nontrivially as the intersection of two other ideals, i.e., if \( I = J \cap K \) with ideals \( J, K \) implies that \( I = J \) or \( I = K \) . It is easy to see that a prime ideal is irreducible (see Exercise 11 in Section 7.4). The ideal \( {\left( x, y\right) }^{2} \) in \( k\left\lbrack {x, y}\right\rbrack \) in Example 2 earlier shows that primary ideals need not be irreducible since it is the intersection of the ideals \( \left( x\right) + {\left( x, y\right) }^{2} = \left( {x,{y}^{2}}\right) \) and \( \left( y\right) + {\left( x, y\right) }^{2} = \left( {y,{x}^{2}}\right) \) . In a Noetherian ring, however, irreducible ideals are necessarily primary: Proposition 20. Let \( R \) be a Noetherian ring. Then (1) every irreducible ideal is primary, and (2) every proper ideal in \( R \) is a finite intersection of irreducible ideals. Proof: To prove (1) let \( Q \) be an irreducible ideal and suppose that \( {ab} \in Q \) and \( b \notin Q \) . It is easy to check that for any fixed \( n \) the set of elements \( x \in R \) with \( {a}^{n}x \in Q \) is an ideal, \( {A}_{n} \), in \( R \) . Clearly \( {A}_{1} \subseteq {A}_{2} \subseteq \ldots \) and since \( R \) is Noetherian this ascending chain of ideals must stabilize, i.e., \( {A}_{n} = {A}_{n + 1} = \ldots \) for some \( n > 0 \) . Consider the two ideals \( I = \left( {a}^{n}\right) + Q \) and \( J = \left( b\right) + Q \) of \( R \), each containing \( Q \) . If \( y \in I \cap J \) then \( y = {a}^{n}z + q \) for some \( z \in R \) and \( q \in Q \) . Since \( {ab} \in Q \), it follows that \( {aJ} \subseteq Q \), and in particular \( {ay} \in Q \) . Then \( {a}^{n + 1}z = {ay} - {aq} \in Q \), so \( z \in {A}_{n + 1} = {A}_{n} \) . But \( z \in {A}_{n} \) means that \( {a}^{n}z \i

Proposition 19. Let \( R \) be a commutative ring with 1 . (1) Prime ideals are primary. (2) The ideal \( Q \) is primary if and only if every zero divisor in \( R/Q \) is nilpotent. (3) If \( Q \) is primary then rad \( Q \) is a prime ideal, and is the unique smallest prime ideal containing \( Q \) . (4) If \( Q \) is an ideal whose radical is a maximal ideal, then \( Q \) is a primary ideal. (5) Suppose \( M \) is a maximal ideal and \( Q \) is an ideal with \( {M}^{n} \subseteq Q \subseteq M \) for some \( n \geq 1 \) . Then \( Q \) is a primary ideal with \( \operatorname{rad}Q = M \) .

The first two statements are immediate from the definition of a primary ideal. For (3), suppose \( {ab} \in \operatorname{rad}Q \) . Then \( {a}^{m}{b}^{m} = {\left( ab\right) }^{m} \in Q \), and since \( Q \) is primary, either \( {a}^{m} \in Q \), in which case \( a \in \operatorname{rad}Q \), or \( {\left( {b}^{m}\right) }^{n} \in Q \) for some positive integer \( n \), in which case \( b \in \operatorname{rad}Q \) . This proves that \( \operatorname{rad}Q \) is a prime ideal, and it follows that \( \operatorname{rad}Q \) is the smallest prime ideal containing \( Q \) (Proposition 12). To prove (4) we pass to the quotient ring \( R/Q \) ; by (2), it suffices to show that every zero divisor in this quotient ring is nilpotent. We are reduced to the situation where \( Q = \left( 0\right) \) and \( M = \operatorname{rad}Q = \operatorname{rad}\left( 0\right) \), which is the nilradical, is a maximal ideal. Since the nilradical is contained in every prime ideal (Proposition 12), it follows that \( M \) is the unique prime ideal, so also the unique maximal ideal. If \( d \) were a zero divisor, then the ideal \( \left( d\right) \) would be a proper ideal, hence contained in a maximal ideal. This implies that \( d \in M \), hence every zero divisor is indeed nilpotent. Finally, suppose \( {M}^{n} \subseteq Q \subseteq M \) for some \( n \geq 1 \) where \( M \) is a maximal ideal. Then \( Q \subseteq M \) so rad \( Q \subseteq \operatorname{rad}M = M \) . Conversely, \( {M}^{n} \subseteq Q \) shows that \( M \subseteq \operatorname{rad}Q \), so \( \operatorname{rad}Q = M \) is a maximal ideal, and \( Q \) is primary by (4).

Corollary 13.1.1. If \( u \in {W}^{1,2}\left( \Omega \right) \) is a weak solution of \( {\Delta u} = f \) with \( f \in \) \( {C}^{k,\alpha }\left( \Omega \right), k \in \mathbb{N},0 < \alpha < 1 \), then \( u \in {C}^{k + 2,\alpha }\left( \Omega \right) \), and for \( {\Omega }_{0} \subset \subset \Omega \) , \[ \parallel u{\parallel }_{{C}^{k + 2,\alpha }\left( {\Omega }_{0}\right) } \leq \operatorname{const}\left( {\parallel f{\parallel }_{{C}^{k,\alpha }\left( \Omega \right) } + \parallel u{\parallel }_{{L}^{2}\left( \Omega \right) }}\right) . \] If \( f \in {C}^{\infty }\left( \Omega \right) \), so is \( u \) . Proof. Since \( u \in {C}^{2,\alpha }\left( \Omega \right) \) by Theorem 13.1.2, we know that it weakly solves \[ \Delta \frac{\partial }{\partial {x}^{i}}u = \frac{\partial }{\partial {x}^{i}}f \] Theorem 13.1.2 then implies \[ \frac{\partial }{\partial {x}^{i}}u \in {C}^{2,\alpha }\left( \Omega \right) \;\left( {i \in \{ 1,\ldots, d\} }\right) , \] and thus \( u \in {C}^{3,\alpha }\left( \Omega \right) \) . The proof is concluded by induction. ## 13.2 The Schauder Estimates In this section, we study differential equations of the type \[ {Lu}\left( x\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{i, j = 1}}^{d}{a}^{ij}\left( x\right) \frac{{\partial }^{2}u\left( x\right) }{\partial {x}^{i}\partial {x}^{j}} + \mathop{\sum }\limits_{{i = 1}}^{d}{b}^{i}\left( x\right) \frac{\partial u\left( x\right) }{\partial {x}^{i}} + c\left( x\right) u\left( x\right) = f\left( x\right) \] (13.2.1) in some domain \( \Omega \subset {\mathbb{R}}^{d} \) . We make the following assumptions: (A) Ellipticity: There exists \( \lambda > 0 \) such that for all \( x \in \Omega ,\xi \in {\mathbb{R}}^{d} \) , \[ \mathop{\sum }\limits_{{i, j = 1}}^{d}{a}^{ij}\left( x\right) {\xi }_{i}{\xi }_{j} \geq \lambda {\left| \xi \right| }^{2} \] Moreover, \( {a}^{ij}\left( x\right) = {a}^{ji}\left( x\right) \) for all \( i, j, x \) . (B) Hölder continuous coefficients: There exists \( K < \infty \) such that \[ {\begin{Vmatrix}{a}^{ij}\end{Vmatrix}}_{{C}^{\alpha }\left( \Omega \right) },{\begin{Vmatrix}{b}^{i}\end{Vmatrix}}_{{C}^{\alpha }\left( \Omega \right) },\parallel c{\parallel }_{{C}^{\alpha }\left( \Omega \right) } \leq K \] for all \( i, j \) . The fundamental estimates of J. Schauder are the following: Theorem 13.2.1. Let \( f \in {C}^{\alpha }\left( \Omega \right) \), and suppose \( u \in {C}^{2,\alpha }\left( \Omega \right) \) satisfies \[ {Lu} = f \] (13.2.2) in \( \Omega \left( {0 < \alpha < 1}\right) \) . For any \( {\Omega }_{0} \subset \subset \Omega \), we then have \[ \parallel u{\parallel }_{{C}^{2,\alpha }\left( {\Omega }_{0}\right) } \leq {c}_{1}\left( {\parallel f{\parallel }_{{C}^{\alpha }\left( \Omega \right) } + \parallel u{\parallel }_{{L}^{2}\left( \Omega \right) }}\right) , \] (13.2.3) with a constant \( {c}_{1} \) depending on \( \Omega ,{\Omega }_{0},\alpha, d,\lambda, K \) . For the proof, we shall need the following lemma: Lemma 13.2.1. Let the symmetric matrix \( {\left( {A}^{ij}\right) }_{i, j = 1,\ldots, d} \) satisfy \[ \lambda {\left| \xi \right| }^{2} \leq \mathop{\sum }\limits_{{i, j = 1}}^{d}{A}^{ij}{\xi }_{i}{\xi }_{j} \leq \Lambda {\left| \xi \right| }^{2}\;\text{ for all }\xi \in {\mathbb{R}}^{d} \] (13.2.4) with \[ 0 < \lambda < \Lambda < \infty \text{.} \] Let \( u \) satisfy \[ \mathop{\sum }\limits_{{i, j = 1}}^{d}{A}^{ij}\frac{{\partial }^{2}u}{\partial {x}^{i}\partial {x}^{j}} = f \] (13.2.5) with \( f \in {C}^{\alpha }\left( \Omega \right) \left( {0 < \alpha < 1}\right) \) . For any \( {\Omega }_{0} \subset \subset \Omega \), we then have \[ \parallel u{\parallel }_{{C}^{2,\alpha }\left( {\Omega }_{0}\right) } \leq {c}_{2}\left( {\parallel f{\parallel }_{{C}^{\alpha }\left( \Omega \right) } + \parallel u{\parallel }_{{L}^{2}\left( \Omega \right) }}\right) . \] (13.2.6) Proof. We shall employ the following notation: \[ A \mathrel{\text{:=}} {\left( {A}^{ij}\right) }_{i, j = 1,\ldots, d},\;{D}^{2}u \mathrel{\text{:=}} {\left( \frac{{\partial }^{2}u}{\partial {x}^{i}\partial {x}^{j}}\right) }_{i, j = 1,\ldots, d}. \] If \( B \) is a nonsingular \( d \times d \) -matrix and if \( y \mathrel{\text{:=}} {Bx}, v \mathrel{\text{:=}} u \circ {B}^{-1} \), i.e., \( v\left( y\right) = u\left( x\right) \) , we have \[ A{D}^{2}u\left( x\right) = A{B}^{t}{D}^{2}v\left( y\right) B, \] and hence \[ \operatorname{Tr}\left( {A{D}^{2}u\left( x\right) }\right) = \operatorname{Tr}\left( {{BA}{B}^{t}{D}^{2}v\left( y\right) }\right) . \] (13.2.7) Since \( A \) is symmetric, we may choose \( B \) such that \( {B}^{t}{AB} \) is the unit matrix. In fact, \( B \) can be chosen as the product of the diagonal matrix \[ D = \left( \begin{array}{lll} {\lambda }_{1}^{-\frac{1}{2}} & & \\ & \ddots & \\ & & {\lambda }_{d}^{-\frac{1}{2}} \end{array}\right) \] \( \left( {{\lambda }_{1},\ldots ,{\lambda }_{d}}\right. \) being the eigenvalues of \( \left. A\right) \) with some orthogonal matrix \( R \) . In this way we obtain the transformed equation \[ {\Delta v}\left( y\right) = f\left( {{B}^{-1}y}\right) . \] (13.2.8) Theorem 13.1.2 then yields \( {C}^{2,\alpha } \) -estimates for \( v \), and these can be transformed back into estimates for \( u = v \circ B \) . The resulting constants will also depend on the bounds \( \lambda ,\Lambda \) for the eigenvalues of \( A \), since these determine the eigenvalues of \( D \) and hence of \( B \) . Proof of Theorem 13.2.1: We shall show that for every \( {x}_{0} \in {\bar{\Omega }}_{0} \) there exists some ball \( B\left( {{x}_{0}, r}\right) \) on which the desired estimate holds. The radius \( r \) of this ball will depend only on \( \operatorname{dist}\left( {{\Omega }_{0},\partial \Omega }\right) \) and the Hölder norms of the coefficients \( {a}^{ij},{b}^{i}, c \) . Since \( {\bar{\Omega }}_{0} \) is compact, it can be covered by finitely many such balls, and this yields the estimate in \( {\Omega }_{0} \) . Thus, let \( {x}_{0} \in {\bar{\Omega }}_{0} \) . We rewrite the differential equation \( {Lu} = f \) as \[ \mathop{\sum }\limits_{{i, j}}{a}^{ij}\left( {x}_{0}\right) \frac{{\partial }^{2}u\left( x\right) }{\partial {x}^{i}\partial {x}^{j}} = \mathop{\sum }\limits_{{i, j}}\left( {{a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) }\right) \frac{{\partial }^{2}u\left( x\right) }{\partial {x}^{i}\partial {x}^{j}} \] \[ - \mathop{\sum }\limits_{i}{b}^{i}\left( x\right) \frac{\partial u\left( x\right) }{\partial {x}^{i}} - c\left( x\right) u\left( x\right) + f\left( x\right) \] \[ = : \varphi \left( x\right) \text{.} \] (13.2.9) If we are able to estimate the \( {C}^{\alpha } \) -norm of \( \varphi \), putting \( {A}^{ij} \mathrel{\text{:=}} {a}^{ij}\left( {x}_{0}\right) \) and applying Lemma 13.2.1 will yield the estimate of the \( {C}^{2,\alpha } \) -norm of \( u \) . The crucial term for the estimate of \( \varphi \) is \( \sum \left( {{a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) }\right) \frac{{\partial }^{2}u}{\partial {x}^{i}\partial {x}^{j}} \) . Let \( B\left( {{x}_{0}, R}\right) \subset \Omega \) . By Lemma 13.1.1 \[ {\left| \mathop{\sum }\limits_{{i, j}}\left( {a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) \right) \frac{{\partial }^{2}u\left( x\right) }{\partial {x}^{i}\partial {x}^{j}}\right| }_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } \] \[ \leq \mathop{\sup }\limits_{{i, j, x \in B\left( {{x}_{0}, R}\right) }}\left| {{a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) }\right| {\left| {D}^{2}u\right| }_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } \] \[ + \mathop{\sum }\limits_{{i, j}}{\left| {a}^{ij}\right| }_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) }\mathop{\sup }\limits_{{B\left( {{x}_{0}, R}\right) }}\left| {{D}^{2}u}\right| . \] (13.2.10) Thus, also \[ {\begin{Vmatrix}\sum \left( {a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) \right) \frac{{\partial }^{2}u}{\partial {x}^{i}\partial {x}^{j}}\end{Vmatrix}}_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } \] \[ \leq \sup \left| {{a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) }\right| \parallel u{\parallel }_{{C}^{2,\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } + {c}_{3}\parallel u{\parallel }_{{C}^{2}\left( {B\left( {{x}_{0}, R}\right) }\right) }, \] (13.2.11) where \( {c}_{3} \) in particular depends on the \( {C}^{\alpha } \) -norm of the \( {a}^{ij} \) . Analogously, \[ {\begin{Vmatrix}\mathop{\sum }\limits_{i}{b}^{i}\left( x\right) \frac{\partial u}{\partial {x}^{i}}\left( x\right) \end{Vmatrix}}_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } \leq {c}_{4}\parallel u{\parallel }_{{C}^{1,\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) }, \] (13.2.12) \[ \parallel c\left( x\right) u\left( x\right) {\parallel }_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } \leq {c}_{5}\parallel u{\parallel }_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) }. \] (13.2.13) Altogether, we obtain \[ \parallel \varphi {\parallel }_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } \leq \mathop{\sup }\limits_{{i, j, x \in B\left( {{x}_{0}, R}\right) }}\left| {{a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) }\right| \parallel u{\parallel }_{{C}^{2,\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) } \] \[ + {c}_{6}\parallel u{\parallel }_{{C}^{2}\left( {B\left( {{x}_{0}, R}\right) }\right) } + \parallel f{\parallel }_{{C}^{\alpha }\left( {B\left( {{x}_{0}, R}\right) }\right) }. \] (13.2.14) By Lemma 13.2.1, from (13.2.9) and (13.2.14) for \( 0 < r < R \), we obtain \[ \parallel u{\parallel }_{{C}^{2,\alpha }\left( {B\left( {{x}_{0}, r}\right) }\right) } \leq {c}_{7}\mathop{\sup }\limits_{{i, j, x \in B\left( {{x}_{0}, R}\right) }}\left| {{a}^{ij}\left( {x}_{0}\right) - {a}^{ij}\left( x\right) }\right| \parallel u{\parallel }_{{C}^{2,\alpha }\left( {B\left( {{x}_{0}, R

Corollary 13.1.1. If \( u \in {W}^{1,2}\left( \Omega \right) \) is a weak solution of \( {\Delta u} = f \) with \( f \in \) \( {C}^{k,\alpha }\left( \Omega \right), k \in \mathbb{N},0 < \alpha < 1 \), then \( u \in {C}^{k + 2,\alpha }\left( \Omega \right) \), and for \( {\Omega }_{0} \subset \subset \Omega \) , \[ \parallel u{\parallel }_{{C}^{k + 2,\alpha }\left( {\Omega }_{0}\right) } \leq \operatorname{const}\left( {\parallel f{\parallel }_{{C}^{k,\alpha }\left( \Omega \right) } + \parallel u{\parallel }_{{L}^{2}\left( \Omega \right) }}\right) . \] If \( f \in {C}^{\infty }\left( \Omega \right) \), so is \( u \) .

Proof. Since \( u \in {C}^{2,\alpha }\left( \Omega \right) \) by Theorem 13.1.2, we know that it weakly solves \[ \Delta \frac{\partial }{\partial {x}^{i}}u = \frac{\partial }{\partial {x}^{i}}f \] Theorem 13.1.2 then implies \[ \frac{\partial }{\partial {x}^{i}}u \in {C}^{2,\alpha }\left( \Omega \right) \;\left( {i \in \{ 1,\ldots, d\} }\right) , \] and thus \( u \in {C}^{3,\alpha }\left( \Omega \right) \) . The proof is concluded by induction.

Theorem 5.9 (Stone [1934]). Every Boolean lattice is isomorphic to the lattice of closed and open subsets of its Stone space. Readers may prove a converse: every Stone space is homeomorphic to the Stone space of its lattice of closed and open subsets. Proof. Let \( L \) be a Boolean lattice and let \( X \) be its Stone space. For every \( a \in L, V\left( a\right) \) is open in \( X \), and is closed in \( X \) since \( X \smallsetminus V\left( a\right) = V\left( {a}^{\prime }\right) \) is open. Conversely, if \( U \in \mathrm{L}\left( X\right) \) is a closed and open subset of \( X \), then \( U \) is a union of basic open sets \( V\left( a\right) \subseteq U \) ; since \( U \) is closed, \( U \) is compact, \( U \) is a finite union \( V\left( {a}_{1}\right) \cup \cdots \cup V\left( {a}_{n}\right) = V\left( {{a}_{1} \vee \cdots \vee {a}_{n}}\right) \), and \( U = V\left( a\right) \) for some \( a \in L \) . Thus \( V \) is a mapping of \( L \) onto \( \mathrm{L}\left( X\right) \) . By 4.9 and 5.7, \( V : L \rightarrow \mathrm{L}\left( X\right) \) is a lattice isomorphism. \( ▱ \) ## Exercises 1. Let \( D \) be the set of all positive divisors of some \( n \in \mathbb{N} \), partially ordered by \( x \leqq y \) if and only if \( x \) divides \( y \) . Show that \( D \) is a distributive lattice. When is \( D \) a Boolean lattice? 2. A cofinite subset of a set \( X \) is a subset \( S \) of \( X \) whose complement \( X \smallsetminus S \) is finite. Show that the subsets of \( X \) that are either finite or cofinite constitute a Boolean lattice. 3. Show that a direct product of Boolean lattices is a Boolean lattice, when ordered componentwise. 4. A central idempotent of a ring \( R \) [with an identity element] is an element \( e \) of \( R \) such that \( {e}^{2} = e \) and \( {ex} = {xe} \) for all \( x \in R \) . Show that the central idempotents of \( R \) constitute a Boolean lattice when ordered by \( e \leqq f \) if and only if \( {ef} = e \) . (Hint: \( e \vee f = e + f - {ef} \) .) 5. Verify that a Boolean ring, partially ordered by \( x \leqq y \) if and only if \( {xy} = x \), is a Boolean lattice, in which \( x \land y = {xy}, x \vee y = x + y + {xy} \), and \( {x}^{\prime } = 1 - x \) . 6. Verify that the addition \( x + y = \left( {{x}^{\prime } \land y}\right) \vee \left( {x \land {y}^{\prime }}\right) \) on a Boolean lattice is associative. 7. Verify that \( \mathrm{L}\left( {\mathrm{R}\left( L\right) }\right) = L \) for every Boolean lattice \( L \), and that \( \mathrm{R}\left( {\mathrm{L}\left( R\right) }\right) = R \) for every Boolean ring \( R \) . 8. Show that a Boolean lattice \( L \) and its Boolean ring \( \mathrm{R}\left( L\right) \) have the same ideals. 9. Construct a purely lattice-theoretic quotient \( L/I \) of a Boolean lattice \( L \) by a lattice ideal \( I \) of \( L \) . 10. Verify that \( \mathrm{R}\left( {\mathop{\prod }\limits_{{i \in I}}{L}_{i}}\right) \cong \mathop{\prod }\limits_{{i \in I}}\mathrm{R}\left( {L}_{i}\right) \) for all Boolean lattices \( {\left( {L}_{i}\right) }_{i \in I} \) . 11. Recall that a closed interval of a lattice \( L \) is a sublattice \( \left\lbrack {a, b}\right\rbrack = \{ x \in L \mid a \leqq \) \( x \leqq b\} \) of \( L \), where \( a \leqq b \) in \( L \) . Show that every closed interval of a Boolean lattice \( L \) is a Boolean lattice (though not a Boolean sublattice of \( L \) ). 12. Show that the irreducible elements of a Boolean lattice are its atoms. A generalized Boolean lattice is a lattice \( L \) with a least element 0 such that every interval \( \left\lbrack {0, x}\right\rbrack \) is a Boolean lattice. 13. Show that the finite subsets of any set \( X \) constitute a generalized Boolean lattice. 14. Show that Propositions 5.3 and 5.4 extend to generalized Boolean lattices, if rings are not required to have an identity element. 15. Show that the identities \( \left( {\mathop{\bigvee }\limits_{{i \in I}}x}\right) \land y = \mathop{\bigvee }\limits_{{i \in I}}\left( {{x}_{i} \land y}\right) \) and \( \left( {\mathop{\bigwedge }\limits_{{i \in I}}x}\right) \vee y = \) \( \mathop{\bigwedge }\limits_{{i \in I}}\left( {{x}_{i} \vee y}\right) \) hold in every Boolean lattice that is complete. 16. Show that the identities \( {\left( \mathop{\bigvee }\limits_{{i \in I}}x\right) }^{\prime } = \mathop{\bigwedge }\limits_{{i \in I}}{x}_{i}^{\prime } \) and \( {\left( \mathop{\bigwedge }\limits_{{i \in I}}x\right) }^{\prime } = \mathop{\bigvee }\limits_{{i \in I}}{x}_{i}^{\prime } \) hold in every Boolean lattice that is complete. 17. Show that a complete Boolean lattice \( L \) is isomorphic to the lattice of all subsets of a set if and only if every element of \( L \) is the supremum of a set of atoms of \( L \) . 18. Show that every compact Hausdorff and totally disconnected topological space is homeomorphic to the Stone space of its lattice of closed and open subsets. ## XV Universal Algebra Universal algebra is the study of algebraic objects in general, also called universal algebras. These general objects were first considered by Whitehead [1898]. Birkhoff [1935], [1944] initiated their systematic study. Varieties are classes of universal algebras defined by identities. Groups, rings, left \( R \) -modules, etc., constitute varieties, and many of their properties are in fact properties of varieties. The main results in this chapter are two theorems of Birkhoff, one that characterizes varieties, one about subdirect decompositions. The chapter draws examples from Chapters I, III, V, VIII, and XIV, and is otherwise independent of previous chapters. ## 1. Universal Algebras A universal algebra is a set with any number of operations. This section gives basic properties, such as the homomorphism and factorization theorems. Definitions. Let \( n \geqq 0 \) be a nonnegative integer. An \( n \) -ary operation \( \omega \) on a set \( X \) is a mapping of \( {X}^{n} \) into \( X \), where \( {X}^{n} \) is the Cartesian product of \( n \) copies of \( X \) ; the number \( n \) is the arity of \( \omega \) . An operation of arity 2 is a binary operation. An operation of arity 1 or unary operation on a set \( X \) is simply a mapping of \( X \) into \( X \) . By convention, the empty cardinal product \( {X}^{0} \) is your favorite one element set, for instance, \( \{ \varnothing \} \) ; hence an operation of arity 0 or constant operation on a set \( X \) merely selects one element of \( X \) . Binary operations predominate in previous chapters, but constant and unary operations were encountered occasionally. There are operations of infinite arity (for instance, infimums and supremums in complete lattices), but many properties in this chapter require finite arity. Order relations and partial operations are excluded for the same reason (a partial operation on a set \( X \) is a mapping of a subset of \( {X}^{n} \) into \( X \) and need not be defined for all \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in {X}^{n} \) ). Universal algebras are classified by their type, which specifies number of operations and arities: Definitions. A type of universal algebras is an ordered pair of a set \( T \) and a mapping \( \omega \mapsto {n}_{\omega } \) that assigns to each \( \omega \in T \) a nonnegative integer \( {n}_{\omega } \), the formal arity of \( \omega \) . A universal algebra, or just algebra, of type \( T \) is an ordered pair of a set \( A \) and a mapping, the type- \( T \) algebra structure on \( A \), that assigns to each \( \omega \in T \) an operation \( {\omega }_{A} \) on \( A \) of arity \( {n}_{\omega } \) . For clarity \( {\omega }_{A} \) is often denoted by just \( \omega \) . For example, rings and lattices are of the same type, which has two elements of arity 2 . Sets are universal algebras of type \( T = \varnothing \) . Groups and semigroups are of the same type, which has one element of arity 2. Groups may also be viewed as universal algebras with one binary operation, one constant operation that selects the identity element, and one unary operation \( x \mapsto {x}^{-1} \) ; the corresponding type has one element of arity 0, one element of arity 1, and one element of arity 2 . Left \( R \) -modules are universal algebras with one binary operation (addition) and one unary operation \( x \mapsto {rx} \) for every \( r \in R \) . These descriptions will be refined in Section 2 when we formally define identities. On the other hand, partially ordered sets and topological spaces are not readily described as universal algebras. Section XVI. 10 explains why, to some extent. Subalgebras of an algebra are subsets that are closed under all operations: Definition. A subalgebra of a universal algebra A of type \( T \) is a subset \( S \) of A such that \( \omega \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in S \) for all \( \omega \in T \) of arity \( n \) and \( {x}_{1},\ldots ,{x}_{n} \in S \) . Let \( S \) be a subalgebra of \( A \) . Every operation \( {\omega }_{A} \) on \( A \) has a restriction \( {\omega }_{S} \) to \( S \) (sends \( {S}^{n} \) into \( S \), if \( \omega \) has arity \( n \) ). This makes \( S \) an algebra of the same type as \( A \), which is also called a subalgebra of \( A \) . Readers will verify that the definition of subalgebras encompasses subgroups, subrings, submodules, etc., provided that groups, rings, modules, etc. are defined as algebras of suitable types. Once started, they may as well prove the following: Proposition 1.1. The intersection of subalgebras of a universal algebra \( A \) is a subalgebra of \( A \) . Proposition 1.2. The union of a nonempty directed family of subalgebras of a universal algebra \( A \) is a subalgebra of \( A \) . In particular, the union of a nonempty chain of subalgebras of a universal algebra \( A \) is a subalgebra of \( A \) . Proposition 1.2 becomes false if infinitary operations are allowed. Homomorphisms are mappings that pres

Theorem 5.9 (Stone [1934]). Every Boolean lattice is isomorphic to the lattice of closed and open subsets of its Stone space.

Let \( L \) be a Boolean lattice and let \( X \) be its Stone space. For every \( a \in L, V\left( a\right) \) is open in \( X \), and is closed in \( X \) since \( X \smallsetminus V\left( a\right) = V\left( {a}^{\prime }\right) \) is open. Conversely, if \( U \in \mathrm{L}\left( X\right) \) is a closed and open subset of \( X \), then \( U \) is a union of basic open sets \( V\left( a\right) \subseteq U \) ; since \( U \) is closed, \( U \) is compact, \( U \) is a finite union \( V\left( {a}_{1}\right) \cup \cdots \cup V\left( {a}_{n}\right) = V\left( {{a}_{1} \vee \cdots \vee {a}_{n}}\right) \), and \( U = V\left( a\right) \) for some \( a \in L \) . Thus \( V \) is a mapping of \( L \) onto \( \mathrm{L}\left( X\right) \) . By 4.9 and 5.7, \( V : L \rightarrow \mathrm{L}\left( X\right) \) is a lattice isomorphism. \( ▱ \)

Lemma 12. Let \( M, B \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) be matrices, with \( M \) irreducible and \( \left| B\right| \leq M \) . Then \( \rho \left( B\right) \leq \rho \left( M\right) \) . In the case of equality \( \left( {\rho \left( B\right) = \rho \left( M\right) }\right) \), the following hold. - \( \left| B\right| = M \) . - For every eigenvector \( x \) of \( B \) associated with an eigenvalue of modulus \( \rho \left( M\right) ,\left| x\right| \) is an eigenvector of \( M \) associated with \( \rho \left( M\right) \) . Proof. In order to establish the inequality, we proceed as above. If \( \lambda \) is an eigenvalue of \( B \), of modulus \( \rho \left( B\right) \), and if \( x \) is a normalized eigenvector, then \( \rho \left( B\right) \left| x\right| \leq \left| B\right| \cdot \left| x\right| \leq \) \( M\left| x\right| \), so that \( {C}_{\rho \left( B\right) } \) is nonempty. Hence \( \rho \left( B\right) \leq R = \rho \left( M\right) \) . Let us investigate the case of equality. If \( \rho \left( B\right) = \rho \left( M\right) \), then \( \left| x\right| \in {C}_{\rho \left( M\right) } \), and therefore \( \left| x\right| \) is an eigenvector: \( M\left| x\right| = \rho \left( M\right) \left| x\right| = \rho \left( B\right) \left| x\right| \leq \left| B\right| \cdot \left| x\right| \) . Hence, \( (M - \) \( \left| B\right| )\left| x\right| \leq 0 \) . Because \( \left| x\right| > 0 \) (from Lemma 11) and \( M - \left| B\right| \geq 0 \), this gives \( \left| B\right| = M \) . ## 8.3.3 The Eigenvalue \( \rho \left( \mathbf{A}\right) \) Is Simple Let \( {P}_{A}\left( X\right) \) be the characteristic polynomial of \( A \) . It is given as the composition of an \( n \) -linear form (the determinant) with polynomial vector-valued functions (the columns of \( X{I}_{n} - A \) ). If \( \phi \) is \( p \) -linear and if \( {V}_{1}\left( X\right) ,\ldots ,{V}_{p}\left( X\right) \) are polynomial vector-valued functions, then the derivative of the polynomial \( P\left( X\right) \mathrel{\text{:=}} \) \( \phi \left( {{V}_{1}\left( X\right) ,\ldots ,{V}_{p}\left( X\right) }\right) \) is given by \[ {P}^{\prime }\left( X\right) = \phi \left( {{V}_{1}^{\prime },{V}_{2},\ldots ,{V}_{p}}\right) + \phi \left( {{V}_{1},{V}_{2}^{\prime },\ldots ,{V}_{p}}\right) + \cdots + \phi \left( {{V}_{1},\ldots ,{V}_{p - 1},{V}_{p}^{\prime }}\right) . \] One therefore has \[ {P}_{A}^{\prime }\left( X\right) = \det \left( {{\mathbf{e}}^{1},{a}_{2},\ldots ,{a}_{n}}\right) + \det \left( {{a}_{1},{\mathbf{e}}^{2},\ldots ,{a}_{n}}\right) + \cdots + \det \left( {{a}_{1},\ldots ,{a}_{n - 1},{\mathbf{e}}^{n}}\right) , \] where \( {a}_{j} \) is the \( j \) th column of \( X{I}_{n} - A \) and \( \left\{ {{\mathbf{e}}^{1},\ldots ,{\mathbf{e}}^{n}}\right\} \) is the canonical basis of \( {\mathbb{R}}^{n} \) . Developing the \( j \) th determinant with respect to the \( j \) th column, one obtains \[ {P}_{A}^{\prime }\left( X\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{P}_{{A}_{j}}\left( X\right) \] (8.1) where \( {A}_{j} \in {\mathbf{M}}_{n - 1}\left( \mathbb{R}\right) \) is obtained from \( A \) by deleting the \( j \) th row and the \( j \) th column. Let us now denote by \( {B}_{j} \in {\mathbf{M}}_{n}\left( \mathbb{R}\right) \) the matrix obtained from \( A \) by replacing the entries of the \( j \) th row and column by zeroes. This matrix is block-diagonal, the two diagonal blocks being \( {A}_{j} \in {\mathbf{M}}_{n - 1}\left( \mathbb{R}\right) \) and \( 0 \in {\mathbf{M}}_{1}\left( \mathbb{R}\right) \) . Hence, the eigenvalues of \( {B}_{j} \) are those of \( {A}_{j} \), together with zero, and therefore \( \rho \left( {B}_{j}\right) = \rho \left( {A}_{j}\right) \) . Furthermore, \( \left| {B}_{j}\right| \leq A \), but \( \left| {B}_{j}\right| \neq A \) because \( A \) is irreducible and \( {B}_{j} \) is block-diagonal, hence reducible. It follows (Lemma 12) that \( \rho \left( {B}_{j}\right) < \rho \left( A\right) \) . Hence \( {P}_{{A}_{j}} \) does not vanish over \( \lbrack \rho \left( A\right) , + \infty ) \) . Because \( {P}_{{A}_{j}}\left( t\right) \approx {t}^{n - 1} \) at infinity, we deduce that \( {P}_{{A}_{j}}\left( {\rho \left( A\right) }\right) > 0 \) . Finally, \( {P}_{A}^{\prime }\left( {\rho \left( A\right) }\right) \) is positive and \( \rho \left( A\right) \) is a simple root. ## 8.4 Cyclic Matrices The following statement completes Theorem 8.2. Theorem 8.3 Under the assumptions of Theorem 8.2, let \( p \) be the cardinality of the set \( {\operatorname{Sp}}_{\max }\left( A\right) \) of eigenvalues of \( A \) of maximal modulus \( \rho \left( A\right) \) . Then we have \( {\operatorname{Sp}}_{\max }\left( A\right) = \rho \left( A\right) {\mathcal{U}}_{p} \), where \( {\mathcal{U}}_{p} \) is the group of pth roots of unity. Every such eigenvalue is simple. The spectrum of \( A \) is invariant under multiplication by \( {\mathcal{U}}_{p} \) . Finally, \( A \) is conjugated via a permutation matrix to the following cyclic form. In this cyclic matrix each element is a block, and the diagonal blocks (which all vanish) are square with nonzero sizes: \[ \left( \begin{matrix} 0 & {M}_{1} & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & & & \ddots & {M}_{p - 1} \\ {M}_{p} & 0 & \cdots & \cdots & 0 \end{matrix}\right) . \] ## Remark The converse is true. The characteristic polynomial of a cyclic matrix is \[ X \mapsto \det \left( {{X}^{p}{I}_{m} - {M}_{1}{M}_{2}\cdots {M}_{p}}\right) \] up to a factor \( {X}^{v} \) (with \( v \) possibly negative). Its spectrum is thus stable under multiplication by \( \exp \left( {2\mathrm{i}\pi /p}\right) \) . Proof. Let us denote by \( X \) the unique nonnegative eigenvector of \( A \) normalized by \( \parallel X{\parallel }_{1} = 1 \) . If \( Y \) is a unitary eigenvector, associated with an eigenvalue \( \mu \) of maximal modulus \( \rho \left( A\right) \), the inequality \( \rho \left( A\right) \left| Y\right| = \left| {AY}\right| \leq A\left| Y\right| \) implies (Lemma 12) \( \left| Y\right| = X \) . Hence there is a diagonal matrix \( D = \operatorname{diag}\left( {{\mathrm{e}}^{\mathrm{i}{\alpha }_{1}},\ldots ,{\mathrm{e}}^{\mathrm{i}{\alpha }_{n}}}\right) \) such that \( Y = {DX} \) . Let us define a unimodular complex number \( {\mathrm{e}}^{\mathrm{i}\gamma } = \mu /\rho \left( A\right) \) and set \( B \mathrel{\text{:=}} {\mathrm{e}}^{-\mathrm{i}\gamma }{D}^{-1}{AD} \) . One has \( \left| B\right| = A \) and \( {BX} = X \) . For every \( j \), one therefore has \[ \left| {\mathop{\sum }\limits_{{k = 1}}^{n}{b}_{jk}{x}_{k}}\right| = \mathop{\sum }\limits_{{k = 1}}^{n}\left| {b}_{jk}\right| {x}_{k} \] Because \( X > 0 \), one deduces that \( B \) is real-valued and nonnegative. Therefore \( B = A \) ; that is, \( {D}^{-1}{AD} = {\mathrm{e}}^{\mathrm{i}\gamma }A \) . The spectrum of \( A \) is thus invariant under multiplication by \( {\mathrm{e}}^{\mathrm{i}\gamma } \) . Let \( \mathcal{U} = \rho {\left( A\right) }^{-1}{\operatorname{Sp}}_{\max }\left( A\right) \), which is included in \( {S}^{1} \), the unit circle. The previous discussion shows that \( \mathcal{U} \) is stable under multiplication. Because \( \mathcal{U} \) is finite, it follows that its elements are roots of unity. The inverse of a \( d \) th root of unity is its own \( \left( {d - 1}\right) \) th power, therefore \( \mathcal{U} \) is stable under inversion. Hence it is a finite subgroup of \( {S}^{1} \) . With \( p \) its cardinal, we have \( \mathcal{U} = {\mathcal{U}}_{p} \) . Let \( {P}_{A} \) be the characteristic polynomial and let \( \omega = \exp \left( {2\mathrm{i}\pi /p}\right) \) . One may apply the first part of the proof to \( \mu = {\omega \rho }\left( A\right) \) . One has thus \( {D}^{-1}{AD} = {\omega A} \), and it follows that \( {P}_{A}\left( X\right) = {\omega }^{n}{P}_{A}\left( {X/\omega }\right) \) . Therefore, multiplication by \( \omega \) sends eigenvalues to eigenvalues of the same multiplicities. In particular, the eigenvalues of maximal modulus are simple. Iterating the conjugation, one obtains \( {D}^{-p}A{D}^{p} = A \) . Let us set \[ {D}^{p} = \operatorname{diag}\left( {{d}_{1},\ldots ,{d}_{n}}\right) \] One has thus \( {d}_{j} = {d}_{k} \), provided that \( {a}_{jk} \neq 0 \) . Because \( A \) is irreducible, one can link any two indices \( j \) and \( k \) by a chain \( {j}_{0} = j,\ldots ,{j}_{r} = k \) such that \( {a}_{{j}_{s - 1},{j}_{s}} \neq 0 \) for every \( s \) . It follows that \( {d}_{j} = {d}_{k} \) for every \( j, k \) . But because one may choose \( {Y}_{1} = {X}_{1} \), that is \( {\alpha }_{1} = 0 \), one also has \( {d}_{1} = 1 \) and hence \( {D}^{p} = {I}_{n} \) . The \( {\alpha }_{j} \) are thus \( p \) th roots of unity. Applying a conjugation by a permutation matrix we may limit ourselves to the case where \( D \) has the block-diagonal form \( \operatorname{diag}\left( {{J}_{0},\omega {J}_{1},\ldots ,{\omega }^{p - 1}{J}_{p - 1}}\right) \), where the \( {J}_{\ell } \) are identity matrices of respective sizes \( {n}_{0},\ldots ,{n}_{p - 1} \) . Decomposing \( A \) into blocks \( {A}_{lm} \) of sizes \( {n}_{\ell } \times {n}_{m} \), one obtains \( {\omega }^{k}{A}_{jk} = {\omega }^{j + 1}{A}_{jk} \) directly from the conjugation identity. Hence \( {A}_{jk} = 0 \), except for the pairs \( \left( {j, k}\right) \) of the form \( \left( {0,1}\right) ,\left( {1,2}\right) ,\ldots ,(p - 2, p - \) \( 1),\left( {p - 1,0}\right) \) . This is the announced cyclic form. ## 8.5 Stochastic Matrices Definition 8.3 A matrix \( M \in {\mathbf{M}}_{n}\left( \mathbb{R}\right) \) is said to be stochastic if \( M \geq 0 \) and if for every \( i = 1,\ldots, n \), one has \[ \mathop{\sum }\limits_{{j = 1}}^{n}{m}_{ij} = 1 \] One says that \( M \) is bistochastic (or doubly stochastic) if both \( M \) and \( {M}^{T} \) are stochastic. Denoting by \( \mathbf{e} \in {\ma

Lemma 12. Let \( M, B \in {\mathbf{M}}_{n}\left( \mathbb{C}\right) \) be matrices, with \( M \) irreducible and \( \left| B\right| \leq M \) . Then \( \rho \left( B\right) \leq \rho \left( M\right) \) .

In order to establish the inequality, we proceed as follows. If \( \lambda \) is an eigenvalue of \( B \), of modulus \( \rho \left( B\right) \), and if \( x \) is a normalized eigenvector, then \[ \rho \left( B\right) \left| x\right| \leq \left| B\right| \cdot \left| x\right| \leq M\left| x\right|, \] so that \( {C}_{\rho \left( B\right) } \) is nonempty. Hence \( \rho \left( B\right) \leq R = \rho \left( M\right) \) . Let us investigate the case of equality. If \( \rho \left( B\right) = \rho \left( M\right) \), then \( \left| x\right| \in {C}_{\rho \left( M\right) } \), and therefore \( \left| x\right| \) is an eigenvector: \[ M\left| x\right| = \rho \left( M\right) \left| x\right| = \rho \left( B\right) \left| x\right| \leq \left| B\right| \cdot \left| x\right|. \] Hence, \[ (M - \left| B\right| )\left| x\right| \leq 0. \] Because \( \left| x\right| > 0 \) (from Lemma 11) and \( M -

Proposition 4.51. If \( S \) is compact, under assumptions (P1),(P2), one has \[ {\partial }_{F}m\left( \bar{x}\right) = \overline{\operatorname{co}}\left\{ {D{f}_{s}\left( \bar{x}\right) : s \in M\left( \bar{x}\right) }\right\} \] \[ {\partial }_{F}p\left( \bar{x}\right) = \mathop{\bigcap }\limits_{{s \in P\left( \bar{x}\right) }}\left\{ {D{f}_{s}\left( \bar{x}\right) }\right\} \] The proof is left as an exercise that the reader can tackle while reading Sect. 4.7.1 ## Exercises 1. Show that if \( X \) is a normed space and \( f \mathrel{\text{:=}} g \circ \ell \), where \( \ell : X \rightarrow Y \) is a continuous and open linear map with values in another normed space and \( g : Y \rightarrow \mathbb{R} \) is locally Lipschitzian, then \( {\partial }_{D}f\left( x\right) = {\partial }_{D}g\left( {\ell \left( x\right) }\right) \circ \ell \) . Can one replace \( \ell \) by a differentiable map? by a differentiable map that is open at \( x \) ? 2. Show by an example that the inclusion \( \partial \left( {g + h}\right) \left( \bar{x}\right) \subset \partial g\left( \bar{x}\right) + \partial h\left( \bar{x}\right) \) is not valid in general. Explain why this inclusion would be more desirable than the reverse one. 3. Prove Proposition 4.34. 4. Prove Corollary 4.35. [Hint: If \( f \mathrel{\text{:=}} g + h \), apply relation (4.20) to \( g = f + \left( {-h}\right) \) .] 5. Give a rule for the subdifferential of a quotient. 6. (a) With the assumptions and notation of Proposition 4.46, show that for all \( v \in X \) one has \( {g}^{D}\left( {\bar{x}, v}\right) = \mathop{\min }\limits_{{i \in I\left( \bar{x}\right) }}{f}_{i}^{D}\left( {\bar{x}, v}\right) ,{h}^{D}\left( {\bar{x}, v}\right) = \mathop{\max }\limits_{{i \in S\left( \bar{x}\right) }}{f}_{i}^{D}\left( {\bar{x}, v}\right) \) . (b) Show that in general, the inclusions of Proposition 4.46 are strict. 7. With the assumptions and notation of Proposition 4.42, show that when \( Y \) is finite-dimensional and \( g \) is differentiable, relation (4.26) can be deduced from (4.27). ## 4.3 Viscosity Subdifferentials In the sequel, given two normed spaces \( X, Y \) and an open subset \( W \) of \( X \), we say that a map \( h : W \rightarrow Y \) is F-smooth at \( \bar{x} \in X \) (resp. H-smooth at \( \bar{x} \) ) if \( h \) is of class \( {C}^{1} \) (resp. \( {D}^{1} \) ) at \( \bar{x} \), i.e., if \( h \) is Fréchet (resp. Hadamard) differentiable on an open neighborhood of \( \bar{x} \) and if \( {h}^{\prime } \) is continuous at \( \bar{x} \) (resp. \( {dh} : \left( {x, v}\right) \mapsto {h}^{\prime }\left( x\right) v \) is continuous at \( \left( {\bar{x}, u}\right) \) for all \( u \in X \) ). We gather both cases by saying that \( h \) is smooth at \( \bar{x} \) . We say that \( h \) is smooth if it is smooth at each point of \( W \) . We say that a Banach space \( X \) is F-smooth (resp. H-smooth) if there is some F-smooth (resp. H-smooth) function \( j \mathrel{\text{:=}} {j}_{X} : X \rightarrow {\mathbb{R}}_{ + } \) such that \( j\left( 0\right) = 0 \) and \[ \left( {{\left( j\left( {x}_{n}\right) \right) }_{n} \rightarrow 0}\right) \Rightarrow \left( {{\left( \begin{Vmatrix}{x}_{n}\end{Vmatrix}\right) }_{n} \rightarrow 0}\right) . \] (4.29) We gather these two cases by saying that \( X \) is smooth and we call \( j \) a forcing function. Note that in replacing \( j \) by \( {j}^{2} \), we get the implication \[ \left( {{\left( \begin{Vmatrix}{x}_{n}\end{Vmatrix}\right) }_{n} \rightarrow 0}\right) \Rightarrow \left( {{\left( {j}^{\prime }\left( {x}_{n}\right) \right) }_{n} \rightarrow 0}\right) . \] Condition (4.29) is more general than the requirement that an equivalent norm on \( X \) be Fréchet (resp. Hadamard) differentiable on \( X \smallsetminus \{ 0\} \) . On a first reading, the reader may assume that \( j \) is the square of such a norm, although such an assumption is not as general. Note that when \( j \) is a forcing function, the function \( k : X \times X \rightarrow \mathbb{R} \) given by \( k\left( {x,{x}^{\prime }}\right) \mathrel{\text{:=}} j\left( {x - {x}^{\prime }}\right) \) is a forcing bifunction in the sense of Sect. 1.6. We first show that the existence of a smooth Lipschitzian bump function ensures condition (4.29). Recall that \( b \) is a bump function if it is nonnegative, not identically equal to zero, and null outside some bounded set (which can be taken to be \( {B}_{X} \) ). Proposition 4.52. If \( X \) has a smooth forcing function, then it has a smooth bump function. If there is on \( X \) a Lipschitzian (resp. smooth) bump function \( {b}_{0} \), then there is a Lipschitzian (resp. smooth) bump function \( b \) such that \( b\left( X\right) \subset \left\lbrack {0,1}\right\rbrack, b\left( 0\right) = 1 \) and \( \left( {x}_{n}\right) \rightarrow 0 \) whenever \( \left( {b\left( {x}_{n}\right) }\right) \rightarrow 1 \), so that \( j \mathrel{\text{:=}} 1 - b \) is a forcing function. If \( {b}_{0} \) is Lipschitzian and smooth, then b can be chosen to be Lipschitzian and smooth. Proof. If \( j \) is a smooth forcing function, then there exists some \( s > 0 \) such that \( j\left( x\right) \geq s \) for all \( x \in X \smallsetminus {B}_{X} \) . Composing \( j \) with a smooth function \( h : \mathbb{R} \rightarrow {\mathbb{R}}_{ + } \) such that \( h\left( 0\right) = 1, h\left( r\right) = 0 \) for \( r \geq s \), we get a smooth bump function \( b \mathrel{\text{:=}} h \circ j \) . Let \( {b}_{0} \) be a Lipschitzian (resp. smooth) bump function with support in \( {B}_{X} \) . We first note that we can assume \( {b}_{0}\left( 0\right) > c \mathrel{\text{:=}} \left( {1/2}\right) \sup {b}_{0} > 0 \) : if it is not the case, we replace \( {b}_{0} \) by \( {b}_{1} \) given by \( {b}_{1}\left( x\right) \mathrel{\text{:=}} {b}_{0}\left( {{kx} + a}\right) \), where \( a \in {B}_{X} \) is such that \( {b}_{0}\left( a\right) > \) \( \left( {1/2}\right) \sup {b}_{0} \) and \( k > \parallel a\parallel + 1 \) . We can also assume that \( {b}_{0} \) attains its maximum at 0 and that \( {b}_{0}\left( 0\right) = 1 \) . If it is not the case, we replace \( {b}_{0} \) by \( \theta \circ {b}_{1} \), where \( \theta \) is a smooth function on \( \mathbb{R} \) satisfying \( \theta \left( r\right) = r \) for \( r \leq c,\theta \left( r\right) = 1 \) for \( r \geq {b}_{0}\left( 0\right) \) . Now, given \( q \in \left( {0,1}\right) \) we set \[ b\left( x\right) \mathrel{\text{:=}} \left( {1 - {q}^{2}}\right) \mathop{\sum }\limits_{{n = 0}}^{\infty }{q}^{2n}{b}_{0}\left( {{q}^{-n}x}\right) , \] with \( {q}^{0} \mathrel{\text{:=}} 1 \), so that \( b\left( 0\right) = 1 \geq b\left( x\right) \) for all \( x, b \) is null on \( X \smallsetminus {B}_{X}, b \) is smooth on \( X \smallsetminus \{ 0\} \) as the sum is locally finite on \( X \smallsetminus \{ 0\} \) . Moreover, since \( {b}_{0}^{\prime } \) is bounded, the series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{q}^{n}{b}_{0}^{\prime }\left( {{q}^{-n}x}\right) \) is uniformly convergent, so that \( b \) is of class \( {\mathrm{C}}^{1} \) (resp. of class \( {\mathrm{D}}^{1} \) ). Furthermore, \( b \) is Lipschitzian on \( X \) when \( {b}_{0} \) is Lipschitzian, and for \( x \in X \smallsetminus {q}^{k}{B}_{X} \) one has \[ b\left( x\right) \leq \left( {1 - {q}^{2}}\right) \mathop{\sum }\limits_{{n = 0}}^{k}{q}^{2n} < 1 = {b}_{0}\left( 0\right) , \] so that \( \left( {x}_{n}\right) \rightarrow 0 \) whenever \( \left( {b\left( {x}_{n}\right) }\right) \rightarrow 1 \) . The last assertion is obvious. The preceding result can be made more precise (at the expense of simplicity). Proposition 4.53. Let \( X \) be a normed space. There exists a Lipschitzian smooth bump function on \( X \) if and only if the following condition is satisfied: H) for all \( c > 1 \), there exists a function \( j : X \rightarrow \mathbb{R} \) that is smooth on \( X \smallsetminus \{ 0\} \), with a derivative that is bounded on every bounded subset of \( X \smallsetminus \{ 0\} \) and such that \[ \forall x \in X,\;\parallel x\parallel \leq j\left( x\right) \leq c\parallel x\parallel . \] (4.30) According to the sense given to the word "smooth," we denote this condition by \( \left( {\mathrm{H}}_{F}\right) \) or \( \left( {\mathrm{H}}_{D}\right) \) whenever it is necessary to be precise. Let us note that replacing \( j \) by \( {j}^{2} \) to ensure smoothness, condition (H) implies condition (4.29). The fact that \( c \) is arbitrarily close to 1 shows that a result in which one uses the differentiability of the norm on \( X \smallsetminus \{ 0\} \) is likely to be valid under assumption (H). We have seen that assumption \( \left( {\mathrm{H}}_{F}\right) \) (resp. \( \left. \left( {\mathrm{H}}_{D}\right) \right) \) is satisfied when the norm of the dual space of \( X \) is locally uniformly rotund (resp. when \( X \) is separable), and one can even take for \( j \) an equivalent norm. Proof. Let us first observe that condition \( \left( \mathrm{H}\right) \) ensures the existence of a Lipschitzian smooth bump function: it suffices to take \( b \mathrel{\text{:=}} k \circ {j}^{2} \), where \( k : \mathbb{R} \rightarrow \mathbb{R} \) is a Lipschitzian smooth function satisfying \( k\left( 0\right) = 1 \) for \( r \in ( - \infty ,\alpha \rbrack \) with \( \alpha \in \left( {0,1}\right) \) and \( k\left( r\right) = 0 \) for \( r \in \lbrack 1,\infty ) \) . Considering separately the case in which the dimension of \( X \) is 1 and the case in which this dimension is greater than 1, we can prove that \( j \) is Lipschitzian on \( {B}_{X} \) . Since \( {j}^{-1}\left( \left\lbrack {0,1}\right\rbrack \right) \subset {B}_{X}, k \circ j \) is Lipschitzian. In order to prove the converse, we first define a function \( {j}_{0} \) satisfying (4.30) on \( {B}_{X} \) . Let \( q > 1 \) with \( {q}^{2} < c \) . Using a translation and composing a smooth Lipschitzian bump function with a smooth function fr

Proposition 4.51. If \( S \) is compact, under assumptions (P1),(P2), one has \[ {\partial }_{F}m\left( \bar{x}\right) = \overline{\operatorname{co}}\left\{ {D{f}_{s}\left( \bar{x}\right) : s \in M\left( \bar{x}\right) }\right\} \] \[ {\partial }_{F}p\left( \bar{x}\right) = \mathop{\bigcap }\limits_{{s \in P\left( \bar{x}\right) }}\left\{ {D{f}_{s}\left( \bar{x}\right) }\right\} \]

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Proposition 6.4.14. Assume that neither \( b/a, c/a \), nor \( c/b \) is the cube of \( a \) rational number. If the elliptic curve \( E \) with affine equation \( {Y}^{2} = {X}^{3} + {\left( 4abc\right) }^{2} \) has zero rank then the equation \( a{x}^{3} + b{y}^{3} + c{z}^{3} \) has no nontrivial rational solutions. Proof. This is essentially a restatement of Corollary 8.1.15, and also immediately follows from the above proposition after rescaling. Note that Proposition 8.4.3 tells us that the elliptic curve \( {Y}^{2} = {X}^{3} - {432}{\left( abc\right) }^{2} \) is 3 -isogenous with the elliptic curve \( {Y}^{2} = {X}^{3} + {\left( 4abc\right) }^{2} \) . The general result on the equation \( a{x}^{3} + b{y}^{3} + c{z}^{3} = 0 \) is a very nice application of 3-descent (see Section 8.4), and I thank T. Fisher for explanations. The relation is the following. Let \( a, b \), and \( c \) be three nonzero rational numbers, and as above, let \( E = {E}_{abc} \) be the elliptic curve with projective equation \( {Y}^{2}Z = {X}^{3} + {\left( 4abc\right) }^{2}{Z}^{3} \) . This curve has the point \( T = \left( {0,{4abc},1}\right) \) as rational point of order 3 . In Section 8.4 we will define a 3-descent map \( \alpha \) from \( E\left( \mathbb{Q}\right) \) to \( {\mathbb{Q}}^{ * }/{\mathbb{Q}}^{*3} \) by setting \( \alpha \left( \mathcal{O}\right) = 1,\alpha \left( T\right) = {\left( abc\right) }^{2} \), and for \( P = \left( {X, Y, Z}\right) \) with \( P \neq \mathcal{O} \) and \( P \neq T \) by setting \( \alpha \left( P\right) = Y/Z - {4abc} \), modulo multiplication by cubes of \( {\mathbb{Q}}^{ * } \) . The fundamental property of \( \alpha \) is that it induces a group homomorphism from \( E\left( \mathbb{Q}\right) /{3E}\left( \mathbb{Q}\right) \) to \( {\mathbb{Q}}^{ * }/{\mathbb{Q}}^{*3} \) (Proposition 8.4.8). Proposition 6.4.15. Let \( \mathcal{C} \) be the cubic curve with projective equation \( a{x}^{3} + \) \( b{y}^{3} + c{z}^{3} = 0 \), let \( E \) be the elliptic curve with projective equation \( {Y}^{2}Z = \) \( {X}^{3} + {\left( 4abc\right) }^{2}{Z}^{3} \), and define \[ \phi \left( {x, y, z}\right) = \left( {-{4abcxyz}, - {4abc}\left( {b{y}^{3} - c{z}^{3}}\right), a{x}^{3}}\right) . \] (1) The map \( \phi \) sends \( \mathcal{C}\left( \mathbb{Q}\right) \) into \( E\left( \mathbb{Q}\right) \) . (2) The image \( \phi \left( {\mathcal{C}\left( \mathbb{Q}\right) }\right) \) is equal to the set of \( P \in E\left( \mathbb{Q}\right) \) such that \( \alpha \left( P\right) = \) \( b/c \in {\mathbb{Q}}^{ * }/{{\mathbb{Q}}^{ * }}^{3} \) . More precisely, if \( P = \left( {X, Y, Z}\right) \in \phi \left( {\mathcal{C}\left( \mathbb{Q}\right) }\right) \) with \( P \) different from \( \mathcal{O} \) and \( T \), and if \( \lambda \in {\mathbb{Q}}^{ * } \) is such that \( Y/Z - {4abc} = \left( {b/c}\right) {\lambda }^{3} \) then \[ \left( {x, y, z}\right) = \left( {{2bc\lambda Z}, - {cX}, b{\lambda }^{2}Z}\right) \in \mathcal{C}\left( \mathbb{Q}\right) \] is a preimage of \( P \) ; in addition, \( \mathcal{O} \in \phi \left( {\mathcal{C}\left( \mathbb{Q}\right) }\right) \) if and only if \( c/b = {\lambda }^{3} \) for some \( \lambda \in {\mathbb{Q}}^{ * } \), and in that case \( \left( {0, - \lambda ,1}\right) \) is a preimage of \( \mathcal{O} \) ; finally, \( T \in \phi \left( {\mathcal{C}\left( \mathbb{Q}\right) }\right) \) if and only if \( b/a = {\lambda }^{3} \) for some \( \lambda \in {\mathbb{Q}}^{ * } \), and in that case \( \left( {-\lambda ,1,0}\right) \) is a preimage of \( T \) . (3) The set \( \mathcal{C}\left( \mathbb{Q}\right) \) is nonempty if and only if the class of \( b/c \) modulo cubes belongs to the image of the 3-descent map \( \alpha \) from \( E\left( \mathbb{Q}\right) \) to \( {\mathbb{Q}}^{ * }/{\mathbb{Q}}^{*3} \) . In addition, \( \mathcal{C}\left( \mathbb{Q}\right) \) is infinite if and only if the class of \( b/c \) modulo cubes is equal to \( \alpha \left( G\right) \) for some nontorsion point \( G \in E\left( \mathbb{Q}\right) \) . Proof. (1). As for Proposition 6.4.13, of which up to rescaling this is a special case, the proof is a simple verification: setting \( \phi \left( {x, y, z}\right) = \left( {X, Y, Z}\right) \) , then if \( a{x}^{3} + b{y}^{3} + c{z}^{3} = 0 \) we check that \( {Y}^{2}Z - {\left( 4abc\right) }^{2}{Z}^{3} = {X}^{3} \) . Furthermore, we cannot have \( X = Y = Z = 0 \) since otherwise \( x = 0 \), so that \( b{y}^{3} + c{z}^{3} = 0 \) , and \( b{y}^{3} - c{z}^{3} = 0 \), and hence \( x = y = z = 0 \), which is excluded. Thus \( \phi \left( {x, y, z}\right) \in E\left( \mathbb{Q}\right) \), proving (1). (2). Note that if \( P = \phi \left( {x, y, z}\right) = \left( {X, Y, Z}\right) \), then \[ c\left( {Y - {4abcZ}}\right) = - {4ab}{c}^{2}\left( {b{y}^{3} - c{z}^{3} + a{x}^{3}}\right) = {8ab}{c}^{3}{z}^{3} = {bZ}{\lambda }^{3} \] with \( \lambda = {2cz}/x \in {\mathbb{Q}}^{ * } \) when \( x \) and \( z \) are nonzero, so that \( \alpha \left( P\right) = b/c \in \) \( {\mathbb{Q}}^{ * }/{\mathbb{Q}}^{*3} \) . Now, \( x \) can be equal to 0 if and only if \( b/c = {\left( -z/y\right) }^{3} \in {\mathbb{Q}}^{*3} \), and in that case we have \( \phi \left( {0, y, z}\right) = \mathcal{O} \) and \( \alpha \left( \mathcal{O}\right) = b/c \in {\mathbb{Q}}^{ * }/{\mathbb{Q}}^{*3} \) . Similarly \( z \) can be equal to 0 if and only if \( b/a = {\left( -x/y\right) }^{3} \in {\mathbb{Q}}^{*3} \), and in that case it is clear that \( \phi \left( {x, y,0}\right) = T \) and \( \alpha \left( T\right) = {\left( abc\right) }^{2} = \left( {b/c}\right) {\left( ac\right) }^{3}\left( {b/a}\right) = b/c \in {\mathbb{Q}}^{ * }/{\mathbb{Q}}^{*3} \) , proving (2). (3). Evidently \( \mathcal{C}\left( \mathbb{Q}\right) \) is nonempty if and only if \( \phi \left( {\mathcal{C}\left( \mathbb{Q}\right) }\right) \) is nonempty, and since clearly the number of preimages of an element of \( \phi \left( {\mathcal{C}\left( \mathbb{Q}\right) }\right) \) is finite, \( \mathcal{C}\left( \mathbb{Q}\right) \) is infinite if and only if \( \phi \left( {\mathcal{C}\left( \mathbb{Q}\right) }\right) \) is infinite. Thus, it follows from (2) that \( \mathcal{C}\left( \mathbb{Q}\right) \) is nonempty if and only if \( b/c \) modulo cubes belongs to the image of \( \alpha \), and it is infinite if and only if \( b/c = \alpha \left( G\right) \) for a point of infinite order, proving (3). Thus, to check whether an equation \( a{x}^{3} + b{y}^{3} + c{z}^{3} = 0 \) has a nontrivial solution we proceed as follows. Using either mwrank or the 2-descent methods, we compute if possible the complete Mordell-Weil group \( E\left( \mathbb{Q}\right) \) of the curve with affine equation \( {Y}^{2} = {X}^{3} + {\left( 4abc\right) }^{2} \), and we also compute the torsion subgroup (which we know will contain the subgroup of order 3 generated by \( T) \) . If \( {P}_{1},\ldots ,{P}_{r} \) form a basis of the free part of \( E\left( \mathbb{Q}\right) \) then \( {P}_{0} = T,{P}_{1},\ldots \) , \( {P}_{r} \) form an \( {\mathbb{F}}_{3} \) -basis of \( E\left( \mathbb{Q}\right) /{3E}\left( \mathbb{Q}\right) \) . We then check whether the class of \( b/c \) modulo cubes belongs to the group generated by the \( \alpha \left( {P}_{i}\right) \in {\mathbb{Q}}^{ * }/{\mathbb{Q}}^{*3} \), which is done using simple linear algebra over \( {\mathbb{F}}_{3} \) . Example. To illustrate, consider the equation \( {x}^{3} + {55}{y}^{3} + {66}{z}^{3} \) . Using Theorem 6.4.2 we check that it is everywhere locally soluble, but none of the results given in Section 6.4.3 allow us to determine whether the equation is globally soluble. Thus we consider the curve \( E \) with affine equation \( {Y}^{2} = {X}^{3} + {\left( 4 \cdot {55} \cdot {66}\right) }^{2} \) . We find that the torsion subgroup has order 3 and is generated by \( {P}_{0} = \) \( T = \left( {0,{14520}}\right) \) . In less than a second the mwrank program tells us that the curve has rank 1, a generator being \( {P}_{1} = \left( {{504},{18408}}\right) \) (so that Proposition 6.4.14 is not applicable). We have (modulo cubes) \( \alpha \left( {P}_{0}\right) = {2}^{2} \cdot {3}^{2} \cdot {5}^{2} \cdot {11} \) , \( \alpha \left( {P}_{1}\right) = 2 \cdot {3}^{2} \), and \( b/c = {2}^{2} \cdot {3}^{2} \cdot 5 \) . Here the linear algebra can be done naïvely: if \( b/c = \alpha \left( {u{P}_{0} + v{P}_{1}}\right) = \alpha {\left( {P}_{0}\right) }^{u}\alpha {\left( {P}_{1}\right) }^{v} \), where \( u \) and \( v \) are defined modulo 3, we see that \( u = 0 \) because of the 11 factor, which is impossible since there is a factor of \( 5 \) in \( b/c \) and none in \( \alpha \left( {P}_{1}\right) \) . This shows that our equation has no nontrivial solutions in \( \mathbb{Q} \), although our curve has nonzero rank. For the convenience of the reader, in the following table we give in a very compact form detailed information on the solubility of the equation \( {x}^{3} + b{y}^{3} + c{z}^{3} = 0 \) for \( 1 \leq b, c \leq {64} \) . The entry in line \( b \) and column \( c \) of the table means the following: - means locally insoluble, L, M, and E mean everywhere locally soluble but not globally soluble (hence a failure of the Hasse principle) obtained using Theorems 6.4.8 and 6.4.10, and Proposition 6.4.15 , used in that order (note that in the range of our table it is sufficient to use Proposition 6.4.14 instead of Proposition 6.4.15, but this is of course not true in general). In every other case, the equation is globally soluble, hence the curve \( {x}^{3} + b{y}^{3} + c{z}^{3} = 0 \) is the projective equation of an elliptic curve, and the entry \( \left( {0,1,2\text{, or 3 in the limits of the table}}\right) \) gives the rank of the curve (see Chapters 7 and 8 for all these notions). In particular, if the entry is equal to 0 this means that the equation has only a finite nonzero number of (projective) solutions, which can all easily be found. By Corollary 8.1.15 th

Proposition 6.4.14. Assume that neither \( b/a, c/a \), nor \( c/b \) is the cube of \( a \) rational number. If the elliptic curve \( E \) with affine equation \( {Y}^{2} = {X}^{3} + {\left( 4abc\right) }^{2} \) has zero rank then the equation \( a{x}^{3} + b{y}^{3} + c{z}^{3} \) has no nontrivial rational solutions.

This is essentially a restatement of Corollary 8.1.15, and also immediately follows from the above proposition after rescaling. Note that Proposition 8.4.3 tells us that the elliptic curve \( {Y}^{2} = {X}^{3} - {432}{\left( abc\right) }^{2} \) is 3-isogenous with the elliptic curve \( {Y}^{2} = {X}^{3} + {\left( 4abc\right) }^{2} \).

Proposition 9.5 THE FAN LEMMA Let \( G \) be a \( k \) -connected graph, let \( x \) be a vertex of \( G \), and let \( Y \subseteq V \smallsetminus \{ x\} \) be a set of at least \( k \) vertices of \( G \) . Then there exists a \( k \) -fan in \( G \) from \( x \) to \( Y \) . We now give the promised application of the Fan Lemma. By Theorem 5.1, in a 2-connected graph any two vertices are connected by two internally disjoint paths; equivalently, any two vertices in a 2-connected graph lie on a common cycle. Dirac (1952b) generalized this latter statement to \( k \) -connected graphs. Theorem 9.6 Let \( S \) be a set of \( k \) vertices in a \( k \) -connected graph \( G \), where \( k \geq 2 \) . Then there is a cycle in \( G \) which includes all the vertices of \( S \) . Proof By induction on \( k \) . We have already observed that the assertion holds for \( k = 2 \), so assume that \( k \geq 3 \) . Let \( x \in S \), and set \( T \mathrel{\text{:=}} S \smallsetminus x \) . Because \( G \) is \( k \) -connected, it is \( \left( {k - 1}\right) \) -connected. Therefore, by the induction hypothesis, there is a cycle \( C \) in \( G \) which includes \( T \) . Set \( Y \mathrel{\text{:=}} V\left( C\right) \) . If \( x \in Y \), then \( C \) includes all the vertices of \( S \) . Thus we may assume that \( x \notin Y \) . If \( \left| Y\right| \geq k \), the Fan Lemma (Proposition 9.5) ensures the existence of a \( k \) -fan in \( G \) from \( x \) to \( Y \) . Because \( \left| T\right| = k - 1 \), the set \( T \) divides \( C \) into \( k - 1 \) edge-disjoint segments. By the Pigeonhole Principle, some two paths of the fan, \( P \) and \( Q \), end in the same  Fig. 9.5. Proof of Theorem 9.6 segment. The subgraph \( C \cup P \cup Q \) contains three cycles, one of which includes \( S = T \cup \{ x\} \) (see Figure 9.5). If \( \left| Y\right| = k - 1 \), the Fan Lemma yields a \( \left( {k - 1}\right) \) -fan from \( x \) to \( Y \) in which each vertex of \( Y \) is the terminus of one path, and we conclude as before. It should be pointed out that the order in which the vertices of \( S \) occur on the cycle whose existence is established in Theorem 9.6 cannot be specified in advance. For example, the 4-connected graph shown in Figure 9.6 has no cycle including the four vertices \( {x}_{1},{y}_{1},{x}_{2},{y}_{2} \) in this exact order, because every \( {x}_{1}{y}_{1} \) -path intersects every \( {x}_{2}{y}_{2} \) -path.  Fig. 9.6. No cycle includes \( {x}_{1},{y}_{1},{x}_{2},{y}_{2} \) in this order ## Exercises *9.2.1 Give a proof of the Fan Lemma (Proposition 9.5). 9.2.2 Show that a 3-connected nonbipartite graph contains at least four odd cycles. \( \star \) 9.2.3 Let \( C \) be a cycle of length at least three in a nonseparable graph \( G \), and let \( S \) be a set of three vertices of \( C \) . Suppose that some component \( H \) of \( G - V\left( C\right) \) is adjacent to all three vertices of \( S \) . Show that there is a 3 -fan in \( G \) from some vertex \( v \) of \( H \) to \( S \) . 9.2.4 Find a 5-connected graph \( G \) and a set \( \left\{ {{x}_{1},{y}_{1},{x}_{2},{y}_{2}}\right\} \) of four vertices in \( G \), such that no cycle of \( G \) contains all four vertices in the given order. (In a 6- connected graph, it can be shown that there is a cycle containing any four vertices in any prescribed order.)  9.2.5 Let \( G \) be a graph, \( x \) a vertex of \( G \), and \( Y \) and \( Z \) subsets of \( V \smallsetminus \{ x\} \), where \( \left| Y\right| < \left| Z\right| \) . Suppose that there are fans from \( x \) to \( Y \) and from \( x \) to \( Z \) . Show that there is a fan from \( x \) to \( Y \cup \{ z\} \) for some \( z \in Z \smallsetminus Y \) . (H. Perfect) ## 9.3 Edge Connectivity We now turn to the notion of edge connectivity. The local edge connectivity between distinct vertices \( x \) and \( y \) is the maximum number of pairwise edge-disjoint \( {xy} \) - paths, denoted \( {p}^{\prime }\left( {x, y}\right) \) ; the local edge connectivity is undefined when \( x = y \) . A nontrivial graph \( G \) is \( k \) -edge-connected if \( {p}^{\prime }\left( {u, v}\right) \geq k \) for any two distinct vertices \( u \) and \( v \) of \( G \) . By convention, a trivial graph is both 0 -edge-connected and 1-edge-connected, but is not \( k \) -edge-connected for any \( k > 1 \) . The edge connectivity \( {\kappa }^{\prime }\left( G\right) \) of a graph \( G \) is the maximum value of \( k \) for which \( G \) is \( k \) -edge-connected. A graph is 1-edge-connected if and only if it is connected; equivalently, the edge connectivity of a graph is zero if and only if it is disconnected. For the four graphs shown in Figure \( {9.1},{\kappa }^{\prime }\left( {G}_{1}\right) = 1,{\kappa }^{\prime }\left( {G}_{2}\right) = 2,{\kappa }^{\prime }\left( {G}_{3}\right) = 3 \), and \( {\kappa }^{\prime }\left( {G}_{4}\right) = 4 \) . Thus, of these four graphs, the only graph that is 4-edge-connected is \( {G}_{4} \) . Graphs \( {G}_{3} \) and \( {G}_{4} \) are 3-edge-connected, but \( {G}_{1} \) and \( {G}_{2} \) are not. Graphs \( {G}_{2},{G}_{3} \), and \( {G}_{4} \) are 2-edge-connected, but \( {G}_{1} \) is not. And, because all four graphs are connected, they are all 1-edge-connected. For distinct vertices \( x \) and \( y \) of a graph \( G \), recall that an edge cut \( \partial \left( X\right) \) separates \( x \) and \( y \) if \( x \in X \) and \( y \in V \smallsetminus X \) . We denote by \( {c}^{\prime }\left( {x, y}\right) \) the minimum cardinality of such an edge cut. With this notation, we may now restate the edge version of Menger's Theorem (7.17). Theorem 9.7 Menger's Theorem (Edge Version) For any graph \( G\left( {x, y}\right) \) , \[ {p}^{\prime }\left( {x, y}\right) = {c}^{\prime }\left( {x, y}\right) \] This theorem was proved in Chapter 7 using flows. It may also be deduced from Theorem 9.1 by considering a suitable line graph (see Exercise 9.3.11). A \( k \) -edge cut is an edge cut \( \partial \left( X\right) \), where \( \varnothing \subset X \subset V \) and \( \left| {\partial \left( X\right) }\right| = k \), that is, an edge cut of \( k \) elements which separates some pair of vertices. Because every nontrivial graph has such edge cuts, it follows from Theorem 9.7 that the edge connectivity \( {\kappa }^{\prime }\left( G\right) \) of a nontrivial graph \( G \) is equal to the least integer \( k \) for which \( G \) has a \( k \) -edge cut. For any particular pair \( x, y \) of vertices of \( G \), the value of \( {c}^{\prime }\left( {x, y}\right) \) can be determined by an application of the Max-Flow Min-Cut Algorithm (7.9). Therefore the parameter \( {\kappa }^{\prime } \) can obviously be determined by \( \left( \begin{array}{l} n \\ 2 \end{array}\right) \) applications of that algorithm. However, the function \( {c}^{\prime } \) takes at most \( n - 1 \) distinct values (Exercise 9.3.13b). Moreover, Gomory and Hu (1961) have shown that \( {\kappa }^{\prime } \) can be computed by just \( n - 1 \) applications of the Max-Flow Min-Cut Algorithm. A description of their approach is given in Section 9.6. ## ESsential Edge Connectivity The vertex and edge connectivities of a graph \( G \), and its minimum degree, are related by the following basic inequalities (Exercise 9.3.2). \[ \kappa \leq {\kappa }^{\prime } \leq \delta \] Thus, for 3-regular graphs, the connectivity and edge connectivity do not exceed three. They are, moreover, always equal for such graphs (Exercise 9.3.5). These two measures of connectivity therefore fail to distinguish between the triangular prism \( {K}_{3}▱{K}_{2} \) and the complete bipartite graph \( {K}_{3,3} \), both of which are 3-regular graphs with connectivity and edge connectivity equal to three. Nonetheless, one has the distinct impression that \( {K}_{3,3} \) is better connected than \( {K}_{3}▱{K}_{2} \) . Indeed, \( {K}_{3}▱{K}_{2} \) has a 3-edge cut which separates the graph into two nontrivial subgraphs, whereas \( {K}_{3,3} \) has no such cut. Recall that a trivial edge cut is one associated with a single vertex. A \( k \) -edge-connected graph is termed essentially \( \left( {k + 1}\right) \) -edge-connected if all of its \( k \) -edge cuts are trivial. For example, \( {K}_{3,3} \) is essentially 4-edge-connected whereas \( {K}_{3}▱{K}_{2} \) is not. If a \( k \) -edge-connected graph has a \( k \) -edge cut \( \partial \left( X\right) \), the graphs \( G/X \) and \( G/\bar{X} \) (obtained by shrinking \( X \) to a single vertex \( x \) and \( \bar{X} \mathrel{\text{:=}} V \smallsetminus X \) to a single vertex \( \bar{x} \), respectively) are also \( k \) -edge-connected (Exercise 9.3.8). By iterating this shrinking procedure, any \( k \) -edge-connected graph with \( k \geq 1 \), can be ’decomposed’ into a set of essentially \( \left( {k + 1}\right) \) -edge-connected graphs. For many problems, it is enough to treat each of these ’components’ separately. (When \( k = 0 \) - that is, when the graph is disconnected - this procedure corresponds to considering each of its components individually.) The notion of essential edge connectivity is particularly useful for 3-regular graphs. For instance, to show that a 3-connected 3-regular graph has a cycle double cover, it suffices to verify that each of its essentially 4-edge-connected components has one; the individual cycle double covers can then be spliced together to yield a cycle double cover of the entire graph (Exercise 9.3.9). ## Connectivity in Digraphs The definitions of connectivity and edge connectivity have straightforward exten

Proposition 9.5 THE FAN LEMMA Let \( G \) be a \( k \) -connected graph, let \( x \) be a vertex of \( G \), and let \( Y \subseteq V \smallsetminus \{ x\} \) be a set of at least \( k \) vertices of \( G \) . Then there exists a \( k \) -fan in \( G \) from \( x \) to \( Y \) .

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Theorem 1. \( {\lambda }_{m}\left( x\right) \) is irreducible in the rational field. Proof. We observe first that \( {\lambda }_{m}\left( x\right) \) has integer coefficients. For, assuming this holds for every \( {\lambda }_{d}\left( x\right), d < m \), and setting \( p\left( x\right) = \mathop{\prod }\limits_{\substack{{d \mid m} \\ {1 \leq d < m} }}{\lambda }_{d}\left( x\right) \), we obtain by the usual division algorithm that \( {x}^{m} - 1 = p\left( x\right) q\left( x\right) + r\left( x\right) \) where \( q\left( x\right) \) and \( r\left( x\right) \) e \( I\left\lbrack x\right\rbrack \) and \( \deg r\left( x\right) < \deg p\left( x\right) \) . On the other hand, we have \( {x}^{m} - 1 = \) \( p\left( x\right) {\lambda }_{m}\left( x\right) \), so by the uniqueness of the quotient and remainder, \( {\lambda }_{m}\left( x\right) = q\left( x\right) \) has integer coefficients. Now suppose that \( {\lambda }_{m}\left( x\right) = \) \( h\left( x\right) k\left( x\right) \) where \( h\left( x\right) \) is irreducible in \( {R}_{0}\left\lbrack x\right\rbrack \) and \( \deg h\left( x\right) \geq 1 \) . By Gauss’ lemma (Vol. I, p. 125) we may assume that \( h\left( x\right) \) and \( k\left( x\right) \) have integer coefficients and leading coefficients 1. Let \( p \) be a prime integer such that \( p \nmid m \) and let \( \zeta \) be a root of \( h\left( x\right) \) . We shall show that \( {\zeta }^{p} \) is a root of \( h\left( x\right) \) . Since \( \left( {p, m}\right) = 1,{\zeta }^{p} \) is a primitive \( m \) -th root of 1 and, if \( {\zeta }^{p} \) is not a root of \( h\left( x\right) ,{\zeta }^{p} \) is a root of \( k\left( x\right) \) ; consequently \( \zeta \) is a root of \( k\left( {x}^{p}\right) \) . Since \( h\left( x\right) \) is irreducible in \( {R}_{0}\left\lbrack x\right\rbrack \) and has \( \zeta \) as a root, \( h\left( x\right) \mid k\left( {x}^{p}\right) \) . It follows (as above) that \( k\left( {x}^{p}\right) = h\left( x\right) l\left( x\right) \), where \( l\left( x\right) \) has integer coefficients and leading coefficient 1. Also we have \( {x}^{m} - 1 = {\lambda }_{m}\left( x\right) p\left( x\right) = \) \( h\left( x\right) k\left( x\right) p\left( x\right) \) and all of these polynomials have integer coefficients and leading coefficients 1 . We now pass to congruences modulo \( p \) or, what is the same thing, to relations in the polynomial ring \( {I}_{p}\left\lbrack x\right\rbrack \) . Then we obtain (5) \[ {x}^{m} - \overrightarrow{1} = \overrightarrow{h}\left( x\right) \overrightarrow{k}\left( x\right) \overrightarrow{p}\left( x\right) \] where in general, if \( f\left( x\right) = {a}_{0}{x}^{n} + {a}_{1}{x}^{n - 1} + \cdots + {a}_{n}{\varepsilon I}\left\lbrack x\right\rbrack \), then \( \bar{f}\left( x\right) = {\bar{a}}_{0}{x}^{n} + {\bar{a}}_{1}{x}^{n - 1} + \cdots + {\bar{a}}_{n},{\bar{a}}_{i} = {a}_{i} + \left( p\right) \) in \( {I}_{p} \) . Similarly, we have \( \bar{k}\left( {x}^{p}\right) = \bar{h}\left( x\right) \bar{l}\left( x\right) \) . On the other hand, using \( {\bar{a}}^{p} = \bar{a} \) for every integer \( a \), we see that \[ \bar{f}{\left( x\right) }^{p} = {\left( {\bar{a}}_{0}{x}^{n} + \cdots + {\bar{a}}_{n}\right) }^{p} = {\bar{a}}_{0}{}^{p}{x}^{pn} + \cdots + {\bar{a}}_{n}{}^{p} \] \[ = {\bar{a}}_{0}{x}^{pn} + \cdots + {\bar{a}}_{n} = \bar{f}\left( {x}^{p}\right) \] for any polynomial \( f\left( x\right) \) . Hence \( \bar{k}{\left( x\right) }^{p} = \bar{k}\left( {x}^{p}\right) = \bar{h}\left( x\right) \bar{l}\left( x\right) \) which implies that \( \left( {h\left( x\right), k\left( x\right) }\right) \neq 1 \) . Then (5) shows that \( {x}^{m} - 1 \) has multiple roots in its splitting field over \( {I}_{p} \) . Since \( p \times m \) this is impossible and so we have proved that \( {\zeta }^{p} \) is a root of \( h\left( x\right) \) for every prime \( p \) satisfying \( p \times m \) . A repetition of this process shows that \( {\zeta }^{r} \) is a root of \( h\left( x\right) \) for every integer \( r \) prime to \( m \) . Since any primitive \( m \) -th root of 1 has the form \( {\zeta }^{r},\left( {r, m}\right) = 1 \) we see that every primitive \( m \) -th root of 1 is a root of \( h\left( x\right) \) . Hence \( h\left( x\right) = {\lambda }_{m}\left( x\right) \) and \( {\lambda }_{m}\left( x\right) \) is irreducible in \( {R}_{0}\left\lbrack x\right\rbrack \) . We now see that \( {\lambda }_{m}\left( x\right) \) is the minimum polynomial over \( {R}_{0} \) of any primitive \( m \) -th root of 1 . Since \( {\mathrm{P}}^{\left( m\right) } = {R}_{0}\left( \zeta \right) ,\zeta \) primitive we have established the formula (6) \[ \left\lbrack {{\mathrm{P}}^{\left( m\right) } : {R}_{0}}\right\rbrack = \varphi \left( m\right) . \] This implies that \( \left( {G : 1}\right) = \varphi \left( m\right) \) for the Galois group \( G \) of \( {\mathrm{P}}^{\left( m\right) }/{R}_{0} \) . Since \( \left( {U\left( m\right) : 1}\right) = \varphi \left( m\right) \) and \( G \) is isomorphic to a subgroup of \( U\left( m\right) \), this proves Theorem 2. Let \( {\mathrm{P}}^{\left( m\right) } \) be the cyclotomic field of order \( m \) over the rationals \( {R}_{0} \) . Then the Galois group of \( {\mathrm{P}}^{\left( m\right) }/{R}_{0} \) is isomorphic to \( U\left( m\right) \), the multiplicative group of units in the ring \( I/\left( m\right) \) . We shall now proceed to determine the structure of the Galois group \( G \) or, what is the same thing, that of \( U\left( m\right) \) . It is easy to see that, if \( m = {p}_{1}{}^{{e}_{1}}{p}_{2}{}^{{e}_{2}}\cdots {p}_{r}{}^{{e}_{r}} \) where the \( {p}_{i} \) are distinct primes, then \( U\left( m\right) \) is isomorphic to the direct product of the \( U\left( {{p}_{i}{}^{{e}_{i}}}\right) \) . For this reason we shall confine our attention to the case \( m = {p}^{e} \) a prime power. Then \( U\left( {p}^{e}\right) \) is a commutative group of order \( \varphi \left( {p}^{e}\right) = \) \( {p}^{e} - {p}^{e - 1} = {p}^{e - 1}\left( {p - 1}\right) \) . We prove first Theorem 3. If \( p \) is an odd prime, then the multiplicative group \( U\left( {p}^{e}\right) \) of units in \( I/\left( {p}^{e}\right) \) is cyclic. Proof. Since the order of this group is \( {p}^{e - 1}\left( {p - 1}\right), U\left( {p}^{e}\right) \) is a direct product of its subgroup \( H \) of order \( {p}^{e - 1} \) consisting of the elements which satisfy \( {x}^{{p}^{s - 1}} = 1 \) and the subgroup \( K \) of order \( p - 1 \) of the elements satisfying \( {x}^{p - 1} = 1 \) . It suffices to show that both \( H \) and \( K \) are cyclic since the direct product of cyclic groups having relatively prime orders is cyclic. If \( e = 1 \), then \( U\left( p\right) = K \) is the multiplicative group of the field \( I/\left( p\right) \) and this is cyclic. Hence we can choose an integer \( a \) such that \( a + \left( p\right) ,{a}^{2} + \) \( \left( p\right) ,\cdots ,{a}^{p - 1} + \left( p\right) \) are distinct in \( I/\left( p\right) \) . Set \( b = {a}^{{p}^{\sigma - 1}} \) . Since \( \left( {a, p}\right) = 1,\left( {b,{p}^{e}}\right) = 1 \) and \( b + \left( {p}^{e}\right) \) and \( a + \left( {p}^{e}\right) {\varepsilon U}\left( {p}^{e}\right) \) . Also \( {b}^{p - 1} = {\left( {a}^{{p}^{e - 1}}\right) }^{p - 1} = {a}^{\varphi \left( {p}^{e}\right) } \equiv 1\left( {\;\operatorname{mod}\;{p}^{e}}\right) \) so \( b + \left( {p}^{e}\right) \) e \( K \) . Since \( b = {a}^{{p}^{q - 1}} \equiv a\left( {\;\operatorname{mod}\;p}\right), b + \left( p\right) ,{b}^{2} + \left( p\right) ,\cdots ,{b}^{p - 1} + \left( p\right) \) are distinct. Hence also \( b + \left( {p}^{e}\right) ,{b}^{2} + \left( {p}^{e}\right) ,\cdots ,{b}^{p - 1} + \left( {p}^{e}\right) \) are distinct. This implies that the order of \( b + \left( {p}^{e}\right) \) is precisely \( p - 1 \) . Since \( \left( {K : 1}\right) = p - 1 \), it follows that \( K \) is cyclic with generator \( b + \left( {p}^{e}\right) \) . It remains to prove that \( H \) is cyclic, and we may assume that \( e \geq 2 \), since, otherwise, \( H = \left( 1\right) \) and the result is clear. Assuming \( e \geq 2 \), we can conclude that \( H \) is a direct product of \( k \geq 1 \) cyclic groups of order \( {p}^{{e}_{i}},{e}_{i} \geq 1 \) . Then the number of solutions of the equation \( {x}^{p} = 1,{x\varepsilon H} \) is \( {p}^{k} \) . Hence it will be enough to show that the number of integers \( n,0 < n < {p}^{e} \) , satisfying \( {n}^{p} \equiv 1\left( {\;\operatorname{mod}\;{p}^{e}}\right) \) does not exceed \( p \) . Now if \( n \) satisfies these conditions, then, since \( {n}^{p} \equiv n\left( {\;\operatorname{mod}\;p}\right) \), we have \( n \equiv 1 \) (mod \( p \) ). Then if \( n \neq 1 \), we may write \( n = 1 + y{p}^{f} + z{p}^{f + 1} \) where \( 1 \leq f \leq e - 1,0 < y < p \), and \( z \) is a non-negative integer. Then \[ {n}^{p} = 1 + \left( \begin{array}{l} p \\ 1 \end{array}\right) \left( {y + {zp}}\right) {p}^{f} + \left( \begin{array}{l} p \\ 2 \end{array}\right) {\left( y + zp\right) }^{2}{p}^{2f} \] \[ + \cdots + {\left( y + zp\right) }^{p}{p}^{pf} \] \[ \equiv 1 + y{p}^{f + 1}\left( {\;\operatorname{mod}\;{p}^{f + 2}}\right) . \] If \( {n}^{p} \equiv 1\left( {\;\operatorname{mod}\;{p}^{e}}\right) \) and \( f < e - 1 \), this gives \( y{p}^{f + 1} \equiv 0({\;\operatorname{mod}\;} \) \( \left. {p}^{f + 2}\right) \) so \( y \equiv 0\left( {\;\operatorname{mod}\;p}\right) \) contrary to \( 0 < y < p \) . Hence we see that, if \( 1 < n < {p}^{e} \) satisfies \( {n}^{p} \equiv 1\left( {\;\operatorname{mod}\;{p}^{e}}\right) \), then \( n = 1 + \) \( y{p}^{e - 1},0 < y < p \) . This gives altogether at most \( p \) solutions including 1 and completes the proof of the theorem. We consider next the case of the prime 2 in the following Theorem 4. \( U\left( 2\right) \) and \( U\left( 4\right) \) are cyclic and, if \( e \geq 3 \), then \( U\left( {2}^{e}\right) \) is a direct product

Theorem 1. \( {\lambda }_{m}\left( x\right) \) is irreducible in the rational field.

We observe first that \( {\lambda }_{m}\left( x\right) \) has integer coefficients. For, assuming this holds for every \( {\lambda }_{d}\left( x\right), d < m \), and setting \( p\left( x\right) = \mathop{\prod }\limits_{\substack{{d \mid m} \\ {1 \leq d < m} }}{\lambda }_{d}\left( x\right) \), we obtain by the usual division algorithm that \( {x}^{m} - 1 = p\left( x\right) q\left( x\right) + r\left( x\right) \) where \( q\left( x\right) \) and \( r\left( x\right) \) are in \( I\left\lbrack x\right\rbrack \) and \( \deg r\left( x\right) < \deg p\left( x\right) \). On the other hand, we have \( {x}^{m} - 1 = p\left( x\right) {\lambda }_{m}\left( x\right) \), so by the uniqueness of the quotient and remainder, \( {\lambda }_{m}\left( x\right) = q\left( x\right) \) has integer coefficients. Now suppose that \( {\lambda }_{m}\left( x\right) = h\left( x\right) k\left( x\right) \) where \( h\left( x\right) \) is irreducible in \( {R}_{0}\left\lbrack x\right\rbrack \) and \( \deg h\left( x\right) \geq 1 \). By Gauss’ lemma (Vol. I, p. 125) we may assume that \( h\left( x\right) \) and \( k\left( x\right) \) have integer coefficients and leading coefficients 1. Let \( p \) be a prime integer such that \( p \nmid m \) and let \( \zeta \) be a root of \( h

Exercise 9.1.4 Let \( \pi \) be a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . Show that \( {\alpha }^{N\left( \pi \right) - 1} \equiv 1\left( {\;\operatorname{mod}\;\pi }\right) \) for all \( \alpha \in \mathbb{Z}\left\lbrack \rho \right\rbrack \) which are coprime to \( \pi \) . Exercise 9.1.5 Let \( \pi \) be a prime not associated to \( \left( {1 - \rho }\right) \) . First show that \( 3 \mid N\left( \pi \right) - 1 \) . If \( \left( {\alpha ,\pi }\right) = 1 \), show that there is a unique integer \( m = 0,1 \), or 2 such that \[ {\alpha }^{\left( {N\left( \pi \right) - 1}\right) /3} \equiv {\rho }^{m}\;\left( {\;\operatorname{mod}\;\pi }\right) \] Let \( N\left( \pi \right) \neq 3 \) . We define the cubic residue character of \( \alpha \left( {\;\operatorname{mod}\;\pi }\right) \) by the symbol \( {\left( \alpha /\pi \right) }_{3} \) as follows: (i) \( {\left( \alpha /\pi \right) }_{3} = 0 \) if \( \pi \mid \alpha \) ; (ii) \( {\alpha }^{\left( {N\left( \pi \right) - 1}\right) /3} \equiv {\left( \alpha /\pi \right) }_{3}\left( {\;\operatorname{mod}\;\pi }\right) \) where \( {\left( \alpha /\pi \right) }_{3} \) is the unique cube root of unity determined by the previous exercise. Exercise 9.1.6 Show that: (a) \( {\left( \alpha /\pi \right) }_{3} = 1 \) if and only if \( {x}^{3} \equiv \alpha \left( {\;\operatorname{mod}\;\pi }\right) \) is solvable in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) ; (b) \( {\left( \alpha \beta /\pi \right) }_{3} = {\left( \alpha /\pi \right) }_{3}{\left( \beta /\pi \right) }_{3} \) ; and (c) if \( \alpha \equiv \beta \left( {\;\operatorname{mod}\;\pi }\right) \), then \( {\left( \alpha /\pi \right) }_{3} = {\left( \beta /\pi \right) }_{3} \) . Let us now define the cubic character \( {\chi }_{\pi }\left( \alpha \right) = {\left( \alpha /\pi \right) }_{3} \) . Exercise 9.1.7 Show that: (a) \( \overline{{\chi }_{\pi }\left( \alpha \right) } = {\chi }_{\pi }{\left( \alpha \right) }^{2} = {\chi }_{\pi }\left( {\alpha }^{2}\right) \) ; and (b) \( \overline{{\chi }_{\pi }\left( \alpha \right) } = {\chi }_{\bar{\pi }}\left( \alpha \right) \) . Exercise 9.1.8 If \( q \equiv 2\left( {\;\operatorname{mod}\;3}\right) \), show that \( {\chi }_{q}\left( \bar{\alpha }\right) = {\chi }_{q}\left( {\alpha }^{2}\right) \) and \( {\chi }_{q}\left( n\right) = 1 \) if \( n \) is a rational integer coprime to \( q \) . This exercise shows that any rational integer is a cubic residue mod \( q \) . If \( \pi \) is prime in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \), we say \( \pi \) is primary if \( \pi \equiv 2\left( {\;\operatorname{mod}\;3}\right) \) . Therefore if \( q \equiv 2\left( {\;\operatorname{mod}\;3}\right) \), then \( q \) is primary in \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . If \( \pi = a + {b\rho } \), then this means \( a \equiv 2\left( {\;\operatorname{mod}\;3}\right) \) and \( b \equiv 0\left( {\;\operatorname{mod}\;3}\right) \) . Exercise 9.1.9 Let \( N\left( \pi \right) = p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) . Among the associates of \( \pi \), show there is a unique one which is primary. We can now state the law of cubic reciprocity: let \( {\pi }_{1},{\pi }_{2} \) be primary. Suppose \( N\left( {\pi }_{1}\right), N\left( {\pi }_{2}\right) \neq 3 \) and \( N\left( {\pi }_{1}\right) \neq N\left( {\pi }_{2}\right) \) . Then \[ {\chi }_{{\pi }_{1}}\left( {\pi }_{2}\right) = {\chi }_{{\pi }_{2}}\left( {\pi }_{1}\right) \] To prove the law of cubic reciprocity, we will introduce Jacobi sums and more general Gauss sums than the ones used in Chapter 7. Let \( {\mathbb{F}}_{p} \) denote the finite field of \( p \) elements. A multiplicative character on \( {\mathbb{F}}_{p} \) is a homomorphism \( \chi : {\mathbb{F}}_{p}^{ \times } \rightarrow {\mathbb{C}}^{ \times } \) . The Legendre symbol \( \left( {a/p}\right) \) is an example of such a character. Another example is the trivial character \( {\chi }_{0} \) defined by \( {\chi }_{0}\left( a\right) = 1 \) for all \( a \in {\mathbb{F}}_{p}^{ \times } \) . It is useful to extend the definition of \( \chi \) to all of \( {\mathbb{F}}_{p} \) . We set \( \chi \left( 0\right) = 0 \) for \( \chi \neq {\chi }_{0} \) and \( {\chi }_{0}\left( 0\right) = 1 \) . For \( a \in {\mathbb{F}}_{p}^{ \times } \), define the Gauss sum \[ {g}_{a}\left( \chi \right) = \mathop{\sum }\limits_{{t \in {\mathbb{F}}_{p}}}\chi \left( t\right) {\zeta }^{at} \] where \( \zeta = {e}^{{2\pi i}/p} \) is a primitive \( p \) th root of unity. We also write \( g\left( \chi \right) \) for \( {g}_{1}\left( \chi \right) \) . Theorem 9.1.10 If \( \chi \neq {\chi }_{0} \), then \( \left| {g\left( \chi \right) }\right| = \sqrt{p} \) . Proof. We first observe that \( a \neq 0 \) and \( \chi \neq {\chi }_{0} \) together imply that \( {g}_{a}\left( \chi \right) = \chi \left( {a}^{-1}\right) g\left( \chi \right) \) because \[ \chi \left( a\right) {g}_{a}\left( \chi \right) = \chi \left( a\right) \mathop{\sum }\limits_{{t \in {\mathbb{F}}_{p}}}\chi \left( t\right) {\zeta }^{at} \] \[ = \mathop{\sum }\limits_{{t \in {\mathbb{F}}_{p}}}\chi \left( {at}\right) {\zeta }^{at} \] \[ = g\left( \chi \right) \text{.} \] With our conventions that \( {\chi }_{0}\left( 0\right) = 1 \), we see for \( a \neq 0 \) , \[ {g}_{a}\left( {\chi }_{0}\right) = \mathop{\sum }\limits_{{t \in {\mathbb{F}}_{p}}}{\zeta }^{at} = 0 \] since this is just the sum of the \( p \) th roots of unity. Finally, \( {g}_{0}\left( \chi \right) = 0 \) if \( \chi \neq {\chi }_{0} \) and \( {g}_{0}\left( {\chi }_{0}\right) = p \) . Now, by our first observation, \[ \mathop{\sum }\limits_{{a \in {\mathbb{F}}_{p}}}{g}_{a}\left( \chi \right) \overline{{g}_{a}\left( \chi \right) } = {\left| g\left( \chi \right) \right| }^{2}\left( {p - 1}\right) . \] On the other hand, \[ \mathop{\sum }\limits_{{a \in {\mathbb{F}}_{p}}}{g}_{a}\left( \chi \right) \overline{{g}_{a}\left( \chi \right) } = \mathop{\sum }\limits_{{s \in {\mathbb{F}}_{p}}}\mathop{\sum }\limits_{{t \in {\mathbb{F}}_{p}}}\chi \left( s\right) \chi \left( t\right) \mathop{\sum }\limits_{{a \in {\mathbb{F}}_{p}}}{\zeta }^{{as} - {at}}. \] If \( s \neq t \), the innermost sum is zero, being the sum of all the \( p \) th roots of unity. If \( s = t \), the sum is \( p \) . Hence \( {\left| g\left( \chi \right) \right| }^{2} = p \) . Let \( {\chi }_{1},{\chi }_{2},\ldots ,{\chi }_{r} \) be characters of \( {\mathbb{F}}_{p} \) . A Jacobi sum is defined by \[ J\left( {{\chi }_{1},\ldots ,{\chi }_{r}}\right) = \mathop{\sum }\limits_{{{t}_{1} + \cdots + {t}_{r} = 1}}{\chi }_{1}\left( {t}_{1}\right) \cdots {\chi }_{r}\left( {t}_{r}\right) \] where the summation is over all solutions of \( {t}_{1} + \cdots + {t}_{r} = 1 \) in \( {\mathbb{F}}_{p} \) . The relationship between Gauss sums and Jacobi sums is given by the following exercise. Exercise 9.1.11 If \( {\chi }_{1},\ldots ,{\chi }_{r} \) are nontrivial and the product \( {\chi }_{1}\cdots {\chi }_{r} \) is also nontrivial, prove that \( g\left( {\chi }_{1}\right) \cdots g\left( {\chi }_{r}\right) = J\left( {{\chi }_{1},\ldots ,{\chi }_{r}}\right) g\left( {{\chi }_{1}\cdots {\chi }_{r}}\right) \) . Exercise 9.1.12 If \( {\chi }_{1},\ldots ,{\chi }_{r} \) are nontrivial, and \( {\chi }_{1}\cdots {\chi }_{r} \) is trivial, show that \[ g\left( {\chi }_{1}\right) \cdots g\left( {\chi }_{r}\right) = {\chi }_{r}\left( {-1}\right) {pJ}\left( {{\chi }_{1},\ldots ,{\chi }_{r - 1}}\right) . \] We are now ready to prove the cubic reciprocity law. It will be convenient to work in the ring \( \Omega \) of all algebraic integers. Lemma 9.1.13 Let \( \pi \) be a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) such that \( N\left( \pi \right) = p \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) . The character \( {\chi }_{\pi } \) introduced above can be viewed as a character of the finite field \( \mathbb{Z}\left\lbrack \rho \right\rbrack /\left( \pi \right) \) of \( p \) elements. \( J\left( {{\chi }_{\pi },{\chi }_{\pi }}\right) = \pi \) . Proof. If \( \chi \) is any cubic character, Exercise 9.1.12 shows that \( g{\left( \chi \right) }^{3} = \) \( {pJ}\left( {\chi ,\chi }\right) \) since \( \chi \left( {-1}\right) = 1 \) . We can write \( J\left( {\chi ,\chi }\right) = a + {b\rho } \) for some \( a, b \in \mathbb{Z} \) . But \[ g{\left( \chi \right) }^{3} = {\left( \mathop{\sum }\limits_{t}\chi \left( t\right) {\zeta }^{t}\right) }^{3} \] \[ \equiv \mathop{\sum }\limits_{t}{\chi }^{3}\left( t\right) {\zeta }^{3t}\;\left( {{\;\operatorname{mod}\;3}\Omega }\right) \] \[ \equiv \mathop{\sum }\limits_{{t \neq 0}}{\zeta }^{3t}\;\left( {{\;\operatorname{mod}\;3}\Omega }\right) \] \[ \equiv - 1\;\left( {{\;\operatorname{mod}\;3}\Omega }\right) \text{.} \] Therefore, \( a + {b\rho } \equiv - 1\left( {{\;\operatorname{mod}\;3}\Omega }\right) \) . In a similar way, \[ g{\left( \bar{\chi }\right) }^{3} \equiv a + b\bar{\rho } \equiv - 1\;\left( {{\;\operatorname{mod}\;3}\Omega }\right) . \] Thus, \( b\sqrt{-3} \equiv 0\left( {{\;\operatorname{mod}\;3}\Omega }\right) \) which means \( - 3{b}^{2}/9 \) is an algebraic integer and by Exercise 3.1.2, it is an ordinary integer. Thus, \( b \equiv 0\left( {\;\operatorname{mod}\;3}\right) \) and \( a \equiv - 1\left( {\;\operatorname{mod}\;3}\right) \) . Also, from Exercise 9.1.12 and Theorem 9.1.10, it is clear that \( {\left| J\left( \chi ,\chi \right) \right| }^{2} = p = J\left( {\chi ,\chi }\right) \overline{J\left( {\chi ,\chi }\right) } \) . Therefore, \( J\left( {\chi ,\chi }\right) \) is a primary prime of norm \( p \) . Set \( J\left( {{\chi }_{\pi },{\chi }_{\pi }}\right) = {\pi }^{\prime } \) . Since \( \pi \bar{\pi } = p = {\pi }^{\prime }\overline{{\pi }^{\prime }} \), we have \( \pi \mid {\pi }^{\prime } \) or \( \pi \mid \overline{{\pi }^{\prime }} \) . We want to eliminate the latter possibility. By definition, \[ J\left( {{\chi }_{\pi },{\chi }_{\pi }}\right) = \mathop{\sum }\l

Exercise 9.1.4 Let \( \pi \) be a prime of \( \mathbb{Z}\left\lbrack \rho \right\rbrack \) . Show that \( {\alpha }^{N\left( \pi \right) - 1} \equiv 1\left( {\;\operatorname{mod}\;\pi }\right) \) for all \( \alpha \in \mathbb{Z}\left\lbrack \rho \right\rbrack \) which are coprime to \( \pi \) .

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Theorem 1. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \) . Then \( {\mathbf{R}}^{2} - J \) has exactly two components. Proof. Let \( N \) be a "strip neighborhood" of \( J \), formed by small convex polyhedral neighborhoods of the edges and vertices of \( J \) . (More precisely, we mean the edges and vertices of a triangulation of \( J \) .) Below and hereafter, pictures of polyhedra will not necessarily look like polyhedra. Only a sample of \( N \) is indicated in Figure 2.1.  Figure 2.1 Lemma 1. \( {\mathbf{R}}^{2} - J \) has at most two components. Proof. Starting at any point \( P \) of \( N - J \), we can work our way around the polygon, along a path in \( N - J \), until we get to either \( {P}_{1} \) or \( {P}_{2} \) . (See Figure 2.2.) From this the lemma follows, because every point \( Q \) of \( {\mathbf{R}}^{2} - J \) can be joined to some point \( P \) of \( N - J \) by a linear segment in \( {\mathbf{R}}^{2} - J \) .  Figure 2.2 It is possible a priori that \( N - J \) has only one component. If so, \( N \) would be a Möbius band. (See Section 21 below.) But this is ruled out by the next lemma. Lemma 2. \( {\mathbf{R}}^{2} - J \) has at least two components. Proof. We choose the axes in general position, in the sense that no horizontal line contains more than one of the vertices of \( J \) . (This can be done, because there are only a finite number of directions that we need to avoid. Hereafter, the phrase "in general position" will be defined in a variety of ways, in a variety of cases. In each case, the intuitive meaning will be the same: general position is a situation which occurs with probability 1 when certain choices are made at random.) For each point \( P \) of \( {\mathbf{R}}^{2} \), let \( {L}_{P} \) be the horizontal line through \( P \) . The index Ind \( P \) of a point \( P \) of \( {\mathbf{R}}^{2} - J \) is defined as follows. (1) If \( {L}_{P} \) contains no vertex of \( J \), then Ind \( P \) is the number of points of \( {L}_{P} \cap J \) that lie to the left of \( P \), reduced modulo 2. Thus Ind \( P \) is 0 or 1 . (2) If \( {L}_{P} \) contains a vertex of \( J \), then Ind \( P \) is the number of points of \( {L}^{\prime } \cap J \), lying to the left of \( P \), reduced modulo 2, where \( {L}^{\prime } \) is a horizontal line lying "slightly above" or "slightly below" \( {L}_{P} \) . Here the phrases in quotation marks mean that no vertex of \( J \) lies on \( {L}^{\prime } \), or between \( {L}_{P} \) and \( {L}^{\prime } \) . It makes no difference whether \( {L}^{\prime } \) lies above or below. The three possibilities for \( J \), relative to \( L \) , are shown in Figure 2.3. In each case, the two possible positions for \( {L}^{\prime } \) give the same index for \( P \) . Evidently the function \[ f : {\mathbf{R}}^{2} - J \rightarrow \{ 0,1\} \] \[ f : P \mapsto \text{Ind}P \]  is a mapping; if Ind \( P = i \), then Ind \( {P}^{\prime } = i \) when \( {P}^{\prime } \) is sufficiently close to \( P \) . The set \( {f}^{-1}\left( 0\right) \) is nonempty; every point above all of \( J \) belongs to \( {f}^{-1}\left( 0\right) \) . To show that \( {f}^{-1}\left( 1\right) \neq \varnothing \), let \( Q \) be a point of \( J \), such that \( {L}_{O} \) contains no vertex of \( J \) . Let \( {P}_{1} \) be the leftmost point of \( J \) on \( {L}_{Q} \) . Let \( P \) be a point of \( {L}_{Q} \), slightly to the right of \( {P}_{1} \), in the sense that \( P \notin J \), and no point between \( {P}_{1} \) and \( P \) belongs to \( J \) . Then Ind \( P = 1 \) . Therefore \( {\mathbf{R}}^{2} - J \) is not connected; it is the union of the disjoint nonempty open sets \( {f}^{-1}\left( 0\right) \) and \( {f}^{-1}\left( 1\right) \) . The bounded component \( I \) of \( {\mathbf{R}}^{2} - J \) is called the interior of \( J \), and the unbounded component \( E \) is called the exterior. Theorem 2. Let \( I \) be the interior of the polygon \( J \) in \( {\mathbf{R}}^{2} \) . Then \( \bar{I} \) is a finite polyhedron. That is, there is a finite complex \( K \) in \( {\mathbf{R}}^{2} \) such that \( \left| K\right| = \bar{I} \) . Proof. Let \( {L}_{1},{L}_{2},\ldots ,{L}_{n} \) be the lines that contain edges of \( J \) . These lines are finite in number, and each intersects the union of the others in a finite number of points. Note that some sets \( {L}_{i} \cap I \) may not be connected; this does not matter. Each line \( {L}_{i} \) decomposes \( {\mathbf{R}}^{2} \) into two closed half-planes \( {H}_{i},{H}_{i}^{\prime } \) ; and any finite intersection of closed half-planes is closed and convex. Therefore \( \mathop{\bigcup }\limits_{{i = 1}}^{n}{L}_{i} \) decomposes \( {\mathbf{R}}^{2} \) into a finite collection of closed convex regions \( {R}_{1},{R}_{2},\ldots ,{R}_{m} \), such that for each \( j \) we have \( \operatorname{Fr}{R}_{j} \subset \) \( \mathop{\bigcup }\limits_{{i = 1}}^{n}{L}_{i} \) . Now \( {R}_{i} \cap J \subset \operatorname{Fr}{R}_{i} \) for each \( j \) . It follows that for each \( j \) we have either \( {R}_{j} \cap \bar{I} \subset \check{J} \) or \( {R}_{j} \subset \bar{I} \) . Thus \( \bar{I} \) is the union of the sets \( {R}_{j} \) that lie in \( \bar{I} \) , and so it is merely a matter of notation to suppose that \[ \bar{I} = \mathop{\bigcup }\limits_{{j = 1}}^{k}{R}_{j} \] For each \( j \leq k,\operatorname{Fr}{R}_{j} \) is the union of a finite number of 1 -simplexes. We choose the triangulations of the sets \( \operatorname{Fr}{R}_{j} \) to be minimal, in the sense that if two edges of \( {R}_{j} \) have an end-point in common, then they are not collinear. For each \( j \), we choose a point \( {w}_{j} \) of \( {R}_{j} - \operatorname{Fr}{R}_{j} \), and for each  Figure 2.4 1-simplex \( v{v}^{\prime } \) of \( \operatorname{Fr}{R}_{j} \) we form the 2-simplex \( {w}_{j}v{v}^{\prime } \) . (See Figure 2.4.) This gives a triangulation of \( {R}_{j} \) . The union of these is a triangulation of \( I \) . We recall that an \( \operatorname{arc}A \) is a 1-cell, that is, the image of a 1-simplex, say, \( \left\lbrack {0,1}\right\rbrack \subset \mathbf{R} \), under a homeomorphism \( f \) . Obviously \( \left\lbrack {0,1}\right\rbrack \) is a 1-manifold with boundary; the entire space \( \left\lbrack {0,1}\right\rbrack \) is a 1-cell neighborhood of each of its points. And Int \( \left\lbrack {0,1}\right\rbrack \) and Bd \( \left\lbrack {0,1}\right\rbrack \) are identifiable. Evidently the open interval \( \left( {0,1}\right) \) lies in Int \( \left\lbrack {0,1}\right\rbrack \) ; it is a Euclidean neighborhood of each of its points. And \( \{ 0,1\} \subset \operatorname{Bd}\left\lbrack {0,1}\right\rbrack \) . The reason is that for each \( x \in \mathbf{R},\mathbf{R} - \{ x\} \) is not connected, while if \( U \) is a connected open set in \( \left\lbrack {0,1}\right\rbrack \), containing 0, then \( U - \{ 0\} \) is connected. Similarly for 1. Therefore Int \( \left\lbrack {0,1}\right\rbrack = \left( {0,1}\right) \) and Bd \( \left\lbrack {0,1}\right\rbrack = \{ 0,1\} \) . It follows immediately that if \( A = f\left( \left\lbrack {0,1}\right\rbrack \right) \) is an arc, with \( P = f\left( 0\right) \) and \( Q = f\left( 1\right) \), then \( \operatorname{Bd}A = \{ P, Q\} \) and Int \( A = A - \{ P, Q\} \) . \( P \) and \( Q \) are called the end-points of \( A \), and \( A \) is called an arc between \( P \) and Q. We recall that a broken line \( B \) is a polyhedral arc. Theorem 3. No broken line separates \( {\mathbf{R}}^{2} \) . That is, if \( B \) is a broken line in \( {\mathbf{R}}^{2} \) , then \( {\mathbf{R}}^{2} - B \) is connected. Proof. Form a strip-neighborhood \( N \) of \( B \) . As in the proof of Lemma 1 in the proof of Theorem 1, each point \( P \) of \( N - B \) can be joined to either \( {P}_{1} \) or \( {P}_{2} \) by a path in \( N - B \) . (See Figure 2.5.) But if \( {P}_{1} \) and \( {P}_{2} \) are near an  Figure 2.5 end-point, as in the figure, then \( {P}_{1} \) can be joined to \( {P}_{2} \) by a path in \( N - B \) . Therefore \( N - B \) is connected. Therefore, as in the proof of Theorem 1, \( {\mathbf{R}}^{2} - B \) is connected. Theorem 4. Let \( X \) be a topological space and let \( U \) be an open set. Then \( \operatorname{Fr}U = \bar{U} - U. \) Proof. By definition, \( \operatorname{Fr}U = \bar{U} \cap \overline{X - U} \) . Therefore \( \operatorname{Fr}U \subset \bar{U} \) . Since \( U \) is open, we have \( U \cap \overline{X - U} = \varnothing \) . Since \( \operatorname{Fr}U \subset \overline{X - U} \), it follows that \( \operatorname{Fr}U \) \( \subset \bar{U} - U \) . Next observe that if \( P \in \bar{U} - U \), then \( P \in \bar{U} \) and \( P \in X - U \subset \overline{X - U} \) . Therefore \( \bar{U} - U \subset \operatorname{Fr}U \) . The theorem follows. Theorem 5. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \), with interior \( I \) and exterior \( E \) . Then every point of \( J \) is a limit point both of \( I \) and of \( E \) . Proof. Let \( F = \operatorname{Fr}I = \bar{I} - I \) . Then \( F \) separates \( {\mathbf{R}}^{2} \) : \[ {\mathbf{R}}^{2} - F = I \cup \left( {{\mathbf{R}}^{2} - \bar{I}}\right) \] and the sets on the right are disjoint, open, and nonempty; \( {\mathbf{R}}^{2} - \bar{I} \) contains \( E;F \subset J \), and \( F \) is closed. If \( F \neq J \), then \( F \) lies in a broken line \( B \subset J \) . Now \[ {\mathbf{R}}^{2}

Theorem 1. Let \( J \) be a polygon in \( {\mathbf{R}}^{2} \) . Then \( {\mathbf{R}}^{2} - J \) has exactly two components.

Proof. Let \( N \) be a "strip neighborhood" of \( J \), formed by small convex polyhedral neighborhoods of the edges and vertices of \( J \) . (More precisely, we mean the edges and vertices of a triangulation of \( J \) .) Below and hereafter, pictures of polyhedra will not necessarily look like polyhedra. Only a sample of \( N \) is indicated in Figure 2.1.  Figure 2.1 Lemma 1. \( {\mathbf{R}}^{2} - J \) has at most two components. Proof. Starting at any point \( P \) of \( N - J \), we can work our way around the polygon, along a path in \( N - J \), until we get to either \( {P}_{1} \) or \( {P}_{2} \) . (See Figure 2.2.) From this the lemma follows, because every point \( Q \) of \( {\mathbf{R}}^{2} - J \) can be joined to some point \( P \) of \( N - J \) by a linear segment in \( {\mathbf{R}}^{2} - J \).  Figure 2.2 It is possible a priori that \( N - J \) has only one component. If so, \( N \) would be a Möbius band. (See Section 21 below.) But this is ruled out by the next lemma. Lemma 2. \( {\mathbf{R}}^{2} - J \) has at least two components. Proof. We choose the axes in general position, in the sense that no horizontal line contains more than one of the vertices of \( J \). (This can be done, because there are only a finite number of directions that we need to avoid.) For each point \( P \) of \( {\mathbf{R}}^{2} \), let \( {L}_{P} \) be the horizontal line through \( P \). The index Ind \( P \) of a point \( P \) of \( {\mathbf{R}}^{2} - J \) is defined as follows: (1) If \( {L}_{P} \) contains no vertex of \( J \), then Ind \( P \) is the number of points of \( {L}_{P} \cap J \) that lie to the left of \( P \), reduced modulo 2; thus Ind \( P \) is 0 or 1; (2) If \( {L}_{P} \) contains a vertex of \( J \), then Ind \( P \) is the number of points of \( {L}^{\prime } \cap J \), lying to the left of

Proposition 8.6. \( {f}_{2k} \circ h = {\left( -1\right) }^{k}\mathop{\sum }\limits_{{l = 0}}^{{2k}}{\left( -1\right) }^{l}{f}_{l}^{i}{f}_{{2k} - l}^{i} \) , \( \operatorname{Pf} \circ h = {\left( -1\right) }^{\left\lbrack n/2\right\rbrack }{f}_{n}^{i} \) . Proof. By Lemma 8.1, \[ {\left| \det \left( x{I}_{n} - M\right) \right| }^{2} = \det \left( {h\left( {x{I}_{n} - M}\right) }\right) = \det \left( {x{I}_{2n} - h\left( M\right) }\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{x}^{2\left( {n - k}\right) }{f}_{2k} \circ h\left( M\right) . \] On the other hand, \[ {\left| \det \left( x{I}_{n} - M\right) \right| }^{2} = {\left| \det \left( x{I}_{n} - i\left( -iM\right) \right) \right| }^{2} = {\left| \mathop{\sum }\limits_{{k = 0}}^{n}{\left( -1\right) }^{k}{\left( -i\right) }^{k}{f}_{k}^{i}\left( M\right) {x}^{n - k}\right| }^{2} \] \[ = {\left| {x}^{n} + i{x}^{n - 1}{f}_{1}^{i}\left( M\right) - {x}^{n - 2}{f}_{2}^{i}\left( M\right) - i{x}^{n - 3}{f}_{3}^{i}\left( M\right) + \cdots \right| }^{2} \] \[ = \mid \left( {{x}^{n} - {x}^{n - 2}{f}_{2}^{i}\left( M\right) + {x}^{n - 4}{f}_{4}^{i}\left( M\right) \cdots }\right) \] \[ + i\left( {{x}^{n - 1}{f}_{1}^{i}\left( M\right) - {x}^{n - 3}{f}_{3}^{i}\left( M\right) + {x}^{n - 5}{f}_{5}^{i}\left( M\right) \cdots }\right) {|}^{2} \] \[ = {\left( {x}^{n} - {x}^{n - 2}{f}_{2}^{i}\left( M\right) + {x}^{n - 4}{f}_{4}^{i}\left( M\right) \cdots \right) }^{2} \] \[ + {\left( {x}^{n - 1}{f}_{1}^{i}\left( M\right) - {x}^{n - 3}{f}_{3}^{i}\left( M\right) + {x}^{n - 5}{f}_{5}^{i}\left( M\right) \cdots \right) }^{2}. \] The coefficient of \( {x}^{{2n} - {2k}} \) in the last equality is \[ \mathop{\sum }\limits_{{k - j\text{ even }}}{\left( -1\right) }^{\left( {k - j}\right) /2}{f}_{k - j}^{i}\left( M\right) {\left( -1\right) }^{\left( {k + j}\right) /2}{f}_{k + j}^{i}\left( M\right) \] \[ + \mathop{\sum }\limits_{{k - j\text{ odd }}}{\left( -1\right) }^{\left( {k - j + 1}\right) /2}{f}_{k - j}^{i}\left( M\right) {\left( -1\right) }^{\left( {k + j + 1}\right) /2}{f}_{k + j}^{i}\left( M\right) \] \[ = {\left( -1\right) }^{k}\left\lbrack {\mathop{\sum }\limits_{{k - j\text{ even }}}{f}_{k - j}^{i}\left( M\right) {f}_{k + j}^{i}\left( M\right) - \mathop{\sum }\limits_{{k - j\text{ odd }}}{f}_{k - j}^{i}\left( M\right) {f}_{k + j}^{i}\left( M\right) }\right\rbrack \] \[ = {\left( -1\right) }^{k}\mathop{\sum }\limits_{{k - j}}{\left( -1\right) }^{k - j}{f}_{k - j}^{i}\left( M\right) {f}_{k + j}^{i}\left( M\right) \] \[ = {\left( -1\right) }^{k}\mathop{\sum }\limits_{l}{\left( -1\right) }^{l}{f}_{l}^{i}\left( M\right) {f}_{{2k} - l}^{i}\left( M\right) . \] This establishes the first identity in the proposition. For the one involving the Pfaffian, it suffices to check the formula in the case when \( M = \left( {i{\lambda }_{1}\ldots i{\lambda }_{n}}\right) \) . Then \[ \operatorname{Pf}\left( {h\left( M\right) }\right) = \operatorname{Pf}\left( \begin{matrix} & & - {\lambda }_{1} & & \\ & & & \ddots & \\ & & & & - {\lambda }_{n} \\ {\lambda }_{1} & & & & \\ & \ddots & & & \\ & & {\lambda }_{n} & & \end{matrix}\right) \] \[ = {\epsilon }^{1\left( {n + 1}\right) 2\left( {n + 2}\right) \ldots n\left( {2n}\right) }{\left( -1\right) }^{n}{\lambda }_{1}\cdots {\lambda }_{n} \] \[ = {\left( -1\right) }^{\left\lbrack n/2\right\rbrack }{\left( -1\right) }^{n}{\lambda }_{1}\cdots {\lambda }_{n} = {\left( -1\right) }^{\left\lbrack n/2\right\rbrack }\det {iM} = {\left( -1\right) }^{\left\lbrack n/2\right\rbrack }{f}_{n}^{i}\left( M\right) . \] EXERCISE 161. Show that any Hermitian inner product is determined by its norm function. Specifically, \[ \operatorname{Re}\langle v, w\rangle = \frac{1}{4}\left( {{\left| v + w\right| }^{2} - {\left| v - w\right| }^{2}}\right) ,\;\operatorname{Im}\langle v, w\rangle = \frac{1}{4}\left( {{\left| v + iw\right| }^{2} - {\left| v - iw\right| }^{2}}\right) . \] EXERCISE 162. Show that a symplectic form on \( V \) induces a Hermitian inner product on \( V \), and that conversely, if \( \langle \) , \( \rangle {isaHermitianinnerproducton} \) \( V \), then \( \sigma \left( {v, w}\right) = - \operatorname{Im}\langle v, w\rangle \) defines a symplectic form on \( V \) . EXERCISE 163. Fill in the details of the proof of Lemma 8.1. ## 9. Chern Classes A complex rank \( n \) vector bundle is a fiber bundle with fiber \( {\mathbb{C}}^{n} \) and structure group \( {GL}\left( {n,\mathbb{C}}\right) \) . Thus, the fiber over each point inherits a complex vector space structure. The realification \( {\xi }_{\mathbb{R}} \) of a complex bundle \( \xi \) and the complexification \( {\xi }_{\mathbb{C}} \) of a real bundle \( \xi \) are defined in the same way as for vector spaces. In particular, \( {\xi }_{\mathbb{R}} \) is orientable, with a canonical orientation. A Hermitian metric on a complex vector bundle \( \xi = \pi : E \rightarrow M \) is a section of the bundle \( \operatorname{Hom}\left( {\xi \otimes \xi ,\mathbb{C}}\right) \) which is a Hermitian inner product on each fiber. Such a metric always exists, since one can choose a Euclidean metric on \( {\xi }_{\mathbb{R}} \) , and this metric induces, by Exercise 161, a Hermitian one on \( \xi \) . A Hermitian connection \( \nabla \) on \( \xi \) is one for which the metric is parallel. In this case, \[ X\langle U, V\rangle = \left\langle {{\nabla }_{X}U, V}\right\rangle + \left\langle {U,{\nabla }_{X}V}\right\rangle ,\;X \in \mathfrak{X}M,\;U, V \in {\Gamma \xi }. \] Just as in the Riemannian case, the curvature tensor \( R \) of a Hermitian connection is skew-adjoint: \[ \langle R\left( {X, Y}\right) U, V\rangle = - \langle U, R\left( {X, Y}\right) V\rangle . \] Thus, given \( p \in M \), and an orthonormal basis \( b : {\mathbb{C}}^{n} \rightarrow {E}_{p},{b}^{-1} \circ R\left( {x, y}\right) \circ b \in \) \( \mathfrak{u}\left( n\right) \) for any \( x, y \in {M}_{p} \) . Let \( {g}_{k}^{i} \) denote the polarization of the polynomial \( {f}_{k}^{i} \) from the previous section. By Proposition \( {1.1},{g}_{k}^{i} \) induces a parallel section \( {\bar{g}}_{k}^{i} \) of \( {\operatorname{End}}_{k}{\left( \xi \right) }^{ * } \), and \( {\bar{g}}_{k}^{i}\left( {R}^{k}\right) \) is a \( {2k} \) -form on \( M \) . By Theorem 1.1, this form is closed, and its cohomology class is independent of the choice of connection. Definition 9.1. The \( k \) -th Chern class \( {c}_{k}\left( \xi \right) \in {H}^{2k}\left( M\right) \) of \( \xi \) is the class determined by the \( {2k} \) -form \[ {c}_{k} = \frac{1}{{\left( 2\pi \right) }^{k}}{\bar{g}}_{k}^{i}\left( {R}^{k}\right) \] \( {c}_{k} \) is called the \( k \) -th Chern form (of the connection). The total Chern class of \( \xi \) is \[ c\left( \xi \right) = {c}_{0}\left( \xi \right) + {c}_{1}\left( \xi \right) + \cdots + {c}_{n}\left( \xi \right) \] where \( {c}_{0}\left( \xi \right) \) denotes the class containing the constant function 1 . EXAMPLE 9.1. A complex line bundle (or, more accurately, its realification) is equivalent to an oriented real plane bundle: To see this, it suffices to exhibit a complex structure \( J \) on an oriented plane bundle \( \xi = \pi : E \rightarrow M \) . Choose a Euclidean metric on \( \xi \), and for nonzero \( u \) in \( E \), define \( {Ju} \) to be the unique vector of norm equal to that of \( u \), such that \( u,{Ju} \) is a positively oriented orthogonal basis of \( {E}_{\pi \left( u\right) } \) . \( J \) is then a complex structure on \( \xi \), and it makes sense to talk about the first Chern class \( {c}_{1}\left( \xi \right) \) of \( \xi \) . Given a Hermitian connection on \( \xi \), the Chern form \( {c}_{1} \) at \( p \in M \) is given by \[ {c}_{1}\left( {x, y}\right) = \frac{1}{2\pi }\operatorname{tr}{iR}\left( {x, y}\right) = \frac{1}{2\pi }\langle {iR}\left( {x, y}\right) u, u\rangle = \frac{1}{2\pi }\langle R\left( {x, y}\right) u, - {iu}\rangle \] \[ = \frac{1}{2\pi }\langle R\left( {x, y}\right) {iu}, u\rangle \in \mathbb{R} \] for unit \( u \) in \( {E}_{p} \) . In terms of the underlying real plane bundle, \[ {c}_{1}\left( {x, y}\right) = \frac{1}{2\pi }\langle R\left( {x, y}\right) {Ju}, u\rangle \] where \( \langle \) , \( \rangle {nowdenotestheEuclideanmetricon}{\xi }_{\mathbb{R}}{inducedbytherealpartof} \) the Hermitian metric on \( \xi \) . By Examples and Remarks 3.1(i), the first Chern class of a complex line bundle equals the Euler class of its realification. More generally, consider a complex rank \( n \) bundle \( \xi = \pi : E \rightarrow M \) with Hermitian connection \( \nabla \) . The real part of the Hermitian metric is a Euclidean metric which is parallel under \( \nabla \) . Thus, \( \nabla \) induces a Riemannian connection \( \widetilde{\nabla } \) on \( {\xi }_{\mathbb{R}} \) . Since \( {iU} \) is parallel along a curve whenever \( U \) is, the complex structure \( J \) is parallel. A Hermitian orthonormal basis \( b : {\mathbb{C}}^{n} \rightarrow {E}_{p} \) induces an isomorphism \( B \) : \( \mathfrak{u}\left( {E}_{p}\right) \rightarrow \mathfrak{u}\left( n\right) \) . There is a corresponding Euclidean orthonormal basis \( b \circ {h}^{-1} \) : \( {\mathbb{R}}^{2n} \rightarrow {E}_{p} \) that induces an isomorphism \( \widetilde{B} : \mathfrak{o}\left( {E}_{p}\right) \rightarrow \mathfrak{o}\left( {2n}\right) \) . Denote by \( \widetilde{h} \) the corresponding homomorphism \( {\widetilde{B}}^{-1} \circ h \circ B : \mathfrak{u}\left( {E}_{p}\right) \rightarrow \mathfrak{o}\left( {E}_{p}\right) \) . If \( R,\widetilde{R} \) denote the curvature tensors of \( \nabla \) and \( \widetilde{\nabla } \), then \( \widetilde{R} = \widetilde{h} \circ R \) . Thus, \[ \widetilde{B}\widetilde{R} = \widetilde{B} \circ \widetilde{h} \circ R = \widetilde{B} \circ {\widetilde{B}}^{-1} \circ h \circ B \circ R = h\left( {BR}\right) , \] and by Proposition 8.6, \[ {\bar{g}}_{2k}\left( {\widetilde{R}}^{2k}\right) = {f}_{2k}\left( {\widetilde{B}\widetilde{R}}\right) = {f}_{2k}

Proposition 8.6. \( {f}_{2k} \circ h = {\left( -1\right) }^{k}\mathop{\sum }\limits_{{l = 0}}^{{2k}}{\left( -1\right) }^{l}{f}_{l}^{i}{f}_{{2k} - l}^{i} \) , \( \operatorname{Pf} \circ h = {\left( -1\right) }^{\left\lbrack n/2\right\rbrack }{f}_{n}^{i} \) .

By Lemma 8.1, \[ {\left| \det \left( x{I}_{n} - M\right) \right| }^{2} = \det \left( {h\left( {x{I}_{n} - M}\right) }\right) = \det \left( {x{I}_{2n} - h\left( M\right) }\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{x}^{2\left( {n - k}\right) }{f}_{2k} \circ h\left( M\right) . \] On the other hand, \[ {\left| \det \left( x{I}_{n} - M\right) \right| }^{2} = {\left| \det \left( x{I}_{n} - i\left( -iM\right) \right) \right| }^{2} = {\left| \mathop{\sum }\limits_{{k = 0}}^{n}{\left( -1\right) }^{k}{\left( -i\right) }^{k}{f}_{k}^{i}\left( M\right) {x}^{n - k}\right| }^{2} \] \[ = {\left| {x}^{n} + i{x}^{n - 1}{f}_{1}^{i}\left( M\right) - {x}^{n - 2}{f}_{2}^{i}\left( M\right) - i{x}^{n - 3}{f}_{3}^{i}\left( M\right) + \cdots \right| }^{2} \] \[ = \mid \left( {{x}^{n} - {x}^{n - 2}{f}_{2}^{i}\left( M\right) + {x}^{n - 4}{f}_{4}^{i}\left( M\right) \cdots }\right) \] \[ + i\left( {{x}^{n - 1}{f}_{1}^{i}\left( M\right) - {x}^{n - 3}{f}_{3}^{i}\left( M\right) + {x}^{n - 5}{f}_{5}^{i}\left( M\right) \cdots }\right) {|}^{2} \] \[ = {\left( {x}^{n} - {x}^{n - 2}{f}_{2}^{i}\left( M\right) + {x}^{n - 4}{f}_{4}^{i}\left( M\right) \cdots \right) }^{2} \] \[ + {\left( {x}^{n - 1}{f}_{1}^{i}\left( M\right) - {x}^{n - 3}{f}_{3}^{i}\left( M\right) + {x}^{n - 5}{f}_{5}^{i}\left( M\right) \cdots \right) }^{2}. \] The coefficient of \( {x}^{{2n} - {2k}} \) in the last equality is \[ \mathop{\sum }\limits_{{k - j\text{ even }}}{\left( -1\right) }^{\left( {k - j}\right) /2}{f}_{k - j}^{i}\left( M\right) {\left( -1\

Theorem 178 I Idempotent, primitive 698 Image 722 Incidence isomorphism 82, 169 isomorphism, general 582, 599 number 79, 82 system 84 Inclusion 4, 20, 269, 271 Indeterminacy 722 Injection 55, 269, 271 Integers, twisted 285 Intersection number 509 Inverse, homotopy 23 Inverse limit, derived 271 of fibrations 432 J \( J \) - adjoint 516 extension 516 homomorphism 504 Jacobi identity (for Samelson products) 470 (for Whitehead products) 478 <table><tr><td>\( \mathbf{K} \)</td></tr><tr><td>Kernel 722</td></tr><tr><td>L</td></tr><tr><td>Ladder 724</td></tr><tr><td>Lens space 91, 94</td></tr><tr><td>Left exact sequence of spaces</td></tr><tr><td>127</td></tr><tr><td>Lifting 4</td></tr><tr><td>, partial 291</td></tr><tr><td>extension problem 34, 291</td></tr><tr><td>Linking number 509</td></tr><tr><td>Local coefficient system 257</td></tr><tr><td>, twisted 285, 306</td></tr><tr><td>in \( \mathbf{G}\left( n\right) \;{306} \)</td></tr><tr><td>Loop 106</td></tr><tr><td>, free 370</td></tr><tr><td>, measured 114</td></tr><tr><td>space 43</td></tr><tr><td>Lower central series 462</td></tr><tr><td>M</td></tr><tr><td>Map</td></tr><tr><td>, attaching 48</td></tr><tr><td>, cellular 76</td></tr><tr><td>, characteristic (of a fibration) 317</td></tr><tr><td>, characteristic (in an \( n \) -cellular extension) 48</td></tr><tr><td>Excision Theorem 57, 580, 598</td></tr><tr><td>, \( n \) -anticonnected 444</td></tr><tr><td>, \( n \) -connected 181</td></tr><tr><td>, simple 447</td></tr><tr><td>, simultaneous attaching 48</td></tr><tr><td>, standard 14</td></tr><tr><td>, \( u \) -admissible 185,626</td></tr><tr><td>Mapping</td></tr><tr><td>cone 23, 106</td></tr><tr><td>cone (of a chain map) 566</td></tr></table> Mapping (cont.) cylinder 23, 106 cylinder, relative 421 fibre 43 Maximal torus 677 Measured loop 114 path 107 Möbius inversion formula 514 Moore space 370 \( \mathbf{N} \) \( n \) -anticonnected map 444 space 418 \( n \) -connected map 181 pair 70 NDR- filtration 28 filtration of a pair 604 pair 22 Nilpotent 462 0 Obstruction 229 , primary 236 , primary (to a homotopy) 240 , primary (to a lifting) 298 , secondary 453 set 234, 450 to a partial lifting 292 Orientable fibration 349, 649 manifold 507 Orientation bundle 263 , coherent 169 , local 258 of a cell 82,168 of a regular cell complex 82 of a sphere 168 Orthogonal group (see also rotation group) 10 P Pair, \( n \) -connected 70 , simple 116 Path 106 , free 106 , measured 107 Poincaré series 322 Polarization 681, 687 Pontryagin algebra 145 of \( J\left( W\right) \;{323} \) of the rotation group 347 of the symplectic group 343 of the unitary group 343, 345 Postnikov invariant 423 system 431 system, fibred 432 system of a space 423 Power-associative 696 Primitive cohomology class 148, 383 element of a coalgebra 152 homology class 143, 383 homotopy class 494 idempotent 698 with respect to a map 504 Principal associated bundle 305, 678 Prism 105, 268, 271 Proclusion 4, 20 Product , basic 511 , cap 349, 508 , Cartesian 19 , categorical 19 , cup 88 , external 120 , in an H-space 117 of CW-complexes 50 of regular cell complexes 88 , weak 28, 208, 419 Projective plane, Cayley 677, 700 Projective space, complex 12 , complex, cohomology of 90 , quaternionic 12 , real 12 , real, cohomology of 89 Pure subgroup 563 \( \mathbf{R} \) Rank (of a basic product) 512 (of a compact Lie group) 678 Reduced join 103, 480 product 326 Reductive cohomology class 147, 383 homology class \( {145},{383} \) Reindexing 573 Relational chain complex 727 Resolution , free 226, 280, 562 , homotopy 421, 426, 446 Resolving sequence 421 for a map 446 Retract 24 , deformation 24 Retraction, deformation 24 Right exact sequence of spaces 127 Rotation group 10 , cohomology of 346 , full 10 , homology of 347 , homotopy groups of 200 S S-map 551 Samelson product 467 and Whitehead product 476 Seam homomorphism 497 Secondary boundary operator 555 cohomology operation 454 obstruction 453 Sectional element 194 Serial number 512 Serre exact sequence 365, 649 Shear map 118, 461 Simple map 447 pair 116 space 116, 421 Simplex, singular 10 Skeleton 50 Smash product (see reduced join) Space , associated compactly generated 18 , cogrouplike 122 , compactly generated 18 , filtered 28 , free 103 , grouplike 118, 461 , \( n \) -anticonnected 418 , \( n \) -connected 178 , simple 116, 421 Spanier-Whitehead category 552 Special triple 691 Spectral sequence 610 , Atiyah-Hirzebruch 640 , Leray-Serre 645 of a fibration 628, 630 of a filtered pair 614, 616 , products in 654 Spherical fibration 195 homology class 143, 247 Spin representation 705 Spinor group 684 Stable cohomology operation 391 cohomotopy module 552 homotopy module 552 homotopy ring 552 homotopy ring, commutativity of 553 Steenrod algebra 393 Steenrod algebra, action of on complex projective space 409 first exceptional Lie group \( {\mathbf{G}}_{2}\;{410} \) real projective space 399 Stiefel manifolds 400 unitary group 410 Steenrod squares 395 Stiefel manifold 11 , cohomology of 348 , complex 12 , homotopy groups of 203 , quaternionic 12 Stiefel-Whitney classes 308 Stratification 459 Structural map 317 Subcomplex 51 Subspace 20 Suspension 105, 496, 544, 724 category 552 , homology 365, 373 Theorem (Freudenthal) 369 Symplectic group 12 , cohomology of 342 , full 12 , homology of 343 \( \mathbf{T} \) Thom class 349 Isomorphism Theorem 350 Topology, weak 27 Total singular complex 10 geometric realization of 94, 208 Total space (of a fibration) 29 Track group 97 Transfer 384 Transgression 496, 648, 724 Transgressive (co)homology class 383 Triad , NDR- 55 , proper 55 Triality, Principle of 712 Trivialization 40 Twisted integers 285 U Unitary cohomology class 567 Unitary group 12 , cohomology of 342 , full 12 , homology of 343 , homotopy properties of 205 , unimodular 12 , unimodular, (co)homology of 345 Universal example \( {30},{165},{175},{232} \) , 485, 491, 534, 559 \( \mathbf{V} \) van Kampen Theorem 58, 94 Vector bundle, orthogonal 305 Vector field problem 201, 400 Vector representation 682 w Wang sequence 319, 336, 671 Weak homotopy equivalence 181, 221 homotopy type 221 product 28, 208, 419 Weakly contractible 220 Weight (of a commutator) 463 (of a monomial) 511 Whitehead product 472, 482 Theorem 181 Whitney classes of a vector bundle 307 , universal 308 Witt formulae 514 ## Graduate Texts in Mathematics ## continued from page ii SACHS/WU. General Relativity for Mathematicians. GRUENBERG/WEIR. Linear Geometry. 2nd ed. EDWARDS. Fermat's Last Theorem. KLINGENBERG. A Course in Differential Geometry. Hartshorne. Algebraic Geometry. MANIN. A Course in Mathematical Logic. GRAVER/WATKINS. Combinatorics with Emphasis on the Theory of Graphs. BROWN/PEARCY. Introduction to Operator Theory I: Elements of Functional Analysis. MASSEY. Algebraic Topology: An Introduction. CROWELL/FOX. Introduction to Knot Theory. KOBLITZ. \( p \) -adic Numbers. \( p \) -adic Analysis, and Zeta-Functions. 2nd ed. LANG. Cyclotomic Fields. ARNOLD. Mathematical Methods in Classical Mechanics. Second Edition. WHITEHEAD. Elements of Homotopy Theory. KARGAPOLOV/MERLZJAKOV. Fundamentals of the Theory of Groups. BOLLOBAS. Graph Theory. EDWARDS. Fourier Series. Vol. I. 2nd ed. WELLS. Differential Analysis on Complex Manifolds. 2nd ed. WATERHOUSE. Introduction to Affine Group Schemes. SERRE. Local Fields. WEIDMANN. Linear Operators in Hilbert Spaces. LANG. Cyclotomic Fields II. MASSEY. Singular Homology Theory. FARKAS/KRA. Riemann Surfaces. Stillwell. Clässical Topology and Combinatorial Group Theory. Hungerford. Algebra. DAVENPORT. Multiplicative Number Theory. 2nd ed. HOCHSCHILD. Basic Theory of Algebraic Groups and Lie Algebras. IITAKA. Algebraic Geometry. HECKE. Lectures on the Theory of Algebraic Numbers. Burris/Sankappanavar. A Course in Universal Algebra. WALTERS. An Introduction to Ergodic Theory. ROBINSON. A Course in the Theory of Groups. FORSTER. Lectures on Riemann Surfaces. BOTT/TU. Differential Forms in Algebraic Topology. WASHINGTON. Introduction to Cyclotomic Fields. IRELAND/ROSEN. A Classical Introduction to Modern Number Theory. Second Edition. Edwards. Fourier Series. Vol. II. 2nd ed. VAN LINT. Introduction to Coding Theory. Brown. Cohomology of Groups. PIERCE. Associative Algebras. LANG. Introduction to Algebraic and Abelian Functions. 2nd ed. Brondsted. An Introduction to Convex Polytopes. BEARDON. On the Geometry of Discrete Groups. DIESTEL. Sequences and Series in Banach Spaces. DUBROVIN/FOMENKO/NOVIKOV. Modern Geometry-Methods and Applications Vol. I. WARNER. Foundations of Differentiable Manifolds and Lie Groups. SHIRYAYEV. Probability, Statistics, and Random Processes. Conway. A Course in Functional Analysis 97 KOBLITZ. Introduction to Elliptic Curves and Modular Forms. 98 Bröcker/Tom DIECK. Representations of compact Lie Groups. 99 Grove/BENSON. Finite Reflection Groups. 2nd ed. 100 Berg/Christensen/Resser. Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. 101 EDWARDS. Galois Theory. 102 VARDARAJAN. Lie Groups, Lie Algebras and Their Representations. 103 LANG. Complex Analysis. 2nd ed. 104 DUBROVIN/FOMENKO/NOVIKOV. Modern Geometry-Methods and Applications Vol. II. 105 LANG. \( S{L}_{2}\left( \mathbf{R}\right) \) . 106 SILVERMAN. The Arithmetic of Elliptic Curves. 107 OLVER. Applications of Lie Groups to Differential Equations. 108 RANGE. Holomorphic Functions and Integral Representations in Several Complex Variables. 109 LEHTO. Univalent Functions and Teichmüller Spaces. 110 LANG. Algebraic Number Theory. 111 Husemöller. Elliptic Cruves. 112 LANG. Elliptic Functions. 113 KARATZAS/SHREVE. Brownian Motion and Stochastic

Theorem 178 I

null

Theorem 2.10 A graph \( G \) is even if and only if \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . Proof Suppose that \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . Then, in particular, \( \left| {\partial \left( v\right) }\right| \) is even for every vertex \( v \) . But, as noted above, \( \partial \left( v\right) \) is just the set of all links incident with \( v \) . Because loops contribute two to the degree, it follows that all degrees are even. Conversely, if \( G \) is even, then Theorem 2.9 implies that all edge cuts are of even cardinality. The operation of symmetric difference of spanning subgraphs was introduced in Section 2.1. The following propositions show how edge cuts behave with respect to symmetric difference. Proposition 2.11 Let \( G \) be a graph, and let \( X \) and \( Y \) be subsets of \( V \) . Then \[ \partial \left( X\right) \bigtriangleup \partial \left( Y\right) = \partial \left( {X\bigtriangleup Y}\right) \] Proof Consider the Venn diagram, shown in Figure 2.9, of the partition of \( V \) \[ \left( {X \cap Y,\;X \smallsetminus Y,\;Y \smallsetminus X,\;\bar{X} \cap \bar{Y}}\right) \] determined by the partitions \( \left( {X,\bar{X}}\right) \) and \( \left( {Y,\bar{Y}}\right) \), where \( \bar{X} \mathrel{\text{:=}} V \smallsetminus X \) and \( \bar{Y} \mathrel{\text{:=}} \) \( V \smallsetminus Y \) . The edges of \( \partial \left( X\right) ,\partial \left( Y\right) \), and \( \partial \left( {X\bigtriangleup Y}\right) \) between these four subsets of \( V \) are indicated schematically in Figure 2.10. It can be seen that \( \partial \left( X\right) \bigtriangleup \partial \left( Y\right) = \) \( \partial \left( {X\bigtriangleup Y}\right) \) .  Fig. 2.9. Partition of \( V \) determined by the partitions \( \left( {X,\bar{X}}\right) \) and \( \left( {Y,\bar{Y}}\right) \) Corollary 2.12 The symmetric difference of two edge cuts is an edge cut. We leave the proof of the second proposition to the reader (Exercise 2.5.1b). Proposition 2.13 Let \( {F}_{1} \) and \( {F}_{2} \) be spanning subgraphs of a graph \( G \), and let \( X \) be a subset of \( V \) . Then \[ {\partial }_{{F}_{1}\bigtriangleup {F}_{2}}\left( X\right) = {\partial }_{{F}_{1}}\left( X\right) \bigtriangleup {\partial }_{{F}_{2}}\left( X\right) \]  Fig. 2.10. The symmetric difference of two cuts ## Bonds A bond of a graph is a minimal nonempty edge cut, that is, a nonempty edge cut none of whose nonempty proper subsets is an edge cut. The bonds of the graph whose edge cuts are depicted in Figure 2.8 are shown in Figure 2.11. The following two theorems illuminate the relationship between edge cuts and bonds. The first can be deduced from Proposition 2.11 (Exercise 2.5.1c). The second provides a convenient way to check when an edge cut is in fact a bond. Theorem 2.14 A set of edges of a graph is an edge cut if and only if it is a disjoint union of bonds. Theorem 2.15 In a connected graph \( G \), a nonempty edge cut \( \partial \left( X\right) \) is a bond if and only if both \( G\left\lbrack X\right\rbrack \) and \( G\left\lbrack {V \smallsetminus X}\right\rbrack \) are connected. Proof Suppose, first, that \( \partial \left( X\right) \) is a bond, and let \( Y \) be a nonempty proper subset of \( X \) . Because \( G \) is connected, both \( \partial \left( Y\right) \) and \( \partial \left( {X \smallsetminus Y}\right) \) are nonempty. It follows that \( E\left\lbrack {Y, X \smallsetminus Y}\right\rbrack \) is nonempty, for otherwise \( \partial \left( Y\right) \) would be a nonempty proper subset of \( \partial \left( X\right) \), contradicting the supposition that \( \partial \left( X\right) \) is a bond. We conclude that \( G\left\lbrack X\right\rbrack \) is connected. Likewise, \( G\left\lbrack {V \smallsetminus X}\right\rbrack \) is connected. Conversely, suppose that \( \partial \left( X\right) \) is not a bond. Then there is a nonempty proper subset \( Y \) of \( V \) such that \( X \cap Y \neq \varnothing \) and \( \partial \left( Y\right) \subset \partial \left( X\right) \) . But this implies (see Figure 2.10) that \( E\left\lbrack {X \cap Y, X \smallsetminus Y}\right\rbrack = E\left\lbrack {Y \smallsetminus X,\bar{X} \cap \bar{Y}}\right\rbrack = \varnothing \) . Thus \( G\left\lbrack X\right\rbrack \) is not connected if \( X \smallsetminus Y \neq \varnothing \) . On the other hand, if \( X \smallsetminus Y = \varnothing \), then \( \varnothing \subset Y \smallsetminus X \subset V \smallsetminus X \) , and \( G\left\lbrack {V \smallsetminus X}\right\rbrack \) is not connected. ## Cuts in Directed Graphs If \( X \) and \( Y \) are sets of vertices (not necessarily disjoint) of a digraph \( D = \left( {V, A}\right) \) , we denote the set of arcs of \( D \) whose tails lie in \( X \) and whose heads lie in \( Y \) by \( A\left( {X, Y}\right) \), and their number by \( a\left( {X, Y}\right) \) . This set of arcs is denoted by \( A\left( X\right) \) when \( Y = X \), and their number by \( a\left( X\right) \) . When \( Y = V \smallsetminus X \), the set \( A\left( {X, Y}\right) \) is called the outcut of \( D \) associated with \( X \), and denoted by \( {\partial }^{ + }\left( X\right) \) . Analogously, the set \( A\left( {Y, X}\right) \) is called the incut of \( D \) associated with \( X \), and denoted by \( {\partial }^{ - }\left( X\right) \) . Observe that \( {\partial }^{ + }\left( X\right) = {\partial }^{ - }\left( {V \smallsetminus X}\right) \) . Note, also, that \( \partial \left( X\right) = {\partial }^{ + }\left( X\right) \cup {\partial }^{ - }\left( X\right) \) . In  the case of loopless digraphs, we refer to \( \left| {{\partial }^{ + }\left( X\right) }\right| \) and \( \left| {{\partial }^{ - }\left( X\right) }\right| \) as the outdegree and indegree of \( X \), and denote these quantities by \( {d}^{ + }\left( X\right) \) and \( {d}^{ - }\left( X\right) \), respectively. A digraph \( D \) is called strongly connected or strong if \( {\partial }^{ + }\left( X\right) \neq \varnothing \) for every nonempty proper subset \( X \) of \( V \) (and thus \( {\partial }^{ - }\left( X\right) \neq \varnothing \) for every nonempty proper subset \( X \) of \( V \), too). ## Exercises \( \star {2.5.1} \) a) Prove Theorem 2.9. b) Prove Proposition 2.13. c) Deduce Theorem 2.14 from Proposition 2.11. \( \star \) 2.5.2 Let \( D \) be a digraph, and let \( X \) be a subset of \( V \) . a) Show that \( \left| {{\partial }^{ + }\left( X\right) }\right| = \mathop{\sum }\limits_{{v \in X}}{d}^{ + }\left( v\right) - a\left( X\right) \) . b) Suppose that \( D \) is even. Using the Principle of Directional Duality, deduce that \( \left| {{\partial }^{ + }\left( X\right) }\right| = \left| {{\partial }^{ - }\left( X\right) }\right| \) . c) Deduce from (b) that every connected even digraph is strongly connected. 2.5.3 Let \( G \) be a graph, and let \( X \) and \( Y \) be subsets of \( V \) . Show that \( \partial \left( {X \cup Y}\right) \bigtriangleup \) \( \partial \left( {X \cap Y}\right) = \partial \left( {X\bigtriangleup Y}\right) \) \( \star \) 2.5.4 Let \( G \) be a loopless graph, and let \( X \) and \( Y \) be subsets of \( V \) . a) Show that: \[ d\left( X\right) + d\left( Y\right) = d\left( {X \cup Y}\right) + d\left( {X \cap Y}\right) + {2e}\left( {X \smallsetminus Y, Y \smallsetminus X}\right) \] b) Deduce the following submodular inequality for degrees of sets of vertices. \[ d\left( X\right) + d\left( Y\right) \geq d\left( {X \cup Y}\right) + d\left( {X \cap Y}\right) \] c) State and prove a directed analogue of this submodular inequality. \( \star \mathbf{2.5.5} \) An odd graph is one in which each vertex is of odd degree. Show that a graph \( G \) is odd if and only if \( \left| {\partial \left( X\right) }\right| \equiv \left| X\right| \left( {\;\operatorname{mod}\;2}\right) \) for every subset \( X \) of \( V \) . \( \star \) 2.5.6 Show that each arc of a strong digraph is contained in a directed cycle. ## 2.5.7 Directed Bond A directed bond of a digraph is a bond \( \partial \left( X\right) \) such that \( {\partial }^{ - }\left( X\right) = \varnothing \) (in other words, \( \partial \left( X\right) \) is the outcut \( {\partial }^{ + }\left( X\right) \) ). a) Show that an arc of a digraph is contained either in a directed cycle, or in a directed bond, but not both. (G.J. Minty) b) Deduce that: i) a digraph is acyclic if and only if every bond is a directed bond, ii) a digraph is strong if and only if no bond is a directed bond. ## \( \star \) 2.5.8 Feedback Arc Set A feedback arc set of a digraph \( D \) is a set \( S \) of arcs such that \( D \smallsetminus S \) is acyclic. Let \( S \) be a minimal feedback arc set of a digraph \( D \) . Show that there is a linear ordering of the vertices of \( D \) such that the arcs of \( S \) are precisely those arcs whose heads precede their tails in the ordering.  2.5.9 Let \( \left( {D, w}\right) \) be a weighted oriented graph. For \( v \in V \), set \( {w}^{ + }\left( v\right) \mathrel{\text{:=}} \sum \{ w\left( a\right) \) : \( \left. {a \in {\partial }^{ + }\left( v\right) }\right\} \) . Suppose that \( {w}^{ + }\left( v\right) \geq 1 \) for all \( v \in V \smallsetminus \{ y\} \), where \( y \in V \) . Show that \( D \) contains a directed path of weight at least one, by proceeding as follows. a) Consider an arc \( \left( {x, y}\right) \in {\partial }^{ - }\left( y\right) \) of maximum weight. Contract this

Theorem 2.10 A graph \( G \) is even if and only if \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) .

Suppose that \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . Then, in particular, \( \left| {\partial \left( v\right) }\right| \) is even for every vertex \( v \) . But, as noted above, \( \partial \left( v\right) \) is just the set of all links incident with \( v \) . Because loops contribute two to the degree, it follows that all degrees are even. Conversely, if \( G \) is even, then Theorem 2.9 implies that all edge cuts are of even cardinality.

Lemma 2.12. If \( \mathrm{F} \) is a Galois extension field of \( \mathrm{K} \) and \( \mathrm{E} \) is a stable intermediate field of the extension, then \( \mathrm{E} \) is Galois over \( \mathrm{K} \) . PROOF. If \( u : E - K \), then there exists \( \sigma \in {\operatorname{Aut}}_{K}F \) such that \( \sigma \left( u\right) \neq u \) since \( F \) is Galois over \( K \) . But \( \sigma \mid E \) e Aut \( {}_{K}E \) by stability. Therefore, \( E \) is Galois over \( K \) by the Remarks after Definition 2.4. Lemma 2.13. If \( \mathrm{F} \) is an extension field of \( \mathrm{K} \) and \( \mathrm{E} \) is an intermediate field of the extension such that \( \mathrm{E} \) is algebraic and Galois over \( \mathrm{K} \), then \( \mathrm{E} \) is stable (relative to \( \mathrm{F} \) and \( \mathrm{K} \) ). REMARK. The hypothesis that \( E \) is algebraic is essential; see Exercise 13. PROOF OF 2.13. If \( u \in E \), let \( f \in K\left\lbrack x\right\rbrack \) be the irreducible polynomial of \( u \) and let \( u = {u}_{1},{u}_{2},\ldots ,{u}_{r} \) be the distinct roots of \( f \) that lie in \( E \) . Then \( r \leq n = \deg f \) by Theorem III.6.7. If \( \tau \) e Aut \( {}_{K}E \), then it follows from Theorem 2.2 that \( \tau \) simply permutes the \( {u}_{i} \) . This implies that the coefficients of the monic polynomial \( g\left( x\right) = \left( {x - {u}_{1}}\right) \) \( \left( {x - {u}_{2}}\right) \cdots \left( {x - {u}_{r}}\right) \in E\left\lbrack x\right\rbrack \) are fixed by every \( \tau \in {\operatorname{Aut}}_{K}E \) . Since \( E \) is Galois over \( K \), we must have \( {g\varepsilon K}\left\lbrack x\right\rbrack \) . Now \( u = {u}_{1} \) is a root of \( g \) and hence \( f \mid g \) (Theorem 1.6(ii)). Since \( g \) is monic and \( \deg g \leq \deg f \), we must have \( f = g \) . Consequently, all the roots of \( f \) are distinct and lie in \( E \) . Now if \( \sigma \in {\operatorname{Aut}}_{K}F \), then \( \sigma \left( u\right) \) is a root of \( f \) by Theorem 2.2, whence \( \sigma \left( u\right) \in E \) . Therefore, \( E \) is stable relative to \( F \) and \( K \) . Let \( E \) be an intermediate field of the extension \( K \subset F \) . A \( K \) -automorphism \( \tau \) e \( {\operatorname{Aut}}_{K}E \) is said to be extendible to \( F \) if there exists \( \sigma \) ε \( {\operatorname{Aut}}_{K}F \) such that \( \sigma \mid E = \tau \) . It is easy to see that the extendible \( K \) -automorphisms form a subgroup of \( {\operatorname{Aut}}_{K}E \) . Recall that if \( E \) is stable, \( {E}^{\prime } = {\operatorname{Aut}}_{E}F \) is a normal subgroup of \( G = {\operatorname{Aut}}_{K}F \) (Lemma 2.11). Consequently, the quotient group \( G/{E}^{\prime } \) is defined. Lemma 2.14. Let \( \mathrm{F} \) be an extension field of \( \mathrm{K} \) and \( \mathrm{E} \) a stable intermediate field of the extension. Then the quotient group \( {Au}{t}_{\mathrm{K}}\mathrm{F}/{Au}{t}_{\mathrm{E}}\mathrm{F} \) is isomorphic to the group of all \( \mathrm{K} \) -automorphisms of \( \mathrm{E} \) that are extendible to \( \mathrm{F} \) . SKETCH OF PROOF. Since \( E \) is stable, the assignment \( \sigma \left| { \rightarrow \sigma }\right| E \) defines a group homomorphism \( {\operatorname{Aut}}_{K}F \rightarrow {\operatorname{Aut}}_{K}E \) whose image is clearly the subgroup of all \( K \) -automorphisms of \( E \) that are extendible to \( F \) . Observe that the kernel is \( {\operatorname{Aut}}_{E}F \) and apply the First Isomorphism Theorem I.5.7. PROOF OF THEOREM 2.5. (Fundamental Theorem of Galois Theory) Theorem 2.7 shows that there is a one-to-one correspondence between closed intermediate fields of the extension and closed subgroups of the Galois group. But in this case all intermediate fields and all subgroups are closed by Lemma 2.10(iii). Statement (i) of the theorem follows immediately from Lemma 2.10(i). (ii) \( F \) is Galois over \( E \) since \( E \) is closed (that is, \( E = {E}^{\prime \prime } \) ). \( E \) is finite dimensional over \( K \) (since \( F \) is) and hence algebraic over \( K \) by Theorem 1.11. Consequently, if \( E \) is Galois over \( K \), then \( E \) is stable by Lemma 2.13. By Lemma 2.11(i) \( {E}^{\prime } = {\operatorname{Aut}}_{E}F \) is normal in \( {\operatorname{Aut}}_{K}F \) . Conversely if \( {E}^{\prime } \) is normal in \( {\operatorname{Aut}}_{K}F \), then \( {E}^{\prime \prime } \) is a stable intermediate field (Lemma 2.11(ii)). But \( E = {E}^{\prime \prime } \) since all intermediate fields are closed and hence \( E \) is Galois over \( K \) by Lemma 2.12. Suppose \( E \) is an intermediate field that is Galois over \( K \) (so that \( {E}^{\prime } \) is normal in \( \left. {{\operatorname{Aut}}_{K}F}\right) \) . Since \( E \) and \( {E}^{\prime } \) are closed and \( {G}^{\prime } = K \) ( \( F \) is Galois over \( K \) ), Lemma 2.10 implies that \( \left| {G/{E}^{\prime }}\right| = \left\lbrack {G : {E}^{\prime }}\right\rbrack = \left\lbrack {{E}^{\prime \prime } : {G}^{\prime }}\right\rbrack = \left\lbrack {E : K}\right\rbrack \) . By Lemma 2.14 \( G/{E}^{\prime } = \) \( {\mathrm{{Aut}}}_{K}F/{\mathrm{{Aut}}}_{E}F \) is isomorphic to a subgroup (of order \( \left\lbrack {E : K}\right\rbrack \) ) of \( {\mathrm{{Aut}}}_{K}E \) . But part (i) of the theorem shows that \( \left| {{\operatorname{Aut}}_{K}E}\right| = \left\lbrack {E : K}\right\rbrack \) (since \( E \) is Galois over \( K \) ). This implies that \( G/{E}^{\prime } \cong {\operatorname{Aut}}_{K}E \) . The modern development of Galois Theory owes a great deal to Emil Artin. Although our treatment is ultimately due to Artin (via I. Kaplansky) his approach differs from the one given here in terms of emphasis. Artin's viewpoint is that the basic object is a given field \( F \) together with a (finite) group \( G \) of automorphisms of \( F \) . One then constructs the subfield \( K \) of \( F \) as the fixed field of \( G \) (the proof that the subset of \( F \) fixed elementwise by \( G \) is a field is a minor variation of the proof of Theorem 2.3). Theorem 2.15. (Artin) Let \( \mathrm{F} \) be a field, \( \mathrm{G} \) a group of automorphisms of \( \mathrm{F} \) and \( \mathrm{K} \) the fixed field of \( \mathrm{G} \) in \( \mathrm{F} \) . Then \( \mathrm{F} \) is Galois over \( \mathrm{K} \) . If \( \mathrm{G} \) is finite, then \( \mathrm{F} \) is a finite dimensional Galois extension of \( \mathbf{K} \) with Galois group \( \mathbf{G} \) . PROOF. In any case \( G \) is a subgroup of \( {\operatorname{Aut}}_{K}F \) . If \( u \in F - K \), then there must be a \( \sigma \in G \) such that \( \sigma \left( u\right) \neq u \) . Therefore, the fixed field of \( {\operatorname{Aut}}_{K}F \) is \( K \), whence \( F \) is Galois over \( K \) . If \( G \) is finite, then Lemma 2.9 (with \( H = 1, J = G \) ) shows that \( \left\lbrack {F : K}\right\rbrack = \left\lbrack {{1}^{\prime } : {G}^{\prime }}\right\rbrack \leq \left\lbrack {G : 1}\right\rbrack = \left| G\right| \) . Consequently, \( F \) is finite dimensional over \( K \) , whence \( G = {G}^{\prime \prime } \) by Lemma 2.10(iii). Since \( {G}^{\prime } = K \) (and hence \( {G}^{\prime \prime } = {K}^{\prime } \) ) by hypothesis, we have \( {\operatorname{Aut}}_{K}F = {K}^{\prime } = {G}^{\prime \prime } = G \) . ## APPENDIX: SYMMETRIC RATIONAL FUNCTIONS Let \( K \) be a field, \( K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) the polynomial domain and \( K\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) the field of rational functions (see the example preceding Theorem 1.5). Since \( K\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is by definition the quotient field of \( K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \), we have \( K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \subset K\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) (under the usual identification of \( f \) with \( f/{1}_{K} \) ). Let \( {S}_{n} \) be the symmetric group on \( n \) letters. A rational function \( \varphi \in K\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is said to be symmetric in \( {x}_{1},\ldots ,{x}_{n} \) over \( K \) if for every \( {\sigma \varepsilon }{S}_{n} \) , \[ \varphi \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) = \varphi \left( {{x}_{\sigma \left( 1\right) },{x}_{\sigma \left( 2\right) },\ldots ,{x}_{\sigma \left( n\right) }}\right) . \] Trivially every constant polynomial is a symmetric function. If \( n = 4 \), then the polynomials \( {f}_{1} = {x}_{1} + {x}_{2} + {x}_{3} + {x}_{4},{f}_{2} = {x}_{1}{x}_{2} + {x}_{1}{x}_{3} + {x}_{1}{x}_{4} + {x}_{2}{x}_{3} + {x}_{2}{x}_{4} + {x}_{3}{x}_{4} \) , \( {f}_{3} = {x}_{1}{x}_{2}{x}_{3} + {x}_{1}{x}_{2}{x}_{4} + {x}_{1}{x}_{3}{x}_{4} + {x}_{2}{x}_{3}{x}_{4} \) and \( {f}_{4} = {x}_{1}{x}_{2}{x}_{3}{x}_{4} \) are all symmetric functions. More generally the elementary symmetric functions in \( {x}_{1},\ldots ,{x}_{n} \) over \( K \) are defined to be the polynomials: \[ {f}_{1} = {x}_{1} + {x}_{2} + \cdots + {x}_{n} = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} \] \[ {f}_{2} = \mathop{\sum }\limits_{{1 \leq i < j \leq n}}{x}_{i}{x}_{j} \] \[ {f}_{3} = \mathop{\sum }\limits_{{1 \leq i < j < k \leq n}}{x}_{i}{x}_{j}{x}_{k} \] \[ {f}_{k} = \mathop{\sum }\limits_{{1 \leq {i}_{1} < \ldots < {i}_{k} \leq n}}{x}_{{i}_{1}}{x}_{{i}_{2}}\cdots {x}_{{i}_{k}} \] \[ \text{.} \] \[ {f}_{n} = {x}_{1}{x}_{2}\cdots {x}_{n} \] The verification that the \( {f}_{i} \) are indeed symmetric follows from the fact that they are simply the coefficients of \( y \) in the polynomial \( g\left( y\right) \in K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \left\lbrack y\right\rbrack \), where \[ g\left( y\right) = \left( {y - {x}_{1}}\right) \left( {y - {x}_{2}}\right) \left( {y - {x}_{3}}\right) \cdots \left( {y - {x}_{n}}\right) \] \[ = {y}^{n} - {f}_{1}{y}^{n - 1} + {f}_{2}{y}^{n - 2} - \cdots + {\left( -1\right) }^{n - 1}{f}_{n - 1}y + {\left( -1\right) }^{

Lemma 2.12. If \( \mathrm{F} \) is a Galois extension field of \( \mathrm{K} \) and \( \mathrm{E} \) is a stable intermediate field of the extension, then \( \mathrm{E} \) is Galois over \( \mathrm{K} \).

If \( u : E - K \), then there exists \( \sigma \in {\operatorname{Aut}}_{K}F \) such that \( \sigma \left( u\right) \neq u \) since \( F \) is Galois over \( K \). But \( \sigma \mid E \) e Aut \( {}_{K}E \) by stability. Therefore, \( E \) is Galois over \( K \) by the Remarks after Definition 2.4.

Corollary 10.7.7. Set \( b = \log \left( {2\pi }\right) - 1 - \gamma /2 \) . Then for all \( s \in \mathbb{C} \) we have the convergent product \[ \zeta \left( s\right) = \frac{{e}^{bs}}{s\left( {s - 1}\right) \Gamma \left( {s/2}\right) }\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\right) {e}^{s/\rho }, \] the product being over all nontrivial zeros of \( \zeta \left( s\right) \) (i.e., such that \( 0 \leq \Re \left( \rho \right) \leq \) 1). Proof. We apply Hadamard's theorem to the function \[ f\left( s\right) = s\left( {1 - s}\right) {\pi }^{-s/2}\Gamma \left( {s/2}\right) \zeta \left( s\right) = 2\left( {1 - s}\right) {\pi }^{-s/2}\Gamma \left( {s/2 + 1}\right) \zeta \left( s\right) . \] Since the zeros of \( \zeta \left( s\right) \) for \( s = - {2k}, k \in {\mathbb{Z}}_{ \geq 1} \), are killed by the poles of \( \Gamma \left( {s/2 + 1}\right) \) and the pole of \( \zeta \left( s\right) \) is killed by \( 1 - s \), it follows that the zeros of \( f\left( s\right) \) are the nontrivial zeros of \( \zeta \left( s\right) \) . Thus for suitable constants \( {a}_{0} \) and \( {a}_{1} \) we have \[ f\left( s\right) = {a}_{0}{e}^{{a}_{1}s}\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\right) {e}^{s/\rho }, \] so that \[ \zeta \left( s\right) = \frac{{a}_{0}{e}^{bs}}{2\left( {1 - s}\right) \Gamma \left( {s/2 + 1}\right) }\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\right) {e}^{s/\rho } \] for \( b = {a}_{1} + \log \left( \pi \right) /2 \) . We deduce that \( {a}_{0} = {2\zeta }\left( 0\right) = - 1 \), and by logarithmic differentiation that \[ \frac{{\zeta }^{\prime }\left( s\right) }{\zeta \left( s\right) } = b - \frac{1}{s - 1} - \frac{{\Gamma }^{\prime }\left( {s/2 + 1}\right) }{{2\Gamma }\left( {s/2 + 1}\right) } + \mathop{\sum }\limits_{\rho }\left( {\frac{1}{s - \rho } + \frac{1}{\rho }}\right) , \] so that \[ \frac{{\zeta }^{\prime }\left( 0\right) }{\zeta \left( 0\right) } = b + 1 - \frac{{\Gamma }^{\prime }\left( 1\right) }{\Gamma \left( 1\right) } \] Using \( {\zeta }^{\prime }\left( 0\right) = - \log \left( {2\pi }\right) /2 \) and \( {\Gamma }^{\prime }\left( 1\right) = - \gamma \) we obtain \( b = \log \left( {2\pi }\right) - 1 - \gamma /2 \) . We are now in a position to give a much better zero-free region than that given by Exercise 64. Theorem 10.7.8. There exists a constant \( C > 0 \) such that \( \zeta \left( s\right) \neq 0 \) for \( t \geq {t}_{0} \) in the region \[ \Re \left( s\right) > 1 - \frac{C}{\log \left( t\right) }. \] Proof. Here we will use the second inequality of Lemma 10.7.2. Fix some \( \sigma > 1 \) (we will see at the end of the proof how to choose it appropriately). Since \( \zeta \left( \sigma \right) = 1/\left( {\sigma - 1}\right) + O\left( 1\right) \) and \( {\zeta }^{\prime }\left( \sigma \right) = - 1/{\left( \sigma - 1\right) }^{2} + O\left( 1\right) \), we have \( - {\zeta }^{\prime }\left( \sigma \right) /\zeta \left( \sigma \right) < 1/\left( {\sigma - 1}\right) + O\left( 1\right) \) . From the above corollary and trivial bounds on \( {\Gamma }^{\prime }\left( s\right) /\Gamma \left( s\right) \) we also have \[ - \frac{{\zeta }^{\prime }\left( s\right) }{\zeta \left( s\right) } = O\left( {\log \left( t\right) }\right) - \mathop{\sum }\limits_{\rho }\left( {\frac{1}{s - \rho } + \frac{1}{\rho }}\right) , \] so if we write \( \rho = \beta + {i\gamma } \) with \( 0 \leq \beta \leq 1 \) and \( \gamma \in \mathbb{R} \) we have \[ - \Re \left( \frac{{\zeta }^{\prime }\left( s\right) }{\zeta \left( s\right) }\right) = O\left( {\log \left( t\right) }\right) - \mathop{\sum }\limits_{\rho }\left( {\frac{\sigma - \beta }{{\left( \sigma - \beta \right) }^{2} + {\left( t - \gamma \right) }^{2}} + \frac{\beta }{{\beta }^{2} + {\gamma }^{2}}}\right) . \] Since \( \sigma > 1 \geq \beta \geq 0 \), we deduce that for all \( s \in \mathbb{C}, - \Re \left( {{\zeta }^{\prime }\left( s\right) /\zeta \left( s\right) }\right) < \) \( O\left( {\log \left( t\right) }\right) \) with \( t = \Im \left( s\right) \) . Now fix some nontrivial zero \( {\rho }_{0} = {\beta }_{0} + i{\gamma }_{0} \) . Then if \( s = \sigma + i{\gamma }_{0} \) (same imaginary part but real part \( \sigma > 1 \) ) we evidently have the stronger inequality \( - \Re \left( {{\zeta }^{\prime }\left( s\right) /\zeta \left( s\right) }\right) < O\left( {\log \left( {\gamma }_{0}\right) }\right) - 1/\left( {\sigma - {\beta }_{0}}\right) \) . Putting all this together in the second inequality of Lemma 10.7.2 applied to \( t = {\gamma }_{0} \) we obtain \[ \frac{3}{\sigma - 1} - \frac{4}{\sigma - {\beta }_{0}} + O\left( {\log \left( {\gamma }_{0}\right) }\right) \geq 0 \] in other words \( 3/\left( {\sigma - 1}\right) - 4/\left( {\sigma - {\beta }_{0}}\right) > - A\log \left( {\gamma }_{0}\right) \) for some constant \( A \) that we may choose strictly positive (since increasing \( A \) gives a worse estimate). Solving for \( 1 - {\beta }_{0} \) gives \[ 1 - {\beta }_{0} \geq \frac{1 - \left( {\sigma - 1}\right) A\log \left( {\gamma }_{0}\right) }{3/\left( {\sigma - 1}\right) + A\log \left( {\gamma }_{0}\right) }. \] Choosing for instance \( \sigma - 1 = 1/\left( {{2A}\log \left( {\gamma }_{0}\right) }\right) \) (this is why we must have \( A > 0) \), we obtain \( 1 - {\beta }_{0} \geq 1/\left( {{14A}\log \left( {\gamma }_{0}\right) }\right) \), proving the theorem. Important Remarks. (1) Using a slight refinement of this proof, it is not difficult to show that in the given region we have \( 1/\zeta \left( s\right) = O\left( {\log \left( t\right) }\right) \), and this zero-free region can be shown to lead to the PNT in the form \[ \pi \left( x\right) = \operatorname{Li}\left( x\right) + O\left( {x\exp \left( {-c\log {\left( x\right) }^{1/2}}\right) }\right) \] for some \( c > 0 \), where \( \operatorname{Li}\left( x\right) \) is as defined before Corollary 10.7.20 below. (2) With much more difficulty one can still improve the zero-free region hence the error term in the PNT. The best-known result is as follows. Set \( g\left( t\right) = \log {\left( t\right) }^{2/3}\log {\left( \log \left( t\right) \right) }^{1/3} \) . There exists \( C > 0 \) such that \( \zeta \left( s\right) = O\left( {g\left( t\right) }\right) \) and \( 1/\zeta \left( s\right) = O\left( {g\left( t\right) }\right) \) uniformly for \( \sigma > 1 - C/g\left( t\right) \), and in particular \( \zeta \left( s\right) \neq 0 \) in that domain. This result is due to N. M. Korobov and I. M. Vinogradov, and is described for instance in [Ell]. It leads to the best known error term for the PNT: \[ \pi \left( x\right) = \operatorname{Li}\left( x\right) + O\left( {x\exp \left( {-c\log {\left( x\right) }^{3/5}\log {\left( \log \left( x\right) \right) }^{-1/5}}\right) }\right) \] for some strictly positive constant \( c \) . This result has remained unchanged for almost half a century, and even the tiny \( \log {\left( \log \left( x\right) \right) }^{-1/5} \) factor has not been improved. ## 10.7.2 Newman's Proof For \( s \in \mathbb{C} \) and \( x \in \mathbb{R} \) we set \[ \Phi \left( s\right) = \mathop{\sum }\limits_{p}\frac{\log p}{{p}^{s}}\;\text{ and }\;\theta \left( x\right) = \mathop{\sum }\limits_{{p \leq x}}\log p. \] The proof proceeds through a series of lemmas. Lemma 10.7.9. The function \( \Phi \left( s\right) - 1/\left( {s - 1}\right) \) is holomorphic in the closed half-plane \( \Re \left( s\right) \geq 1 \) . Proof. It is clear that the series for \( \Phi \left( s\right) \) converges absolutely for \( \Re \left( s\right) > 1 \) and normally for \( \Re \left( s\right) \geq 1 + \varepsilon \) for any fixed \( \varepsilon > 0 \), hence defines an analytic function in \( \Re \left( s\right) > 1 \) . For \( \Re \left( s\right) > 1 \) the absolutely convergent Euler product representation for \( \zeta \left( s\right) \) implies that \[ - \frac{{\zeta }^{\prime }\left( s\right) }{\zeta \left( s\right) } = \mathop{\sum }\limits_{p}\frac{\log p}{{p}^{s} - 1} = \Phi \left( s\right) + \mathop{\sum }\limits_{p}\frac{\log p}{{p}^{s}\left( {{p}^{s} - 1}\right) }. \] The rightmost sum converges absolutely for \( \Re \left( s\right) > 1/2 \), proving that \( \Phi \left( s\right) \) extends meromorphically to \( \Re \left( s\right) > 1/2 \) with poles only at the pole \( s = 1 \) of \( \zeta \left( s\right) \) and at the zeros of \( \zeta \left( s\right) \) . At \( s = 1 \) we have a simple pole with residue 1. Furthermore, by Corollary 10.7.4 we know that \( \zeta \left( s\right) \) does not vanish for \( \Re \left( s\right) \geq 1 \), so that \( \Phi \left( s\right) - 1/\left( {s - 1}\right) \) is holomorphic for \( \Re \left( s\right) \geq 1 \) . Lemma 10.7.10. We have \( \theta \left( x\right) = O\left( x\right) \) . Proof. For a positive integer \( n \) we have \[ {2}^{2n} = \mathop{\sum }\limits_{{0 \leq k \leq {2n}}}\left( \begin{matrix} {2n} \\ k \end{matrix}\right) \geq \left( \begin{matrix} {2n} \\ n \end{matrix}\right) \geq \mathop{\prod }\limits_{{n < p \leq {2n}}}p = {e}^{\theta \left( {2n}\right) - \theta \left( n\right) }. \] Since \( \theta \left( x\right) \) changes by \( O\left( {\log \left( x\right) }\right) \) when \( x \) changes by a bounded amount, we deduce that \( \theta \left( x\right) - \theta \left( {x/2}\right) \leq {Cx} \) for any \( C > \log 2 \) and \( x \geq {x}_{0} = {x}_{0}\left( C\right) \) . Summing this inequality for \( x, x/2,\ldots, x/{2}^{r} \), where \( x/{2}^{r} \geq {x}_{0} > x/{2}^{r + 1} \), we obtain \( \theta \left( x\right) \leq {2Cx} + O\left( 1\right) \), proving the lemma. Lemma 10.7.11. The integral \[ {\int }_{1}^{\infty }\frac{\theta \left( x\right) - x}{{x}^{2}}{dx} \] converges. Proof. For \( \Re \left( s\right) > 1 \) we have by Stieltjes integration \[ \Phi \left( s\right) = \mathop{\sum }\limits_{p}\frac{\log p}{{p}^{s}} = {\int }_{1}^{\infty }\frac{{d\theta }\left( x\right) }{{x}^{s}} = s{\int }_{1}^{\infty }\frac{\theta \left

Corollary 10.7.7. Set \( b = \log \left( {2\pi }\right) - 1 - \gamma /2 \) . Then for all \( s \in \mathbb{C} \) we have the convergent product \[ \zeta \left( s\right) = \frac{{e}^{bs}}{s\left( {s - 1}\right) \Gamma \left( {s/2}\right) }\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\right) {e}^{s/\rho }, \] the product being over all nontrivial zeros of \( \zeta \left( s\right) \) (i.e., such that \( 0 \leq \Re \left( \rho \right) \leq \) 1).

We apply Hadamard's theorem to the function \[ f\left( s\right) = s\left( {1 - s}\right) {\pi }^{-s/2}\Gamma \left( {s/2}\right) \zeta \left( s\right) = 2\left( {1 - s}\right) {\pi }^{-s/2}\Gamma \left( {s/2 + 1}\right) \zeta \left( s\right) . \] Since the zeros of \( \zeta \left( s\right) \) for \( s = - {2k}, k \in {\mathbb{Z}}_{ \geq 1} \), are killed by the poles of \( \Gamma \left( {s/2 + 1}\right) \) and the pole of \( \zeta \left( s\right) \) is killed by \( 1 - s \), it follows that the zeros of \( f\left( s\right) \) are the nontrivial zeros of \( \zeta \left( s\right) \) . Thus for suitable constants \( {a}_{0} \) and \( {a}_{1} \) we have \[ f\left( s\right) = {a}_{0}{e}^{{a}_{1}s}\mathop{\prod }\limits_{\rho }\left( {1 - \frac{s}{\rho }}\right) {e}^{s/\rho }, \] so that \[ \zeta \left( s\right) = \frac{{a}_{0}{e}^{bs}}{2\left( {1 - s}\right) {\pi }^{-s/2}\Gamma

Theorem 2. Let \( \chi \) be a nontrivial Dirichlet character modulo m. Then \( L\left( {1,\chi }\right) \neq 0 \) . Proof. Having already proved that \( L\left( {1,\chi }\right) \neq 0 \) if \( \chi \) is complex we assume \( \chi \) is real. Assume \( L\left( {1,\chi }\right) = 0 \) and consider the function \[ \psi \left( s\right) = \frac{L\left( {s,\chi }\right) L\left( {s,{\chi }_{0}}\right) }{L\left( {{2s},{\chi }_{0}}\right) } \] The zero of \( L\left( {s,\chi }\right) \) at \( s = 1 \) cancels the simple pole of \( L\left( {s,{\chi }_{0}}\right) \) so the numerator is analytic on \( \sigma > 0 \) . The denominator is nonzero and analytic for \( \sigma > \frac{1}{2} \) . Thus \( \psi \left( s\right) \) is analytic on \( \sigma > \frac{1}{2} \) . Moreover, since \( L\left( {{2s},{\chi }_{0}}\right) \) has a pole at \( s = \frac{1}{2} \) we have \( \psi \left( s\right) \rightarrow 0 \) as \( s \rightarrow \frac{1}{2} \) . We assume temporarily that \( s \) is real and \( s > 1 \) . Then \( \psi \left( s\right) \) has an infinite product expansion \[ \psi \left( s\right) = \mathop{\prod }\limits_{p}{\left( 1 - \chi \left( p\right) {p}^{-s}\right) }^{-1}{\left( 1 - {\chi }_{0}\left( p\right) {p}^{-s}\right) }^{-1}\left( {1 - {\chi }_{0}\left( p\right) {p}^{-{2s}}}\right) \] \[ = \mathop{\prod }\limits_{{p \nmid m}}\frac{\left( 1 - {p}^{-{2s}}\right) }{\left( {1 - {p}^{-s}}\right) \left( {1 - \chi \left( p\right) {p}^{-s}}\right) }. \] If \( \chi \left( p\right) = - 1 \) the \( p \) -factor is equal to 1 . Thus \[ \psi \left( s\right) = \mathop{\prod }\limits_{{\chi \left( p\right) = 1}}\frac{1 + {p}^{-s}}{1 - {p}^{-s}} \] where the product is over all \( p \) such that \( \chi \left( p\right) = 1 \) . Now, \[ \frac{1 + {p}^{-s}}{1 - {p}^{-s}} = \left( {1 + {p}^{-s}}\right) \left( {\mathop{\sum }\limits_{{k = 0}}^{\infty }{p}^{-{ks}}}\right) = 1 + 2{p}^{-s} + 2{p}^{-{2s}} + \cdots + . \] Applying Lemma 3 we find that \( \psi \left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{n}^{-s} \) where \( {a}_{n} \geq 0 \) and the series converges for \( s > 1 \) . Note that \( {a}_{1} = 1 \) . (It is possible, but unnecessary to give an explicit formula for \( {a}_{n} \) ). We once again consider \( \psi \left( s\right) \) as a function of a complex variable and expand it in a power series about \( s = 2,\psi \left( s\right) = \mathop{\sum }\limits_{{m = 0}}^{\infty }{b}_{m}{\left( s - 2\right) }^{m} \) . Since \( \psi \left( s\right) \) is analytic for \( \sigma > \frac{1}{2} \) the radius of convergence of this power series is at least \( \frac{3}{2} \) . To compute the \( {b}_{m} \) we use Taylor’s theorem, i.e., \( {b}_{m} = {\psi }^{\left( m\right) }\left( 2\right) /m \) ! where \( {\psi }^{\left( m\right) }\left( s\right) \) is the \( m \) th derivative of \( \psi \left( s\right) \) . Since \( \psi \left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{n}^{-s} \) we find \( {\psi }^{\left( m\right) }\left( 2\right) = \) \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{\left( -\ln n\right) }^{m}{n}^{-2} = {\left( -1\right) }^{m}{c}_{m} \) with \( {c}_{m} \geq 0 \) . Thus \( \psi \left( s\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{m}{\left( 2 - s\right) }^{m} \) with \( {c}_{m} \) nonnegative and \( {c}_{0} = \psi \left( 2\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{n}^{-2} \geq {a}_{1} = \overline{1} \) . It follows that for real \( s \) in the interval \( \left( {\frac{1}{2},2}\right) \) we have \( \bar{\psi }\left( s\right) \geq 1 \) . This contradicts \( \psi \left( s\right) \rightarrow 0 \) as \( s \rightarrow \frac{1}{2} \), and so \( L\left( {1,\chi }\right) \neq 0 \) . We are now in a position to prove Proposition 16.4.1. Suppose \( \chi \) is a nontrivial Dirichlet character. We want to show \( G\left( {s,\chi }\right) \) remains bounded as \( s \rightarrow 1 \) through real values \( s > 1 \) . Since \( L\left( {1,\chi }\right) \neq 0 \) there is a disc \( D \) about \( L\left( {1,\chi }\right) \) such \( 0 \notin D \) . Let \( \ln z \) be a single-valued branch of the logarithm defined on \( D \) . There is a \( \delta > 0 \) such that \( L\left( {s,\chi }\right) \in D \) for \( s \in \left( {1,1 + \delta }\right) \) . Consider \( \ln L\left( {s,\chi }\right) \) and \( G\left( {s,\chi }\right) \) for \( s \) in this interval. The exponential of both functions is \( L\left( {s,\chi }\right) \) . Thus there is an integer \( N \) such that \( G\left( {s,\chi }\right) = {2\pi iN} + \ln L\left( {s,\chi }\right) \) for \( s \in \left( {1,1 + \delta }\right) \) . This implies \( \mathop{\lim }\limits_{{s \rightarrow 1}}G\left( {s,\chi }\right) \) exists and is equal to \( {2\pi iN} + \ln L\left( {1,\chi }\right) \) . Since \( G\left( {s,\chi }\right) \) has a limit as \( s \rightarrow 1 \) it clearly remains bounded. ## §6 Evaluating \( L\left( {s,\chi }\right) \) at Negative Integers In the last section we showed how to analytically continue \( L\left( {s,\chi }\right) \) into the region \( \{ s \in \mathbb{C} \mid \sigma > 0\} \) . Riemann showed how to analytically continue these functions to the whole complex plane. As noted earlier this fact has important consequences for number theory. For example, the values \( L\left( {1 - k,\chi }\right) \), where \( k \) is a positive integer, are closely related to the Bernoulli numbers. A knowledge of these numbers has deep connections with the theory of cyclotomic fields. We will analytically continue \( L\left( {s,\chi }\right) \) and evaluate the numbers \( L\left( {1 - k,\chi }\right) \) following a method due to D. Goss [141]. Before beginning we need to discuss some properties of the \( \Gamma \) -function. This is defined by \[ \Gamma \left( s\right) = {\int }_{0}^{\infty }{e}^{-t}{t}^{s - 1}{dt} \] (i) It is not hard to see that the integral converges and defines an analytic function on the region \( \{ s \in \mathbb{C} \mid \sigma > 0\} \) . For \( \sigma > 1 \) we integrate by parts and find \[ \Gamma \left( s\right) = - {\left. {e}^{-t}{t}^{s - 1}\right| }_{0}^{\infty } + \left( {s - 1}\right) {\int }_{0}^{\infty }{e}^{-t}{t}^{s - 2}{dt} \] It follows that \( \Gamma \left( s\right) = \left( {s - 1}\right) \Gamma \left( {s - 1}\right) \) for \( \sigma > 1 \) . Since \( \Gamma \left( 1\right) = {\int }_{0}^{\infty }{e}^{-t}{dt} \) \( = 1 \) we see \( \Gamma \left( {n + 1}\right) = n \) ! for positive integers \( n \) . The functional equation \( \Gamma \left( s\right) = \left( {s - 1}\right) \Gamma \left( {s - 1}\right) \) enables us to analytically continue \( \Gamma \left( s\right) \) by a step by step process. If \( \sigma > - 1 \) we define \( {\Gamma }_{1}\left( s\right) \) by \[ {\Gamma }_{1}\left( s\right) = \frac{1}{s}\Gamma \left( {s + 1}\right) \] (ii) For \( \sigma > 0,{\Gamma }_{1}\left( s\right) = \Gamma \left( s\right) \) . Moreover, \( {\Gamma }_{1}\left( s\right) \) is analytic on \( \sigma > - 1 \) except for a simple pole at \( s = 0 \) . Similarly, if \( k \) is a positive integer we define \[ {\Gamma }_{k}\left( s\right) = \frac{1}{s\left( {s + 1}\right) \cdots \left( {s + k - 1}\right) }\Gamma \left( {s + k}\right) . \] \( {\Gamma }_{k}\left( s\right) \) is analytic on \( \{ s \in \mathbb{C} \mid \sigma > - k\} \) except for simple poles at \( s = 0, - 1,\ldots \) , \( 1 - k \) and \( {\Gamma }_{k}\left( s\right) = \Gamma \left( s\right) \) for \( \sigma > 0 \) . These functions fit together to give an analytic continuation of \( \Gamma \left( s\right) \) to the whole complex plane with poles at the nonpositive integers and nowhere else. From now on \( \Gamma \left( s\right) \) will denote this extended function. We remark, without proof, that \( \Gamma {\left( s\right) }^{-1} \) is entire. We will now show how to analytically continue \( \zeta \left( s\right) \) by the same process. It is necessary to express \( \zeta \left( s\right) \) as an integral. In Equation (i) substitute \( {nt} \) for \( t \) . We find, for \( \sigma > 1 \) \[ {n}^{-s}\Gamma \left( s\right) = {\int }_{0}^{\infty }{e}^{-{nt}}{t}^{s - 1}{dt} \] (iii) Sum both sides of (iii) for \( n = 1,2,3,\ldots \) . It is not hard to justify interchanging the sum and the integral. The result is \[ \Gamma \left( s\right) \zeta \left( s\right) = {\int }_{0}^{\infty }\frac{{e}^{-t}}{1 - {e}^{-t}}{t}^{s - 1}{dt}. \] (iv) If we tried to integrate by parts at this stage we would be blocked by the fact that \( 1 - {e}^{-t} \) is zero when \( t = 0 \) . To get around this we use a trick. In (iv) substitute \( {2t} \) for \( t \) . We find \[ {2}^{1 - s}\Gamma \left( s\right) \zeta \left( s\right) = 2{\int }_{0}^{\infty }\frac{{e}^{-{2t}}}{1 - {e}^{-{2t}}}{t}^{s - 1}{dt}. \] (v) Define \( {\zeta }^{ * }\left( s\right) = \left( {1 - {2}^{1 - s}}\right) \zeta \left( s\right) \) and \( R\left( x\right) = x/\left( {1 - x}\right) - 2\left( {{x}^{2}/\left( {1 - {x}^{2}}\right) }\right) \) . Subtracting (v) from (iv) yields \[ \Gamma \left( s\right) {\zeta }^{ * }\left( s\right) = {\int }_{0}^{\infty }R\left( {e}^{-t}\right) {t}^{s - 1}{dt}. \] (vi) What has been gained? A simple algebraic manipulation shows \( R\left( x\right) = \) \( x/\left( {1 + x}\right) \) . Thus \( R\left( {e}^{-t}\right) = {e}^{-t}/\left( {1 + {e}^{-t}}\right) \) has a denominator that does not vanish at \( t = 0 \) . The integral in Equation (vi) thus converges for \( \sigma > 0 \) and this equation provides a continuation for \( \zeta \left( s\right) \) to the region \( \{ s \in \mathbb{C} \mid \sigma > 0\} \) . Let \( {R}_{0}\left( t\right) = R\left( {e}^{-t}\right) \) and for \( m \geq 1,{R}_{m}\left( t\right) = \left( {{d}^{m}/d{t}^{m}}\right) R\left( {e}^{-t}\right) \) . It is easy to see that \( {R}_{m}\left( t\right) = {e}^{-t}{P}_{m}\left( {e}^{-t}\right) {\left( 1 + {e}^{-t}\right) }^{-{2}^{m}} \) where \( {P}_{m} \) is a polynomial. It follows that \( {R}_{m}\left( 0\right) \) is finite and \( {R}_{m

Theorem 2. Let \( \chi \) be a nontrivial Dirichlet character modulo m. Then \( L\left( {1,\chi }\right) \neq 0 \) .

Having already proved that \( L\left( {1,\chi }\right) \neq 0 \) if \( \chi \) is complex, we assume \( \chi \) is real. Assume \( L\left( {1,\chi }\right) = 0 \) and consider the function \[ \psi \left( s\right) = \frac{L\left( {s,\chi }\right) L\left( {s,{\chi }_{0}}\right) }{L\left( {{2s},{\chi }_{0}}\right) } \] The zero of \( L\left( {s,\chi }\right) \) at \( s = 1 \) cancels the simple pole of \( L\left( {s,{\chi }_{0}}\right) \) so the numerator is analytic on \( \sigma > 0 \) . The denominator is nonzero and analytic for \( \sigma > \frac{1}{2} \) . Thus \( \psi \left( s\right) \) is analytic on \( \sigma > \frac{1}{2} \) . Moreover, since \( L\left( {{2s},{\chi }_{0}}\right) \) has a pole at \( s = \frac{1}{2} \) we have \( \psi \left( s\right) \rightarrow 0 \) as \( s \rightarrow \frac{1}{2} \) . We assume temporarily that \( s \) is real and \( s > 1 \) . Then \( \psi \left( s\right) \) has an infinite product expansion \[ \psi \left( s\right) = \mathop{\prod }\limits_{p}{\left( 1 - \chi \left( p\right) {p}^{-s}\right) }^{-1}{\left( 1 - {\chi }_{0}\left( p\right) {p}^{-s}\right) }^{-1}\left( {1 - {\chi }_{0}\left( p\right) {p}^{-{2s}}}\right) \] \[ = \mathop{\prod }\limits_{{p \nmid m}}\frac{\left( 1 - {p}^{-{2s}}\right) }{\left( {1 - {p}^{-s}}\right) \left( {1 - \chi \left( p\right) {p}^{-s}}\right) }. \] If \( \chi \left( p\right) = - 1 \) the \( p \) -factor is equal to 1 . Thus \[ \psi \left( s\right) = \mathop{\prod }\limits_{{\chi \left( p\right) = 1}}\frac{1 + {p}^{-s}}{1 - {p}^{-s}} \] where the product is over all \( p \) such that \( \chi \left( p\right) = 1 \) . Now, \[ \frac{1 + {p}^{-s}}{1 - {p}^{-s}} = \left( {1 + {p}^{-s}}\right) \left( {\mathop{\sum }\limits_{{k = 0}}^{\infty }{p}^{-{ks}}}\right) = 1 + 2{p}^{-s} + 2{p}^{-{2s}} + \cdots + . \] Applying Lemma 3 we find that \( \psi \left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{n}^{-s} \) where \( {a}_{n} \geq 0 \) and the series converges for \( s > 1 \) . Note that \( {a}_{1} = 1 \) . (It is possible, but unnecessary to give an explicit formula for \( {a}_{n} ).\) We once again consider \(

Theorem 4.2.2 Let \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{n} \) be a set of generators for a finitely generated \( \mathbb{Z} \) -module \( M \), and let \( N \) be a submodule. (a) \( \exists {\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{m} \) in \( N \) with \( m \leq n \) such that \[ N = \mathbb{Z}{\beta }_{1} + \mathbb{Z}{\beta }_{2} + \cdots + \mathbb{Z}{\beta }_{m} \] and \( {\beta }_{i} = \mathop{\sum }\limits_{{j \geq i}}{p}_{ij}{\alpha }_{j} \) with \( 1 \leq i \leq m \) and \( {p}_{ij} \in \mathbb{Z} \) . (b) If \( m = n \), then \( \left\lbrack {M : N}\right\rbrack = {p}_{11}{p}_{22}\cdots {p}_{nn} \) . Proof. (a) We will proceed by induction on the number of generators of a \( \mathbb{Z} \) -module. This is trivial when \( n = 0 \) . We can assume that we have proved the above statement to be true for all \( \mathbb{Z} \) -modules with \( n - 1 \) or fewer generators, and proceed to prove it for \( n \) . We define \( {M}^{\prime } \) to be the submodule generated by \( {\alpha }_{2},{\alpha }_{3},\ldots ,{\alpha }_{n} \) over \( \mathbb{Z} \), and define \( {N}^{\prime } \) to be \( N \cap {M}^{\prime } \) . Now, if \( n = 1 \), then \( {M}^{\prime } = 0 \) and there is nothing to prove. If \( N = {N}^{\prime } \), then the statement is true by our induction hypothesis. So we assume that \( N \neq {N}^{\prime } \) and consider \( A \), the set of all integers \( k \) such that \( \exists {k}_{2},{k}_{3},\ldots ,{k}_{n} \) with \( k{\alpha }_{1} + {k}_{2}{\alpha }_{2} + \cdots + {k}_{n}{\alpha }_{n} \in N \) . Since \( N \) is a submodule, we deduce that \( A \) is a subgroup of \( \mathbb{Z} \) . All additive subgroups of \( \mathbb{Z} \) are of the form \( m\mathbb{Z} \) for some integer \( m \), and so \( A = {k}_{11}\mathbb{Z} \) for some \( {k}_{11} \) . Then let \( {\beta }_{1} = {k}_{11}{\alpha }_{1} + {k}_{12}{\alpha }_{2} + \cdots + {k}_{1n}{\alpha }_{n} \in N \) . If we have some \( \alpha \in N \) , then \[ \alpha = \mathop{\sum }\limits_{{i = 1}}^{n}{h}_{i}{\alpha }_{i} \] with \( {h}_{i} \in \mathbb{Z} \) and \( {h}_{1} \in A \) so \( {h}_{1} = a{k}_{11} \) . Therefore, \( \alpha - a{\beta }_{1} \in {N}^{\prime } \) . By the induction hypothesis, there exist \[ {\beta }_{i} = \mathop{\sum }\limits_{{j \geq i}}{k}_{ij}{\alpha }_{j} \] \( i = 2,3\ldots, m \), which generate \( {N}^{\prime } \) over \( \mathbb{Z} \) and which satisfy all the conditions above. It is clear that adding \( {\beta }_{1} \) to this list gives us a set of generators of \( N \) . (b) Consider \( \alpha \), an arbitrary element of \( M \) . Then \( \alpha = \sum {c}_{i}{\alpha }_{i} \) . Recalling that \[ {\beta }_{i} = \mathop{\sum }\limits_{{j \geq i}}{p}_{ij}{\alpha }_{j} \] we write \( {c}_{1} = {p}_{11}{q}_{1} + {r}_{1} \), with \( 0 \leq {r}_{1} < {p}_{11} \) . Then \( \alpha - {q}_{1}{\beta }_{1} = \sum {c}_{i}^{\prime }{\alpha }_{i} \) where \( 0 \leq {c}_{1}^{\prime } < {p}_{11} \) . Note that \( \alpha \equiv \alpha - {q}_{1}{\beta }_{1}\left( {\;\operatorname{mod}\;N}\right) \) . Next we write \( {c}_{2}^{\prime } = {p}_{22}{q}_{2} + {r}_{2} \), where \( 0 \leq {r}_{2} < {p}_{22} \), and note that \[ \alpha \equiv \alpha - {q}_{1}{\beta }_{1} - {q}_{2}{\beta }_{2}\;\left( {\;\operatorname{mod}\;N}\right) \] It is clear by induction that we can continue this process to arrive at an expression \( {\alpha }^{\prime } = \sum {k}_{i}{\alpha }_{i} \) with \( 0 \leq {k}_{i} < {p}_{ii} \) and \( \alpha \equiv {\alpha }^{\prime }\left( {\;\operatorname{mod}\;N}\right) \) . It remains only to show that if we have \( \alpha = \sum {c}_{i}{\alpha }_{i} \) and \( \beta = \sum {d}_{i}{\alpha }_{i} \) where \( {c}_{i} \neq {d}_{i} \) for at least one \( i \) and \( 0 \leq {c}_{i},{d}_{i} < {p}_{ii} \), then \( \alpha \) and \( \beta \) are distinct mod \( N \) . Suppose that this is not true, and that \[ \sum {c}_{i}{\alpha }_{i} \equiv \sum {d}_{i}{\alpha }_{i}\;\left( {\;\operatorname{mod}\;N}\right) \] where \( {c}_{i} \neq {d}_{i} \) for at least one \( i \) . Suppose \( {c}_{i} = {d}_{i} \) for \( i < r \) and \( {c}_{r} \neq {d}_{r} \) . Then \( \sum \left( {{c}_{i} - {d}_{i}}\right) {\alpha }_{i} \in N \), so \[ \mathop{\sum }\limits_{{i \geq r}}\left( {{c}_{i} - {d}_{i}}\right) {\alpha }_{i} = \mathop{\sum }\limits_{{i \geq r}}{k}_{i}{\beta }_{i} = \mathop{\sum }\limits_{{i \geq r}}{k}_{i}\left( {\mathop{\sum }\limits_{{j \geq i}}{p}_{ij}{\alpha }_{j}}\right) . \] Since \( {c}_{r},{d}_{r} \) are both less than \( {p}_{rr} \), we have \( {c}_{r} = {d}_{r} \), a contradiction. Thus, each coset in \( M/N \) has a unique representative \[ \alpha = \sum {c}_{i}{\alpha }_{i} \] with \( 0 \leq {c}_{i} < {p}_{ii} \), and there are \( {p}_{11}{p}_{22}\cdots {p}_{nn} \) of them. So \( \left\lbrack {M : N}\right\rbrack = \) \( {p}_{11}{p}_{22}\cdots {p}_{nn} \) Exercise 4.2.3 Show that \( {\mathcal{O}}_{K} \) has an integral basis. Exercise 4.2.4 Show that \( \det \left( {\operatorname{Tr}\left( {{\omega }_{i}{\omega }_{j}}\right) }\right) \) is independent of the choice of integral basis. We are justified now in making the following definition. If \( K \) is an algebraic number field of degree \( n \) over \( \mathbb{Q} \), define the discriminant of \( K \) as \[ {d}_{K} \mathrel{\text{:=}} \det {\left( {\omega }_{i}^{\left( j\right) }\right) }^{2}, \] where \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{n} \) is an integral basis for \( K \) . Exercise 4.2.5 Show that the discriminant is well-defined. In other words, show that given \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{n} \) and \( {\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{n} \), two integral bases for \( K \), we get the same discriminant for \( K \) . We can generalize the notion of a discriminant for arbitrary elements of \( K \) . Let \( K/\mathbb{Q} \) be an algebraic number field, a finite extension of \( \mathbb{Q} \) of degree \( n \) . Let \( {\sigma }_{1},{\sigma }_{2},\ldots ,{\sigma }_{n} \) be the embeddings of \( K \) . For \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \in K \) we can define \( {d}_{K/\mathbb{Q}}\left( {{a}_{1},\ldots ,{a}_{n}}\right) = {\left\lbrack \det \left( {\sigma }_{i}\left( {a}_{j}\right) \right) \right\rbrack }^{2} \) . Exercise 4.2.6 Show that \[ {d}_{K/\mathbb{Q}}\left( {1, a,\ldots ,{a}^{n - 1}}\right) = \mathop{\prod }\limits_{{i > j}}{\left( {\sigma }_{i}\left( a\right) - {\sigma }_{j}\left( a\right) \right) }^{2}. \] We denote \( {d}_{K/\mathbb{Q}}\left( {1, a,\ldots ,{a}^{n - 1}}\right) \) by \( {d}_{K/\mathbb{Q}}\left( a\right) \) . Exercise 4.2.7 Suppose that \( {u}_{i} = \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{ij}{v}_{j} \) with \( {a}_{ij} \in \mathbb{Q},{v}_{j} \in K \) . Show that \( {d}_{K/\mathbb{Q}}\left( {{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) = {\left( \det \left( {a}_{ij}\right) \right) }^{2}{d}_{K/\mathbb{Q}}\left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) . \) For a module \( M \) with submodule \( N \), we can define the index of \( N \) in \( M \) to be the number of elements in \( M/N \), and denote this by \( \left\lbrack {M : N}\right\rbrack \) . Suppose \( \alpha \) is an algebraic integer of degree \( n \), generating a field \( K \) . We define the index of \( \alpha \) to be the index of \( \mathbb{Z} + \mathbb{Z}\alpha + \cdots + \mathbb{Z}{\alpha }^{n - 1} \) in \( {\mathcal{O}}_{K} \) . Exercise 4.2.8 Let \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \in {\mathcal{O}}_{K} \) be linearly independent over \( \mathbb{Q} \) . Let \( N = \mathbb{Z}{a}_{1} + \mathbb{Z}{a}_{2} + \cdots + \mathbb{Z}{a}_{n} \) and \( m = \left\lbrack {{\mathcal{O}}_{K} : N}\right\rbrack \) . Prove that \[ {d}_{K/\mathbb{Q}}\left( {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right) = {m}^{2}{d}_{K} \] ## 4.3 Examples Example 4.3.1 Suppose that the minimal polynomial of \( \alpha \) is Eisensteinian with respect to a prime \( p \), i.e., \( \alpha \) is a root of the polynomial \[ {x}^{n} + {a}_{n - 1}{x}^{n - 1} + \cdots + {a}_{1}x + {a}_{0}, \] where \( p \mid {a}_{i},0 \leq i \leq n - 1 \) and \( {p}^{2} \nmid {a}_{0} \) . Show that the index of \( \alpha \) is not divisible by \( p \) . Solution. Let \( M = \mathbb{Z} + \mathbb{Z}\alpha + \cdots + \mathbb{Z}{\alpha }^{n - 1} \) . First observe that since \[ {\alpha }^{n} + {a}_{n - 1}{\alpha }^{n - 1} + \cdots + {a}_{1}\alpha + {a}_{0} = 0, \] then \( {\alpha }^{n}/p \in M \subseteq {\mathcal{O}}_{K} \) . Also, \( \left| {{\mathrm{N}}_{K}\left( \alpha \right) }\right| = {a}_{0} ≢ 0\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . We will proceed by contradiction. Suppose \( p \mid \left\lbrack {{\mathcal{O}}_{K} : M}\right\rbrack \) . Then there is an element of order \( p \) in the group \( {\mathcal{O}}_{K}/M \), meaning \( \exists \xi \in {\mathcal{O}}_{K} \) such that \( \xi \notin M \) but \( {p\xi } \in M \) . Then \[ {p\xi } = {b}_{0} + {b}_{1}\alpha + \cdots + {b}_{n - 1}{\alpha }^{n - 1}, \] where not all the \( {b}_{i} \) are divisible by \( p \), for otherwise \( \xi \in M \) . Let \( j \) be the least index such that \( p \nmid {b}_{j} \) . Then \[ \eta = \xi - \left( {\frac{{b}_{0}}{p} + \frac{{b}_{1}}{p}\alpha + \cdots + \frac{{b}_{j - 1}}{p}{\alpha }^{j - 1}}\right) \] \[ = \frac{{b}_{j}}{p}{\alpha }^{j} + \frac{{b}_{j + 1}}{p}{\alpha }^{j + 1} + \cdots + \frac{{b}_{n}}{p}{\alpha }^{n} \] is in \( {\mathcal{O}}_{K} \), since both \( \xi \) and \[ \frac{{b}_{0}}{p} + \frac{{b}_{1}}{p}\alpha + \cdots + \frac{{b}_{n}}{p}{\alpha }^{j - 1} \] are in \( {\mathcal{O}}_{K} \) . If \( \eta \in {\mathcal{O}}_{K} \), then of course \( \eta {\alpha }^{n - j - 1} \) is also in \( {\mathcal{O}}_{K} \), and \[ \eta {\alpha }^{n - j - 1} = \frac{{b}_{j}}{p}{\alpha }^{n - 1} + \frac{{\alpha }^{n}}{p}\left( {{b}_{j + 1} + {b}_{j + 2}\alpha + \cdots + {b}_{n}{\alpha }^{n - j - 2}}\right) . \] Since both \( {\alpha }^{n}/p \) and \( \left( {{b}_{j + 1} + {b}_{j + 2}\alpha + \cdots + {b}_{n}{\alpha }^{n - j - 2}}\right)

Theorem 4.2.2 Let \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{n} \) be a set of generators for a finitely generated \( \mathbb{Z} \) -module \( M \), and let \( N \) be a submodule. (a) \( \exists {\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{m} \) in \( N \) with \( m \leq n \) such that \[ N = \mathbb{Z}{\beta }_{1} + \mathbb{Z}{\beta }_{2} + \cdots + \mathbb{Z}{\beta }_{m} \] and \( {\beta }_{i} = \mathop{\sum }\limits_{{j \geq i}}{p}_{ij}{\alpha }_{j} \) with \( 1 \leq i \leq m \) and \( {p}_{ij} \in \mathbb{Z} \) . (b) If \( m = n \), then \( \left\lbrack {M : N}\right\rbrack = {p}_{11}{p}_{22}\cdots {p}_{nn} \) .

(a) We will proceed by induction on the number of generators of a \( \mathbb{Z} \) -module. This is trivial when \( n = 0 \) . We can assume that we have proved the above statement to be true for all \( \mathbb{Z} \) -modules with \( n - 1 \) or fewer generators, and proceed to prove it for \( n \) . We define \( {M}^{\prime } \) to be the submodule generated by \( {\alpha }_{2},{\alpha }_{3},\ldots ,{\alpha }_{n} \) over \( \mathbb{Z} \), and define \( {N}^{\prime } \) to be \( N \cap {M}^{\prime } \) . Now, if \( n = 1 \), then \( {M}^{\prime } = 0 \) and there is nothing to prove. If \( N = {N}^{\prime } \), then the statement is true by our induction hypothesis. So we assume that \( N \neq {N}^{\prime } \) and consider \( A \), the set of all integers \( k \) such that \( \exists {k}_{2},{k}_{3},\ldots ,{k}_{n} \) with \( k{\alpha }_{1} + {k}_{2}{\alpha }_{2} + \cdots + {k}_{n}{\alpha }_{n} \in N \) . Since \( N \) is a submodule, we deduce that \( A \) is a subgroup of \( \mathbb{Z} \) . All additive subgroups of \( \mathbb{Z} \) are of the form \( m\mathbb{Z} \) for some integer \( m \), and so \( A = {k}_{11}\mathbb{Z} \) for some \( {k}_{11} \) . Then let \( {\beta }_{1} = {k}_{11}{\alpha }_{1} + {k}_{12}{\alpha }_{2} + \cdots + {k}_{1n}{\alpha }_{n} \in N \) . If we have some \( \alpha \in N \) , then \[ \alpha = \mathop{\sum }\limits_{{i = 1}}^{n}{h}_{i}{\alpha }_{i} \] with \( {h}_{i} \in \mathbb{Z} \) and

Lemma 14.1.7. Let \( f : X \rightarrow Y \) be a map between two metric spaces. (i) Suppose \( \omega : \lbrack 0,\infty ) \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is a function such that \( d\left( {f\left( x\right), f\left( y\right) }\right) \leq \omega \left( {d\left( {x, y}\right) }\right) \) for every \( x, y \in X \), and \( \omega \left( s\right) \rightarrow 0 \) as \( s \rightarrow {0}^{ + } \) . Then \( f \) is uniformly continuous and \( \omega \geq {\omega }_{f} \) . (ii) \( f \) is \( K \) -Lipschitz if and only if \( {\omega }_{f}\left( s\right) \leq {Ks} \) for all \( s > 0 \) . (iii) If \( f \) is uniformly continuous and \( X \) is metrically convex, then \( {\omega }_{f}\left( s\right) < \infty \) for all \( s > 0 \) . Proof. We do (iii) and leave the other statements as an exercise. We need to show that for \( s > 0 \) there is \( {C}_{s} > 0 \) such that \( d\left( {f\left( x\right), f\left( y\right) }\right) \leq {C}_{s} \) whenever \( d\left( {x, y}\right) \leq s \) . From the definition of uniform continuity there exists \( {\delta }_{1} > 0 \) such that if \( 0 < \) \( d\left( {a, b}\right) < {\delta }_{1} \), then \( d\left( {f\left( a\right), f\left( b\right) }\right) < 1 \) . Let \( N = {N}_{s} \in \mathbb{N} \) be such that \( s/N < {\delta }_{1} \) . By the metric convexity of \( X \) one can find points \( x = {x}_{0},{x}_{1},\ldots ,{x}_{N} = y \) in \( X \) such that \( d\left( {{x}_{j},{x}_{j + 1}}\right) < d\left( {x, y}\right) /N < s/N < {\delta }_{1} \) for \( 0 \leq j \leq N - 1 \) . Therefore, by the triangle inequality, \[ d\left( {f\left( x\right), f\left( y\right) }\right) \leq \mathop{\sum }\limits_{{j = 0}}^{{N - 1}}d\left( {f\left( {x}_{j}\right), f\left( {x}_{j + 1}\right) }\right) \leq N, \] and our claim holds with \( {C}_{s} = {N}_{s} \) . By Lemma 14.1.7 (ii), the modulus of continuity of a Lipschitz map is controlled by a linear function. Roughly speaking, one could interpret this by saying that the Lipschitz behavior of a map is closer to a linear behavior than the uniform behavior; hence it seems natural to attempt to Lipschitz-ize a uniformly continuous map. The next result [53] does this in an explicit manner. Proposition 14.1.8 (Corson and Klee [53]). Let \( f : X \rightarrow Y \) be a uniformly continuous map. If \( X \) is metrically convex, then for every \( \theta > 0 \) there exists a constant \( {K}_{\theta } > 0 \) such that \( d\left( {f\left( x\right), f\left( y\right) }\right) \leq {K}_{\theta }d\left( {x, y}\right) \) whenever \( d\left( {x, y}\right) \geq \theta \) . Proof. Fix \( \theta > 0 \) . Given \( x, y \) in \( X \) with \( d\left( {x, y}\right) \geq \theta \), let \( m \) be the smallest integer such that \( d\left( {x, y}\right) /m < \theta \) . By the metric convexity of \( X \) we may choose points \( x = {x}_{0},{x}_{1},\ldots ,{x}_{m} = y \) in \( X \) with \( d\left( {{x}_{j},{x}_{j + 1}}\right) < \theta \) . The triangle inequality, Lemma 14.1.7(iii), and our choice of \( m \) yield \[ \left. {d\left( {f\left( x\right), f\left( y\right) }\right) \leq \mathop{\sum }\limits_{{j = 0}}^{{m - 1}}{df}\left( {x}_{j}\right), f\left( {x}_{j + 1}\right) }\right) \leq m{\omega }_{f}\left( \theta \right) \leq \frac{2{\omega }_{f}\left( \theta \right) }{\theta }d\left( {x, y}\right) . \] Definition 14.1.9. Given a map \( f : X \rightarrow Y \) between two metric spaces \( X \) and \( Y \), for \( \theta > 0 \) let us define the (possibly infinite) number \[ {\operatorname{Lip}}_{\theta }\left( f\right) = \sup \left\{ {\frac{d\left( {f\left( x\right), f\left( y\right) }\right) }{d\left( {x, y}\right) } : x, y \in X, d\left( {x, y}\right) \geq \theta }\right\} . \] Obviously, \( {\operatorname{Lip}}_{\theta }\left( f\right) \) decreases as \( \theta \) increases. Put \[ {\operatorname{Lip}}_{\infty }\left( f\right) \mathrel{\text{:=}} \mathop{\inf }\limits_{{\theta > 0}}{\operatorname{Lip}}_{\theta }\left( f\right) = \mathop{\lim }\limits_{{\theta \rightarrow \infty }}{\operatorname{Lip}}_{\theta }\left( f\right) \] to denote the (possibly zero) asymptotic Lipschitz constant of \( f \) . The map \( f \) is coarse Lipschitz if \( {\operatorname{Lip}}_{\infty }\left( f\right) < \infty \), i.e., there is \( \theta > 0 \) for which \( {\operatorname{Lip}}_{\theta }\left( f\right) < \infty \) . In this case it is said that \( f \) satisfies a Lipschitz condition for large distances. When \( {\operatorname{Lip}}_{\theta }\left( f\right) < \infty \) for all \( \theta > 0 \), we say that \( f \) is Lipschitz at large distances. Definition 14.1.10. A map \( f : X \rightarrow Y \) between metric spaces is a coarse Lispchitz embedding if there exist constants \( 0 < {c}_{1} < {c}_{2} \) and \( \theta > 0 \) such that \[ {c}_{1}d\left( {x, y}\right) \leq d\left( {f\left( x\right), f\left( y\right) }\right) \leq {c}_{2}d\left( {x, y}\right) ,\;\forall x, y \in X\text{ with }d\left( {x, y}\right) \geq \theta . \] (14.5) Note that a coarse Lipschitz embedding need not be injective. Coarse Lipschitz embeddings are the large-scale analogue of Lipschitz embed-dings. Of course, this notion is of interest only for unbounded metric spaces. Definition 14.1.11. A metric space \( \left( {X, d}\right) \) is said to be bounded if there exists \( r > 0 \) such that \( d\left( {x, y}\right) \leq r \) for all \( x, y \in X \) . Otherwise, it is called unbounded. A coarse Lipschitz embedding does not observe the fine structure of a metric space in a neighborhood of a point, since it need not be continuous; it captures only the macroscopic structure of the space, i.e., where distances are large. This is in stark contrast to a uniform homeomorphism, which observes only the local structure of the space. Example 14.1.12. Consider \( \mathbb{R} \) with its standard metric \( d \) induced by the absolute value, and define new metrics on \( \mathbb{R} \) by \[ \rho \left( {x, y}\right) = \min \{ \left| {x - y}\right| ,1\} ,\;\forall \left( {x, y}\right) \in {\mathbb{R}}^{2},\;\sigma \left( {x, y}\right) = \left\{ \begin{array}{ll} \left| {x - y}\right| + 1 & x \neq y, \\ 0 & x = y. \end{array}\right. \] Then the identity map id: \( \left( {\mathbb{R}, d}\right) \rightarrow \left( {\mathbb{R},\rho }\right) \) is a uniform homeomorphism but not a coarse Lipschitz embedding, whereas id: \( \left( {\mathbb{R}, d}\right) \rightarrow \left( {\mathbb{R},\sigma }\right) \) is a coarse Lipschitz embedding that is not a uniform homeomorphism. Remark 14.1.13. In geometric group theory it is customary to define a quasi-isometric embedding as an embedding \( f : X \rightarrow Y \) such that for all \( x, y \in X \) the inequalities \[ \frac{1}{A}d\left( {x, y}\right) - B \leq d\left( {f\left( x\right), f\left( y\right) }\right) \leq {Ad}\left( {x, y}\right) + B \] hold for some constants \( A \geq 1 \) and \( B \geq 0 \) . Lemma 14.1.15 states that when the domain space is metrically convex, then an embedding is quasi-isometric if and only if it is a coarse Lipschitz embedding. This terminology will be preferred in this book, since in nonlinear Banach space theory a quasi-isometry means that for every \( \epsilon > 0 \) there exists a Lipschitz isomorphism with distortion constant \( 1 + \epsilon \) . Let us record an elementary fact that will be handy. Lemma 14.1.14. Let \( X \) be an unbounded metrically convex space. Given \( x \in X \) and \( r > 0 \) there is \( y \in X \) such that \( d\left( {x, y}\right) = r \) . Proof. Since \( X \) is unbounded, there is \( z \in X \) such that \( d\left( {x, z}\right) = R > r \) . Let \( t = r/R \in \left( {0,1}\right) \) . The metric convexity of \( X \) yields \( y \in X \) such that \( d\left( {x, y}\right) = {td}\left( {x, z}\right) = r. \) Lemma 14.1.15. Let \( f : X \rightarrow Y \) be a map between two unbounded metric spaces. (i) Suppose that for some constants \( A \geq 1 \) and \( B \geq 0 \) the inequalities \[ \frac{1}{A}d\left( {x, y}\right) - B \leq d\left( {f\left( x\right), f\left( y\right) }\right) \leq {Ad}\left( {x, y}\right) + B \] hold for all \( x, y \in X \) . Then \( f \) is a coarse Lipschitz embedding. (ii) If \( X \) is metrically convex and \( f \) is a coarse Lipschitz embedding, then there exist constants \( A \geq 1 \) and \( B \geq 0 \) such that \[ \frac{1}{A}d\left( {x, y}\right) - B \leq d\left( {f\left( x\right), f\left( y\right) }\right) \leq {Ad}\left( {x, y}\right) + B,\;\forall x, y \in X. \] (14.6) Proof. (i) Suppose \( d\left( {x, y}\right) > \theta \) for a given \( \theta > 0 \) . Then, \[ d\left( {f\left( x\right), f\left( y\right) }\right) \leq {Ad}\left( {x, y}\right) + B \leq {Ad}\left( {x, y}\right) + \frac{B}{\theta }d\left( {x, y}\right) = \left( {A + \frac{B}{\theta }}\right) d\left( {x, y}\right) . \] If, on the other hand, we impose that \( d\left( {x, y}\right) > \theta > {AB} \), we also have \[ d\left( {f\left( x\right), f\left( y\right) }\right) \geq \frac{1}{A}d\left( {x, y}\right) - B \geq \frac{1}{A}d\left( {x, y}\right) - \frac{B}{\theta }d\left( {x, y}\right) = \left( {\frac{1}{A} - \frac{B}{\theta }}\right) d\left( {x, y}\right) . \] Summing up, for all \( x, y \in X \) such that \( d\left( {x, y}\right) > \theta > {AB} \) , \[ \left( {\frac{1}{A} - \frac{B}{\theta }}\right) d\left( {x, y}\right) \leq d\left( {f\left( x\right), f\left( y\right) }\right) \leq \left( {A + \frac{B}{\theta }}\right) d\left( {x, y}\right) . \] (ii) Now assume (14.5) holds for some constants \( 0 < {c}_{1} < {c}_{2} \) and some \( \theta > 0 \) . If \( d\left( {x, y}\right) \geq \theta \), then (14.6) is trivially satisfied for every \( B \geq 0 \), so suppose \( d\left( {x, y}\right) < \theta \) . By Lemma 14.1.14 we can pick \( z \in X \) with \( d\left( {y, z}\right) = {2\theta } \), and so \( d\left( {x, z}\right) > \theta \) . Then, \[ d\left( {f\left( x\right), f\left( y\right) }\right) \leq d\left( {f\left( x\right), f\left( z\right) }\right)

Lemma 14.1.7 (iii). If \( f \) is uniformly continuous and \( X \) is metrically convex, then \( {\omega }_{f}\left( s\right) < \infty \) for all \( s > 0 \).

We need to show that for \( s > 0 \) there is \( {C}_{s} > 0 \) such that \( d\left( {f\left( x\right), f\left( y\right) }\right) \leq {C}_{s} \) whenever \( d\left( {x, y}\right) \leq s \). From the definition of uniform continuity there exists \( {\delta }_{1} > 0 \) such that if \( 0 < d\left( {a, b}\right) < {\delta }_{1} \), then \( d\left( {f\left( a\right), f\left( b\right) }\right) < 1 \). Let \( N = {N}_{s} \in \mathbb{N} \) be such that \( s/N < {\delta }_{1} \). By the metric convexity of \( X \) one can find points \( x = {x}_{0},{x}_{1},\ldots ,{x}_{N} = y \) in \( X \) such that \( d\left( {{x}_{j},{x}_{j + 1}}\right) < d\left( {x, y}\right) /N < s/N < {\delta }_{1} \) for \( 0 \leq j \leq N - 1 \). Therefore, by the triangle inequality, \[ d\left( {f\left( x\right), f\left( y\right) }\right) \leq \mathop{\sum }\limits_{{j = 0}}^{{N - 1}}d\left( {f\left( {x}_{j}\right), f\left( {x}_{j + 1}\right) }\right) \leq N, \] and our claim holds with \( {C}_{s} = {N}_{s} \).

Proposition 4.4. For all ideals \( \mathfrak{a},\mathfrak{b} \) of \( R \) and \( \mathfrak{A} \) of \( {S}^{-1}R \) : (1) \( {\mathfrak{a}}^{E} = {S}^{-1}R \) if and only if \( \mathfrak{a} \cap S \neq \varnothing \) ; (2) if \( \mathfrak{a} = {\mathfrak{A}}^{C} \), then \( \mathfrak{A} = {\mathfrak{a}}^{E} \) ; (3) \( {\left( \mathfrak{a} + \mathfrak{b}\right) }^{E} = {\mathfrak{a}}^{E} + {\mathfrak{b}}^{E},{\left( \mathfrak{a} \cap \mathfrak{b}\right) }^{E} = {\mathfrak{a}}^{E} \cap {\mathfrak{b}}^{E} \), and \( {\left( \mathfrak{a}\mathfrak{b}\right) }^{E} = {\mathfrak{a}}^{E}{\mathfrak{b}}^{E} \) . The proofs make good exercises. Proposition 4.5. Let \( S \) be a proper multiplicative subset of \( R \) . Contraction and expansion induce a one-to-one correspondence between prime ideals of \( {S}^{-1}R \) and prime ideals of \( R \) disjoint from \( S \) . Proof. If \( \mathfrak{p} \subseteq R \smallsetminus S \) is a prime ideal of \( R \), then \( a/s \in {\mathfrak{p}}^{E} \) implies \( a/s = b/t \) for some \( b \in \mathfrak{p}, t \in S \), at \( u = {bsu} \in \mathfrak{p} \) for some \( u \in S \), and \( a \in \mathfrak{p} \), since \( \mathfrak{p} \) is prime and \( {tu} \notin \mathfrak{p} \) ; thus, \( a/s \in {\mathfrak{p}}^{E} \) if and only if \( a \in \mathfrak{p} \) . Hence \( {\mathfrak{p}}^{E} \) is a prime ideal of \( {S}^{-1}R : 1/1 \notin {\mathfrak{p}}^{E} \), and if \( \left( {a/s}\right) \left( {b/t}\right) \in {\mathfrak{p}}^{E} \) and \( a/s \notin {\mathfrak{p}}^{E} \), then \( {ab} \in \mathfrak{p} \) , \( a \notin \mathfrak{p}, b \in \mathfrak{p} \), and \( b/t \in {\mathfrak{p}}^{E} \) . Also \( \mathfrak{p} = {\left( {\mathfrak{p}}^{E}\right) }^{C} \) . Conversely, if \( \mathfrak{P} \) is a prime ideal of \( {S}^{-1}R \), then \( {\mathfrak{P}}^{C} \) is a prime ideal of \( R \) : \( 1 \notin {\mathfrak{P}}^{C} \), since \( 1/1 \notin \mathfrak{P} \), and if \( {ab} \in {\mathfrak{P}}^{C} \) and \( a \notin {\mathfrak{P}}^{C} \), then \( \left( {a/1}\right) \left( {b/1}\right) \in \mathfrak{P} \) , \( a/1 \notin \mathfrak{P}, b/1 \in \mathfrak{P} \), and \( b \in {\mathfrak{P}}^{C} \) . Moreover, \( {\left( {\mathfrak{P}}^{C}\right) }^{E} = \mathfrak{P} \), by 4.4. \( ▱ \) Proposition 4.6. Let \( S \) be a proper multiplicative subset of \( R \) . Contraction and expansion induce a one-to-one correspondence, which preserves radicals, between primary ideals of \( {S}^{-1}R \) and primary ideals of \( R \) disjoint from \( S \) . This is proved like 4.5. The following properties also make good exercises. Proposition 4.7. Let \( S \) be a proper multiplicative subset of \( R \) . (1) If \( R \) is Noetherian, then \( {S}^{-1}R \) is Noetherian. (2) If \( E \) is integral over \( R \), then \( {S}^{-1}E \) is integral over \( {S}^{-1}R \) . (3) If \( R \) is an integrally closed domain, then so is \( {S}^{-1}R \) . Localization. If \( \mathfrak{p} \) is a prime ideal of \( R \), then \( R \smallsetminus \mathfrak{p} \) is a proper multiplicative subset of \( R \) . Definition. The localization of a commutative ring \( R \) at a prime ideal \( \mathfrak{p} \) is the ring of fractions \( {R}_{\mathfrak{p}} = {\left( R \smallsetminus \mathfrak{p}\right) }^{-1}R \) . Every commutative ring is isomorphic to a ring of fractions (see the exercises); but not every ring is isomorphic to a localization. Proposition 4.8. If \( \mathfrak{p} \) is a prime ideal of \( R \), then \( {R}_{\mathfrak{p}} \) has only one maximal ideal, \( \mathfrak{M} = {\mathfrak{p}}^{E} = \left\{ {a/s \in {R}_{\mathfrak{p}} \mid a \in \mathfrak{p}}\right\} \) ; moreover, \( x \in {R}_{\mathfrak{p}} \) is a unit if and only if \( x \notin \mathfrak{M} \) . Proof. If \( a/s \in \mathfrak{M} \), then \( a/s = b/t \) for some \( b \in \mathfrak{p}, t \notin \mathfrak{p} \), at \( u = {bsu} \in \mathfrak{p} \) for some \( u \notin \mathfrak{p} \), and \( a \in \mathfrak{p} \) since \( \mathfrak{p} \) is a prime ideal and \( {tu} \notin \mathfrak{p} \) . Thus \( a/s \in \mathfrak{M} \) if and only if \( a \in \mathfrak{p} \) . Now, \( x = a/s \in {R}_{\mathfrak{p}} \) is a unit if and only if \( x \notin \mathfrak{M} \) : if \( a \notin \mathfrak{p} \), then \( x \) is a unit, and \( {x}^{-1} = s/a \) ; conversely, if \( x \) is a unit, then \( {ab}/{st} = 1 \) for some \( b, t \in R, t \notin \mathfrak{p},{abu} = {stu} \notin \mathfrak{p} \) for some \( u \notin \mathfrak{p} \), and \( a \notin \mathfrak{p} \) . Hence the ideal \( \mathfrak{M} \) of \( {R}_{\mathfrak{p}} \) is a maximal ideal. \( ▱ \) Definition. A commutative ring is local when it has only one maximal ideal. For instance, valuation rings are local, by VI.6.1; \( {R}_{\mathfrak{p}} \) is local, by 4.8. In a local ring \( R \) with maximal ideal \( \mathfrak{m} \), every \( x \in R \smallsetminus \mathfrak{m} \) is a unit (see the exercises). Homomorphisms. Localization transfers properties from local rings to more general rings. We illustrate this with some nifty homomorphism properties. Theorem 4.9. Every homomorphism of a ring \( R \) into an algebraically closed field \( L \) can be extended to every integral extension \( E \) of \( R \) . Proof. If \( R \) is a field, then \( E \) is a field, by 3.3, and \( E \) is an algebraic extension of \( R \) ; we saw that 4.9 holds in that case.  Now, let \( R \) be local and let \( \varphi : R \rightarrow L \) be a homomorphism whose kernel is the maximal ideal \( \mathfrak{m} \) of \( R \) . Then \( \varphi \) factors through the projection \( R \rightarrow R/\mathfrak{m} \) and induces a homomorphism \( \psi : R/\mathfrak{m} \rightarrow L \) . By 3.5,3.7, there is a maximal ideal \( \mathfrak{M} \) of \( E \) that lies over \( \mathfrak{m} \) . By 3.4, the field \( R/\mathfrak{m} \) may be identified with a subfield of \( E/\mathfrak{M} \) . Then \( E/\mathfrak{M} \) is algebraic over \( R/\mathfrak{m} \) and \( \psi : R/\mathfrak{m} \rightarrow L \) can be extended to \( E/\mathfrak{M} \) . Hence \( \varphi \) can be extended to \( E \) .  Finally, let \( \varphi : R \rightarrow L \) be any homomorphism. Then \( \mathfrak{p} = \operatorname{Ker}\varphi \) is a prime ideal of \( R \) and \( S = R \smallsetminus \mathfrak{p} \) is a proper multiplicative subset of \( R \) and of \( E \) . By 4.2, \( \varphi = \psi \circ \iota \) for some homomorphism \( \psi : {S}^{-1}R = {R}_{\mathfrak{p}} \rightarrow L \), namely, \( \psi \left( {a/s}\right) = \varphi \left( a\right) \varphi {\left( s\right) }^{-1} \) . Then Ker \( \psi = \left\{ {a/s \in {R}_{\mathfrak{p}} \mid a \in \operatorname{Ker}\varphi = \mathfrak{p}}\right\} \) is the maximal ideal of \( {R}_{\mathfrak{p}} \) . Therefore \( \psi \) extends to \( {S}^{-1}E \), which is integral over \( {S}^{-1}R \) by 4.7; hence \( \varphi \) extends to \( E \) . \( ▱ \) Theorem 4.10. Every homomorphism of a field \( K \) into an algebraically closed field \( L \) can be extended to every finitely generated ring extension of \( K \) . Proof. Let \( \varphi : K \rightarrow L \) be a homomorphism and let \( R = K\left\lbrack {{\alpha }_{1},\ldots ,{\alpha }_{m}}\right\rbrack \) be a finitely generated ring extension of \( K \) . First, assume that \( R \) is a field. We may assume that \( R \) is not algebraic over \( K \) . Let \( {\beta }_{1},\ldots ,{\beta }_{n} \) be a transcendence base of \( R \) over \( K \) . Every \( \alpha \in R \) is algebraic over \( K\left( {{\beta }_{1},\ldots ,{\beta }_{n}}\right) \), so that \( {\gamma }_{k}{\alpha }^{k} + \cdots + {\gamma }_{0} = 0 \) for some \( k > 0 \) and \( {\gamma }_{0},\ldots ,{\gamma }_{k} \in K\left( {{\beta }_{1},\ldots ,{\beta }_{n}}\right) ,{a}_{k} \neq 0 \) . Since we may multiply \( {\gamma }_{0},\ldots ,{\gamma }_{k} \) by a common denominator in \( K\left( {{\beta }_{1},\ldots ,{\beta }_{n}}\right) \cong K\left( {{X}_{1},\ldots ,{X}_{n}}\right) \), we may assume that \( {\gamma }_{0},\ldots ,{\gamma }_{k} \in D = K\left\lbrack {{\beta }_{1},\ldots ,{\beta }_{n}}\right\rbrack \) . Dividing by \( {\gamma }_{k} \neq 0 \) then shows that \( \alpha \) is integral over \( D\left\lbrack {1/{\gamma }_{k}}\right\rbrack \) . Applying this to \( {\alpha }_{1},\ldots ,{\alpha }_{m} \) shows that \( {\alpha }_{1},\ldots ,{\alpha }_{m} \) are integral over \( D\left\lbrack {1/{\delta }_{1},\ldots ,1/{\delta }_{m}}\right\rbrack \) for some nonzero \( {\delta }_{1},\ldots ,{\delta }_{m} \in D \) . Hence \( {\alpha }_{1},\ldots ,{\alpha }_{m} \) are integral over \( D\left\lbrack {1/\delta }\right\rbrack \), where \( \delta = {\delta }_{1}\cdots {\delta }_{m} \in D,\delta \neq 0 \) . Then \( R \) is integral over \( D\left\lbrack {1/\delta }\right\rbrack \) . Now, \( \varphi \) extends to a homomorphism \[ \psi : D = K\left\lbrack {{\beta }_{1},\ldots ,{\beta }_{n}}\right\rbrack \cong K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \rightarrow L\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack . \] Let \( g = \psi \left( \delta \right) \) . Since the algebraically closed field \( L \) is infinite, we have \( g\left( {{x}_{1},\ldots ,{x}_{n}}\right) \neq 0 \) for some \( {x}_{1},\ldots ,{x}_{n} \in L \), as readers will show. Let \( \chi = \) \( \widehat{x} \circ \psi \), where \( \widehat{x} : L\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \rightarrow L \) is the evaluation homomorphism \( f \mapsto \) \( f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) . Then \( \mathfrak{p} = \operatorname{Ker}\chi \) is a prime ideal of \( D \) and 4.2 extends \( \chi \) to the local ring \( {D}_{\mathfrak{p}} \) . By 4.3 we may assume that \( {D}_{\mathfrak{p}} = K{\left\lbrack {\beta }_{1},\ldots ,{\beta }_{n}\right\rbrack }_{\mathfrak{p}} \subseteq \) \( K\left( {{\beta }_{1},\ld

Proposition 4.4. For all ideals \( \mathfrak{a},\mathfrak{b} \) of \( R \) and \( \mathfrak{A} \) of \( {S}^{-1}R \) : (1) \( {\mathfrak{a}}^{E} = {S}^{-1}R \) if and only if \( \mathfrak{a} \cap S \neq \varnothing \) ; (2) if \( \mathfrak{a} = {\mathfrak{A}}^{C} \), then \( \mathfrak{A} = {\mathfrak{a}}^{E} \) ; (3) \( {\left( \mathfrak{a} + \mathfrak{b}\right) }^{E} = {\mathfrak{a}}^{E} + {\mathfrak{b}}^{E},{\left( \mathfrak{a} \cap \mathfrak{b}\right) }^{E} = {\mathfrak{a}}^{E} \cap {\mathfrak{b}}^{E} \), and \( {\left( \mathfrak{a}\mathfrak{b}\right) }^{E} = {\mathfrak{a}}^{E}{\mathfrak{b}}^{E} \) .

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Proposition 11.101. \( \mathfrak{B} \) is generated by \( {\mathfrak{B}}_{0} \) and any set of representatives for \( {W}^{\prime } \) in \( N \) . ## Exercises 11.102. Show that there is a short exact sequence \( 1 \rightarrow {\mathfrak{B}}^{\prime } \rightarrow \mathfrak{B} \rightarrow {W}^{\prime } \rightarrow 1 \) . 11.103. Let \( {W}^{\prime \prime } \) be the submonoid of \( {W}^{\prime } \) consisting of those \( w \in {W}^{\prime } \) such that \( w\mathfrak{C} \subseteq \mathfrak{C} \) . [Equivalently, if we identify \( E \) with a vector space \( V \) and \( {W}^{\prime } \) with a lattice \( L \) in \( V \), then \( {W}^{\prime \prime } = L \cap \overline{\mathfrak{D}} \), where \( \mathfrak{D} \) is the direction of \( \mathfrak{C} \) .] Show that (11.13) remains valid if \( {W}^{\prime } \) is replaced by \( {W}^{\prime \prime } \) . Remark 11.104. Readers familiar with ascending HNN extensions (see [228, Section 1.2]) should find the situation in these exercises familiar. This suggests that \( \mathfrak{B} \) is, in some sense that we will not make precise, a "generalized ascending HNN extension with base group \( {\mathfrak{B}}_{0} \) ." ## 11.8.6 Example Let \( K \) be a field with a discrete valuation, and consider the Euclidean BN-pair in \( G \mathrel{\text{:=}} {\mathrm{{SL}}}_{n}\left( K\right) \) constructed in Section 6.9. Its Weyl group \( W \) is the Euclidean reflection group studied in Section 10.1.7. And as we saw in Section 10.1.8, \( W \) is the Coxeter group of type \( {\widetilde{\mathrm{A}}}_{n - 1} \) and is the affine Weyl group of the root system of type \( {\mathrm{A}}_{n - 1} \) . We already know that there is a second BN-pair in \( G \) , obtained by forgetting that \( K \) has a valuation and applying Section 6.5 ; its Weyl group is the symmetric group on \( n \) letters, which is the finite reflection group \( \bar{W} \) associated to \( W \) . Recall that \( \bar{W} \) is the (ordinary) Weyl group of the root system of type \( {\mathrm{A}}_{n - 1} \) . In view of the relationship between the Weyl groups of our two BN-pairs, the following result is not surprising: Proposition 11.105. Let \( X \) be the Euclidean building \( \left| {\Delta \left( {G, B}\right) }\right| \) associated to \( G = {\mathrm{{SL}}}_{n}\left( K\right) \) . (1) There is a sector \( \mathfrak{C} \) in the fundamental apartment \( E = \left| \sum \right| \) such that the stabilizer \( \mathfrak{B} \) of \( {\mathfrak{C}}_{\infty } \) is the upper-triangular subgroup of \( G \) . (2) The apartment system \( \mathcal{A} \) associated to \( \left( {G, B, N}\right) \) is good. The subcomplex \( {\Delta }_{\infty }\left( \mathcal{A}\right) \) of \( {\Delta }_{\infty } \) is therefore isomorphic to the spherical building associated to \( G \) in Section 6.5. (3) \( \mathcal{A} \) is the complete apartment system if and only if \( K \) is complete with respect to the given valuation. Sketch of proof. Identify the fundamental apartment \( \sum \) with the complex \( \sum \left( {W, V}\right) \) studied in Section 10.1.6. [Recall that we gave an explicit way of making this identification in Section 10.1.7.] Then \( E = \left| \sum \right| \) is identified with \( V \) . As "fundamental sector" \( \mathfrak{C} \) we take the subset of \( V \) defined by \( {x}_{1} < \cdots < {x}_{n} \) . Its closure is a subcomplex of \( E \) whose vertices, from the point of view of \( A \) -lattices, are the classes \( \left\lbrack \left\lbrack {{\pi }^{{a}_{1}}{e}_{1},\ldots ,{\pi }^{{a}_{n}}{e}_{n}}\right\rbrack \right\rbrack \) with \( {a}_{1} \leq \cdots \leq {a}_{n} \) . The group \( {\mathfrak{B}}_{0} \) that fixes \( \mathfrak{C} \) pointwise is therefore given by \[ {\mathfrak{B}}_{0} = \mathop{\bigcap }\limits_{{d \in D}}d \cdot {\mathrm{{SL}}}_{n}\left( A\right) \cdot {d}^{-1} \] where \( D \) is the set of matrices in \( {\mathrm{{GL}}}_{n}\left( K\right) \) of the form \( \operatorname{diag}\left( {{\pi }^{{a}_{1}},\ldots ,{\pi }^{{a}_{n}}}\right) \) with \( {a}_{1} \leq \cdots \leq {a}_{n} \) . An easy computation shows that this intersection is the upper-triangular subgroup of \( {\mathrm{{SL}}}_{n}\left( A\right) \) . Now apply formula (11.13) to get \( {\mathfrak{B}}^{\prime } = \mathop{\bigcup }\limits_{{t \in T}}t{\mathfrak{B}}_{0}{t}^{-1} \), where \( T \) is the diagonal subgroup of \( G \) . Another easy computation then shows that this union consists of all upper-triangular matrices in \( G \) whose diagonal entries are units in \( A \) . In view of (11.12), we conclude that \( \mathfrak{B} \) is indeed the full upper-triangular subgroup of \( G \) . Statement (1) is now proved, and (2) follows immediately from (1) and Proposition 11.100. Turning now to (3), suppose first that \( K \) is not complete. Let \( \widehat{K} \) be the completion of \( K \), and let \( \widehat{G},\widehat{B} \), and \( \widehat{N} \) be the analogues of \( G, B \), and \( N \) over \( \widehat{K} \) . Then \( G \) is dense in \( \widehat{G} \), and \( \widehat{B} \) is an open subgroup of \( \widehat{G} \) ; it follows that \( \Delta \left( {G, B}\right) = \Delta \left( {\widehat{G},\widehat{B}}\right) \) (Section 6.10.1). On the other hand, it is easy to see that \( \widehat{G}/\widehat{N} \) is strictly bigger than \( G/N \), so we definitely get more apartments using \( \left( {\widehat{G},\widehat{B},\widehat{N}}\right) \) than we get from \( \left( {G, B, N}\right) \) . The apartment system associated to \( \left( {G, B, N}\right) \) is therefore not complete. Finally, suppose \( K \) is complete, and let \( {E}^{\prime } \) be an arbitrary apartment in the complete system. We will show that \( {E}^{\prime } \in \mathcal{A} \) by constructing a \( g \in G \) such that \( {E}^{\prime } = {gE} \), where \( E \) is the fundamental apartment. We may assume that \( {E}^{\prime } \) contains the fundamental chamber \( C \), in which case we will find the desired \( g \) in \( B \) . Let \( \phi : E \rightarrow {E}^{\prime } \) be the isomorphism that fixes \( E \cap {E}^{\prime } \) . In view of Theorem 11.43, every bounded subset of \( {E}^{\prime } \) is contained in an apartment in \( \mathcal{A} \) . So if we exhaust \( E \) by an increasing sequence of bounded sets \( {F}_{i} \) containing \( C \) , then we can find \( {b}_{i} \in B \) such that \( {b}_{i} \) maps \( {F}_{i} \) into \( {\left. {E}^{\prime }\text{by the map}\phi \right| }_{{F}_{i}} \) . We claim that the \( {F}_{i} \) and \( {b}_{i} \) can be chosen such that \( {b}_{i + 1} \equiv {b}_{i}{\;\operatorname{mod}\;{\pi }^{i}} \) . Accepting this for the moment, we can easily complete the proof. For the completeness of \( K \) implies that \( {b}_{i} \) converges to some \( b \in B \) as \( i \rightarrow \infty \), whence \( {E}^{\prime } = {bE} \) and we are done. It remains to prove the claim. By looking at the stabilizers of the vertices of \( E \), one sees first that the \( {F}_{i} \) can be chosen such that any element of \( G \) that fixes \( {F}_{i} \) pointwise is in \( {\mathrm{{SL}}}_{n}\left( A\right) \) and is congruent to a diagonal matrix \( {\;\operatorname{mod}\;{\pi }^{i}} \) . In particular, for any choice of the \( {b}_{i} \) we will have \( {b}_{i + 1}^{-1}{b}_{i} \) congruent to a diagonal matrix \( {\;\operatorname{mod}\;{\pi }^{i}} \) . Now assume inductively that \( {b}_{1},\ldots ,{b}_{i} \) have been chosen and that they satisfy the required congruences. Let \( b \) be any element of \( B \) such that \( {\left. b\right| }_{{F}_{i + 1}} = {\left. \phi \right| }_{{F}_{i + 1}} \) . Then there is a diagonal matrix \( t \in {\mathrm{{SL}}}_{n}\left( A\right) \) such that \( {b}^{-1}{b}_{i} \equiv t{\;\operatorname{mod}\;{\pi }^{i}} \) . Since \( t \) fixes \( E \) pointwise, we can complete the inductive step by setting \( {b}_{i + 1} \mathrel{\text{:=}} {bt} \) . ## 11.9 Classification Recall from Chapter 9 that Tits classified thick, irreducible, spherical buildings of rank \( \geq 3 \), and that they are (roughly) in \( 1 - 1 \) correspondence with simple algebraic groups or classical groups or mixed groups over an arbitrary field. [More generally, the classification extends to Moufang buildings of rank 2.] Tits later [258] classified thick Euclidean buildings of rank \( \geq 4 \) . [More generally, the classification extends to thick Euclidean buildings of rank 3 such that the spherical building at infinity is Moufang.] The result this time is that they are (roughly) in 1-1 correspondence with absolutely simple algebraic groups or classical groups or mixed groups defined over a field that is complete with respect to a discrete valuation. See Tits [252, Section 4] or Weiss [283, Chapter 28] for the list of such groups in case the residue field is finite (which holds if and only if the corresponding building is locally finite). The existence of a building associated to a group as above had already been proved by Bruhat and Tits \( \left\lbrack {{59},{60}}\right\rbrack \) ; see Section C. 11 below. To achieve the classification, then, Tits essentially had to find an inverse to this construction and produce, from a Euclidean building, an algebraic group over a field with discrete valuation. For overviews of the classification theorem, see Ronan [200, Section 10.5] or Weiss [282]. For a complete proof, see Weiss [283]. We will not try to say much about the proof except to indicate very briefly why one might expect a Euclidean building \( X \) to yield an algebraic group over a field with discrete valuation. The starting point for the proof is the consideration of the spherical building at infinity. This has rank \( \geq 3 \) [or else it has rank 2 and is Moufang], so one can assume that it is known and, typically, corresponds to an algebraic group over a field \( K \) . This field is visible in the root groups \( {U}_{\alpha } \), many of which are isomorphic to the additive group of \( K \) (see Section 7.9).

Proposition 11.101. \( \mathfrak{B} \) is generated by \( {\mathfrak{B}}_{0} \) and any set of representatives for \( {W}^{\prime } \) in \( N \) .

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Corollary 1.10. If \( k \) is a finite field, then \( {k}^{ * } \) is cyclic. An element \( \zeta \) in a field \( k \) such that there exists an integer \( n \geqq 1 \) such that \( {\zeta }^{n} = 1 \) is called a root of unity, or more precisely an \( n \) -th root of unity. Thus the set of \( n \) -th roots of unity is the set of roots of the polynomial \( {X}^{n} - 1 \) . There are at most \( n \) such roots, and they obviously form a group, which is cyclic by Theorem 1.9. We shall study roots of unity in greater detail later. A generator for the group of \( n \) -th roots of unity is called a primitive \( n \) -th root of unity. For example, in the complex numbers, \( {e}^{{2\pi i}/n} \) is a primitive \( n \) -th root of unity, and the \( n \) -th roots of unity are of type \( {e}^{{2\pi iv}/n} \) with \( 1 \leqq v \leqq n. \) The group of roots of unity is denoted by \( \mathbf{\mu } \) . The group of roots of unity in a field \( K \) is denoted by \( \mathbf{\mu }\left( K\right) \) . A field \( k \) is said to be algebraically closed if every polynomial in \( k\left\lbrack X\right\rbrack \) of degree \( \geqq 1 \) has a root in \( k \) . In books on analysis, it is proved that the complex numbers are algebraically closed. In Chapter V we shall prove that a field \( k \) is always contained in some algebraically closed field. If \( k \) is algebraically closed then the irreducible polynomials in \( k\left\lbrack X\right\rbrack \) are the polynomials of degree 1 . In such a case, the unique factorization of a polynomial \( f \) of degree \( \geqq 0 \) can be written in the form \[ f\left( X\right) = c\mathop{\prod }\limits_{{i = 1}}^{r}{\left( X - {\alpha }_{i}\right) }^{{m}_{i}} \] with \( c \in k, c \neq 0 \) and distinct roots \( {\alpha }_{1},\ldots ,{\alpha }_{r} \) . We next develop a test when \( {m}_{i} > 1 \) . Let \( A \) be a commutative ring. We define a map \[ D : A\left\lbrack X\right\rbrack \rightarrow A\left\lbrack X\right\rbrack \] of the polynomial ring into itself. If \( f\left( X\right) = {a}_{n}{X}^{n} + \cdots + {a}_{0} \) with \( {a}_{i} \in A \), we define the derivative \[ {Df}\left( X\right) = {f}^{\prime }\left( X\right) = \mathop{\sum }\limits_{{v = 1}}^{n}v{a}_{v}{X}^{v - 1} = n{a}_{n}{X}^{n - 1} + \cdots + {a}_{1}. \] One verifies easily that if \( f, g \) are polynomials in \( A\left\lbrack X\right\rbrack \), then \[ {\left( f + g\right) }^{\prime } = {f}^{\prime } + {g}^{\prime },\;{\left( fg\right) }^{\prime } = {f}^{\prime }g + f{g}^{\prime }, \] and if \( a \in A \), then \[ {\left( af\right) }^{\prime } = a{f}^{\prime }. \] Let \( K \) be a field and \( f \) a non-zero polynomial in \( K\left\lbrack X\right\rbrack \) . Let \( a \) be a root of \( f \) in \( K \) . We can write \[ f\left( X\right) = {\left( X - a\right) }^{m}g\left( X\right) \] with some polynomial \( g\left( X\right) \) relatively prime to \( X - a \) (and hence such that \( g\left( a\right) \neq 0 \) ). We call \( m \) the multiplicity of \( a \) in \( f \), and say that \( a \) is a multiple root if \( m > 1 \) . Proposition 1.11. Let \( K, f \) be as above. The element a of \( K \) is a multiple root of \( f \) if and only if it is a root and \( {f}^{\prime }\left( a\right) = 0 \) . Proof. Factoring \( f \) as above, we get \[ {f}^{\prime }\left( X\right) = {\left( X - a\right) }^{m}{g}^{\prime }\left( X\right) + m{\left( X - a\right) }^{m - 1}g\left( X\right) . \] If \( m > 1 \), then obviously \( {f}^{\prime }\left( a\right) = 0 \) . Conversely, if \( m = 1 \) then \[ {f}^{\prime }\left( X\right) = \left( {X - a}\right) {g}^{\prime }\left( X\right) + g\left( X\right) \] whence \( {f}^{\prime }\left( a\right) = g\left( a\right) \neq 0 \) . Hence if \( {f}^{\prime }\left( a\right) = 0 \) we must have \( m > 1 \), as desired. Proposition 1.12. Let \( f \in K\left\lbrack X\right\rbrack \) . If \( K \) has characteristic 0, and \( f \) has degree \( \geqq 1 \), then \( {f}^{\prime } \neq 0 \) . Let \( K \) have characteristic \( p > 0 \) and \( f \) have degree \( \geqq 1 \) . Then \( {f}^{\prime } = 0 \) if and only if, in the expression for \( f\left( X\right) \) given by \[ f\left( X\right) = \mathop{\sum }\limits_{{v = 0}}^{n}{a}_{v}{X}^{v} \] \( p \) divides each integer \( v \) such that \( {a}_{v} \neq 0 \) . Proof. If \( K \) has characteristic 0, then the derivative of a monomial \( {a}_{v}{X}^{v} \) such that \( v \geqq 1 \) and \( {a}_{v} \neq 0 \) is not zero since it is \( v{a}_{v}{X}^{v - 1} \) . If \( K \) has characteristic \( p > 0 \), then the derivative of such a monomial is 0 if and only if \( p \mid v \), as contended. Let \( K \) have characteristic \( p > 0 \), and let \( f \) be written as above, and be such that \( {f}^{\prime }\left( X\right) = 0 \) . Then one can write \[ f\left( X\right) = \mathop{\sum }\limits_{{\mu = 0}}^{d}{b}_{\mu }{X}^{p\mu } \] with \( {b}_{\mu } \in K \) . Since the binomial coefficients \( \left( \begin{array}{l} p \\ v \end{array}\right) \) are divisible by \( p \) for \( 1 \leqq v \leqq p - 1 \) we see that if \( K \) has characteristic \( p \), then for \( a, b \in K \) we have \[ {\left( a + b\right) }^{p} = {a}^{p} + {b}^{p} \] Since obviously \( {\left( ab\right) }^{p} = {a}^{p}{b}^{p} \), the map \[ x \mapsto {x}^{p} \] is a homomorphism of \( K \) into itself, which has trivial kernel, hence is injective. Iterating, we conclude that for each integer \( r \geqq 1 \), the map \( x \mapsto {x}^{{p}^{r}} \) is an endomorphism of \( K \), called the Frobenius endomorphism. Inductively, if \( {c}_{1},\ldots ,{c}_{n} \) are elements of \( K \), then \[ {\left( {c}_{1} + \cdots + {c}_{n}\right) }^{p} = {\mathrm{c}}_{1}^{p} + \cdots + {\mathrm{c}}_{n}^{p}. \] Applying these remarks to polynomials, we see that for any element \( a \in K \) we have \[ {\left( X - a\right) }^{{p}^{r}} = {X}^{{p}^{r}} - {a}^{{p}^{r}} \] If \( c \in K \) and the polynomial \[ {X}^{{p}^{r}} - c \] has one root \( a \) in \( K \), then \( {a}^{{p}^{r}} = c \) and \[ {X}^{{p}^{r}} - c = {\left( X - a\right) }^{{p}^{r}}. \] Hence our polynomial has precisely one root, of multiplicity \( {p}^{r} \) . For instance, \( {\left( X - 1\right) }^{{p}^{r}} = {X}^{{p}^{r}} - 1 \) . ## §2. POLYNOMIALS OVER A FACTORIAL RING Let \( A \) be a factorial ring, and \( K \) its quotient field. Let \( a \in K, a \neq 0 \) . We can write \( a \) as a quotient of elements in \( A \), having no prime factor in common. If \( p \) is a prime element of \( A \), then we can write \[ a = {p}^{r}b \] where \( b \in K, r \) is an integer, and \( p \) does not divide the numerator or denominator of \( b \) . Using the unique factorization in \( A \), we see at once that \( r \) is uniquely determined by \( a \), and we call \( r \) the order of \( a \) at \( p \) (and write \( r = {\operatorname{ord}}_{p}a \) ). If \( a = 0 \), we define its order at \( p \) to be \( \infty \) . If \( a,{a}^{\prime } \in K \) and \( a{a}^{\prime } \neq 0 \), then \[ {\operatorname{ord}}_{p}\left( {a{a}^{\prime }}\right) = {\operatorname{ord}}_{p}a + {\operatorname{ord}}_{p}{a}^{\prime } \] This is obvious. Let \( f\left( X\right) \in K\left\lbrack X\right\rbrack \) be a polynomial in one variable, written \[ f\left( X\right) = {a}_{0} + {a}_{1}X + \cdots + {a}_{n}{X}^{n}. \] If \( f = 0 \), we define \( {\operatorname{ord}}_{p}f \) to be \( \infty \) . If \( f \neq 0 \), we define \( {\operatorname{ord}}_{p}f \) to be \[ {\operatorname{ord}}_{p}f = \min {\operatorname{ord}}_{p}{a}_{i} \] the minimum being taken over all those \( i \) such that \( {a}_{i} \neq 0 \) . If \( r = {\operatorname{ord}}_{p}f \), we call \( u{p}^{r} \) a \( p \) -content for \( f \), if \( u \) is any unit of \( A \) . We define the content of \( f \) to be the product. \[ \prod {p}^{{\operatorname{ord}}_{p}f} \] the product being taken over all \( p \) such that \( {\operatorname{ord}}_{p}f \neq 0 \), or any multiple of this product by a unit of \( A \) . Thus the content is well defined up to multiplication by a unit of \( A \) . We abbreviate content by cont. If \( b \in K, b \neq 0 \), then \( \operatorname{cont}\left( {bf}\right) = b\operatorname{cont}\left( f\right) \) . This is clear. Hence we can write \[ f\left( X\right) = c \cdot {f}_{1}\left( X\right) \] where \( c = \operatorname{cont}\left( f\right) \), and \( {f}_{1}\left( X\right) \) has content 1 . In particular, all coefficients of \( {f}_{1} \) lie in \( A \), and their g.c.d. is 1 . We define a polynomial with content 1 to be a primitive polynomial. Theorem 2.1. (Gauss Lemma). Let \( A \) be a factorial ring, and let \( K \) be its quotient field. Let \( f, g \in K\left\lbrack X\right\rbrack \) be polynomials in one variable. Then \[ \operatorname{cont}\left( {fg}\right) = \operatorname{cont}\left( f\right) \operatorname{cont}\left( g\right) \] Proof. Writing \( f = c{f}_{1} \) and \( g = d{g}_{1} \) where \( c = \operatorname{cont}\left( f\right) \) and \( d = \operatorname{cont}\left( g\right) \) , we see that it suffices to prove: If \( f, g \) have content 1, then \( {fg} \) also has content 1, and for this, it suffices to prove that for each prime \( p \) , \( {\operatorname{ord}}_{p}\left( {fg}\right) = 0 \) . Let \[ f\left( X\right) = {a}_{n}{X}^{n} + \cdots + {a}_{0},\;{a}_{n} \neq 0, \] \[ g\left( X\right) = {b}_{m}{X}^{m} + \cdots + {b}_{0},\;{b}_{m} \neq 0, \] be polynomials of content 1 . Let \( p \) be a prime of \( A \) . It will suffice to prove that \( p \) does not divide all coefficients of \( {fg} \) . Let \( r \) be the largest integer such that \( 0 \leqq r \leqq n,{a}_{r} \neq 0 \), and \( p \) does not divide \( {a}_{r} \) . Similarly, let \( {b}_{s} \) be the coefficient of \( g \) farthest to the left, \( {b}_{s} \neq 0 \), such that \( p \) does not divide \( {b}_{s} \) . Consider the coefficient of \( {X}^{r + s} \) in \( f\left( X\right) g\left( X\right) \) . This coefficient is e

Corollary 1.10. If \( k \) is a finite field, then \( {k}^{ * } \) is cyclic.

An element \( \zeta \) in a field \( k \) such that there exists an integer \( n \geqq 1 \) such that \( {\zeta }^{n} = 1 \) is called a root of unity, or more precisely an \( n \)-th root of unity. Thus the set of \( n \)-th roots of unity is the set of roots of the polynomial \( {X}^{n} - 1 \). There are at most \( n \) such roots, and they obviously form a group, which is cyclic by Theorem 1.9. A generator for the group of \( n \)-th roots of unity is called a primitive \( n \)-th root of unity. For example, in the complex numbers, \( {e}^{{2\pi i}/n} \) is a primitive \( n \)-th root of unity, and the \( n \)-th roots of unity are of type \( {e}^{{2\pi iv}/n} \) with \( 1 \leqq v \leqq n \). The group of roots of unity is denoted by \( \mathbf{\mu} \). The group of roots of unity in a field \( K \) is denoted by \( \mathbf{\mu}(K) \). In a finite field \( k \), consider the group \( {k}^{ * } \) which consists of all non-zero elements of \( k \). This group is finite and abelian. By Theorem 1.9, any finite subgroup of the multiplicative group of a field is cyclic. Therefore, the group \( {k}^{ * } \) is cyclic, meaning there exists an element \( g \in {k}^{ * } \) such that every element in \( {k}^{ * } \) can be written as a power of \( g \). Hence, \( {k}^{ * } \) is cyclic.

Proposition 1.1. Let \( \alpha : S \rightarrow T \) be a morphism of algebraic sets. If \( A \) is an irreducible subset of \( S \), then \( \alpha \left( A\right) \) is an irreducible subset of \( T \) . Proof. This follows direcly from the definition of irreducibility; using only the continuity of \( \alpha \) . Lemma 1.2. Let \( F \) be a field, \( S \) an irreducible affine algebraic \( F \) -set, \( B \) an integral domain F-algebra. Then \( \mathcal{P}\left( S\right) \otimes B \) is an integral domain. Proof. Let \( u \) and \( v \) be elements of \( \mathcal{P}\left( S\right) \otimes B \) such that \( {uv} = 0 \) . Write \[ u = \mathop{\sum }\limits_{{i = 1}}^{n}{u}_{i} \otimes {b}_{i}\;\text{ and }\;v = \mathop{\sum }\limits_{{i = 1}}^{n}{v}_{i} \otimes {b}_{i}, \] where the \( {b}_{i} \) ’s are \( F \) -linearly independent elements of \( B \), and the \( {u}_{i} \) ’s and \( {v}_{i} \) ’s belong to \( \mathcal{P}\left( S\right) \) . Every element \( s \) of \( S \), via evaluation at \( s \), defines a \( B \) - algebra homomorphism \( {s}^{B} \) from \( \mathcal{P}\left( S\right) \otimes B \) to \( B \) . If there is an index \( i \) such that \( {v}_{i}\left( s\right) \neq 0 \), then \( {s}^{B}\left( v\right) \neq 0 \), whence \( {s}^{B}\left( u\right) = 0 \), so that \( {u}_{j}\left( s\right) = 0 \) for each \( j \) . Therefore, in any case, we have \( {u}_{j}\left( s\right) {v}_{i}\left( s\right) = 0 \) for all indices \( i \) and \( j \) and all elements \( s \) of \( S \) . Thus, \( {u}_{j}{v}_{i} = 0 \) for all indices \( i \) and \( j \) . If \( v \neq 0 \) then one of the \( {v}_{i} \) ’s must be different from 0, and it follows that \( u = 0 \) . Proposition 1.3. If \( S \) and \( T \) are irreducible algebraic sets, so is \( S \times T \) . Proof. We know from the beginning of this Section that \( \mathcal{P}\left( S\right) \) and \( \mathcal{P}\left( T\right) \) are integral domains. By Lemma 1.2, it follows that \( \mathcal{P}\left( S\right) \otimes \mathcal{P}\left( T\right) \) is an integral domain. Since this is \( \mathcal{P}\left( {S \times T}\right) \), we conclude that \( S \times T \) is irreducible. \( ▱ \) Theorem 1.4. Let \( G \) be an affine algebraic group. The irreducible components of \( G \) are mutually disjoint. The component \( {G}_{1} \) containing the neutral element of \( G \) is a closed normal subgroup of \( G \), and the irreducible components of \( G \) are the cosets of \( {G}_{1} \) in \( G \) . Moreover, \( {G}_{1} \) is the only irreducible closed subgroup of finite index in \( G \) . Proof. Suppose that \( U \) and \( V \) are irreducible components of \( G \) and that each contains the neutral element of \( G \) . The product set \( {UV} \) in \( G \) is the image of \( U \times V \) under the composition map of \( G \) . By Propositions 1.3 and 1.1, it is therefore irreducible. Since it contains \( U \) and \( V \), we conclude that \[ U = {UV} = V\text{.} \] Thus, only one of the irreducible components of \( G \) contains the neutral element. This defines \( {G}_{1} \) . We have just seen that \( {G}_{1}{G}_{1} = {G}_{1} \) . Since the inversion of \( G \) is a homeomorphism, \( {G}_{1}^{-1} \) is an irreducible component of \( G \) . Since it contains the neutral element, it therefore coincides with \( {G}_{1} \) . Thus, \( {G}_{1} \) is a subgroup of \( G \) . By considering the translation actions of \( G \) on itself, we see immediately that the set of left cosets of \( {G}_{1} \) in \( G \), as well as the set of right cosets, coincides with the set of irreducible components of \( G \) . This evidently implies that \( {G}_{1} \) is a normal subgroup of finite index in \( G \) . Finally, let \( K \) be any closed irreducible subgroup of finite index in \( G \) . Clearly, the set of left (or right) cosets of \( K \) in \( G \) coincides with the set of irreducible components of \( G \), i.e., with the set of cosets of \( {G}_{1} \) . Hence, \( K = {G}_{1} \) . 2. Let \( F \) be a field, \( V \) an \( F \) -space. The exterior algebra \( \bigwedge \left( V\right) \) built over \( V \) is defined as the factor algebra of the tensor algebra \( \otimes \left( V\right) {\;\operatorname{mod}\;\text{the ideal gen-}} \) erated by the squares of the elements of \( V \) . If \( G \) is a group and \( V \) is a \( G \) - module, then \( / \smallsetminus \left( V\right) \) inherits the structure of a \( G \) -module via the tensor product construction, and \( G \) acts on \( V \) by \( F \) -algebra automorphisms respecting the grading of \( V \) by its homogeneous components \( \mathop{\bigwedge }\limits^{k}\left( V\right) \) \( \left( {k = 0,1,\ldots }\right) \) . A module of this type plays the decisive role in the proof of the following theorem. Theorem 2.1. Let \( G \) be an affine algebraic \( F \) -group, \( H \) an algebraic subgroup of \( G \) . There is a finite subset \( E \) of \( \mathcal{P}\left( G\right) \), and an element \( f \) of \( \mathcal{P}\left( G\right) \) whose restriction to \( H \) is a group homomorphism from \( H \) to \( {F}^{ * } \), such that (1) \( x \cdot e = f\left( x\right) \) e for every \( x \) in \( H \) and every \( e \) in \( E \) (2) if \( x \) is an element of \( G \) such that \( x \cdot e \) belongs to Fe for every element \( e \) of \( E \) then \( x \) belongs to \( H \) . Proof. Let \( I \) denote the annihilator of \( H \) in \( \mathcal{P}\left( G\right) \) . There is a finite-dimensional left \( G \) -stable sub \( F \) -space \( V \) of \( \mathcal{P}\left( G\right) \) such that \( V \cap I \) generates \( I \) as an ideal. Let \( d \) denote the dimension of \( V \cap I \), and consider the action of \( G \) on \( \mathop{\bigwedge }\limits^{d}\left( V\right) \) . Let \( S \) denote the canonical image of \( \mathop{\bigwedge }\limits^{d}\left( {V \cap I}\right) \) in \( \mathop{\bigwedge }\limits^{d}\left( V\right) \) . Clearly, \( S \) is 1-dimensional, and we write \( S = {Fs} \), fixing any non-zero element \( s \) of \( S \) . Since \( H \cdot I \subset I \), it is clear that \( S \) is an \( H \) -stable subspace of \( \mathop{\bigwedge }\limits^{d}\left( V\right) \) . Thus, for \( x \) in \( H \), we have \( x \cdot s = g\left( x\right) s \), where \( g \) is a group homomorphism from \( H \) to \( {F}^{ * } \) . Now choose \( \sigma \) from \( {\left( \bigwedge {}^{d}\left( V\right) \right) }^{ \circ } \) such that \( \sigma \left( s\right) = 1 \), and let \( f \) be the representative function \( \sigma /s \) on \( G \) . It is clear from the comodule form of the construction of tensor products of \( G \) -modules that \( f \) is an element of \( \mathcal{P}\left( G\right) \) . Evidently, the restriction of \( f \) to \( H \) coincides with \( g \) . Now let \( \left( {{\sigma }_{1},\ldots ,{\sigma }_{p}}\right) \) be an \( F \) -basis of the annihilator of \( S \) in \( {\left( \bigwedge {}^{d}\left( V\right) \right) }^{ \circ } \) , and consider the elements \( {\sigma }_{i}/s \) of \( \mathcal{P}\left( G\right) \) . For every element \( x \) of \( H \), we have \[ x \cdot \left( {{\sigma }_{i}/s}\right) = {\sigma }_{i}/\left( {x \cdot s}\right) = g\left( x\right) {\sigma }_{i}/s. \] Conversely, suppose that \( x \) is an element of \( G \) such that \( x \cdot \left( {{\sigma }_{i}/s}\right) \) is an \( F \) - multiple of \( {\sigma }_{i}/s \) for each \( i \) . Evaluating at the neutral element of \( G \), we obtain \( {\sigma }_{i}\left( {x \cdot s}\right) = 0 \) for each \( i \) . This shows that \( x \cdot s \) belongs to \( S \), so that \( x \cdot S = S \) . Let \( t \) be an element of \( V \cap I \) . Then, in \( \mathop{\bigwedge }\limits^{{d + 1}}\left( V\right) \), we have \( {tS} = \left( 0\right) \), whence also \( x \cdot \left( {tS}\right) = \left( 0\right) \) . But \[ x \cdot \left( {tS}\right) = \left( {x \cdot t}\right) \left( {x \cdot S}\right) = \left( {x \cdot t}\right) S. \] Thus, we have \( \left( {x \cdot t}\right) S = \left( 0\right) \), which means that \( x \cdot t \) belongs to \( V \cap I \) . Since \( V \cap I \) generates \( I \) as an ideal, our result shows that \( x \cdot I \subset I \), whence \( x \) belongs to \( H \) . This proves the theorem, with \( E = \left( {{\sigma }_{1}/s,\ldots ,{\sigma }_{p}/s}\right) \) . If \( e \) is an element of \( \mathcal{P}\left( G\right) \), and \( g \) is a group homomorphism from \( H \) to \( {F}^{ * } \) such that \( x \cdot e = g\left( x\right) e \) for every element \( x \) of \( H \), then \( e \) is called a semi-invariant of \( H \), and \( g \) is called the weight of \( e \) . Theorem 2.2. Let \( H \) be a normal algebraic subgroup of the algebraic group \( G \) . There is a finite subset \( Q \) of \( \mathcal{P}\left( G\right) \) such that the left element-wise fixer of \( Q \) in \( G \) is precisely \( H \) . Proof. Let \( E \) be a finite set of semi-invariants of \( H \), such as given by Theorem 2.1, and let \( g \) denote the common weight of the elements of \( E \) . Let \( J \) be the smallest left \( G \) -stable subspace of \( \mathcal{P}\left( G\right) \) that contains \( E \) . Since \( \mathcal{P}\left( G\right) \) is locally finite as a \( G \) -module, \( J \) is of finite dimension over the base field \( F \) . Those elements of \( J \) which are \( H \) -semi-invariants of weight \( g \) evidently constitute a sub \( H \) -module, \( {J}_{1} \) say, of \( J \) . If \( x \) is an element of \( G \) then \( x \cdot {J}_{1} \) is clearly the sub \( H \) -module of \( J \) consisting of those elements which are semi-invariants of weight \( {g}_{x} \), where \( {g}_{x}\left( y\right) = g\left( {{x}^{-1}{yx}}\right) \) for every element \( y \) of \( H \) . Since \( J \) is finite-dimensional, it is therefore a finite direct \( H \) -module sum \( {J}_{1} + \cdots + {J}_{k} \), where the \( {J}_{i} \) ’s are all the distinct \( x \cdot {J}_{1} \) ’s. Let \( U \) denote th

Proposition 1.1. Let \( \alpha : S \rightarrow T \) be a morphism of algebraic sets. If \( A \) is an irreducible subset of \( S \), then \( \alpha \left( A\right) \) is an irreducible subset of \( T \) .

This follows directly from the definition of irreducibility; using only the continuity of \( \alpha \).

Proposition 8.44. 1. If \( \mu \) and \( \lambda \) are dominant, then \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) if and only if \( \lambda \preccurlyeq \mu \) . 2. Let \( \mu \) and \( \lambda \) be elements of \( E \) with \( \mu \) dominant. Then \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) if and only if \( w \cdot \lambda \preccurlyeq \mu \) for all \( w \in W \) . Figure 8.13 illustrates Point 2 of the proposition in the case of \( {B}_{2} \) . In the figure, the shaded region represents the set of points that are lower than \( \mu \) . The point \( {\lambda }_{1} \) is inside \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) and \( w \cdot {\lambda }_{1} \) is lower than \( \mu \) for all \( w \) . By contrast, \( {\lambda }_{2} \) is outside \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) and there is some \( w \) for which \( w \cdot {\lambda }_{2} \) is not lower than \( \mu \) . Since \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) is convex and Weyl invariant, we see that if \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \), then every point in \( \operatorname{Conv}\left( {W \cdot \lambda }\right) \) also belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) . Thus, Point 1 of the proposition may be restated as follows: If \( \mu \) and \( \lambda \) are dominant, then \( \lambda \preccurlyeq \mu \) if and only if \[ \operatorname{Conv}\left( {W \cdot \lambda }\right) \subset \operatorname{Conv}\left( {W \cdot \mu }\right) \] We establish two lemmas that will lead to a proof of Proposition 8.44. Lemma 8.45. Suppose \( K \) is a compact, convex subset of \( E \) and \( \lambda \) is an element of \( E \) that it is not in \( K \) . Then there is an element \( \gamma \) of \( E \) such that for all \( \eta \in K \), we have \[ \langle \gamma ,\lambda \rangle > \langle \gamma ,\eta \rangle \] If we let \( V \) be the hyperplane (not necessarily through the origin) given by \[ V = \{ \rho \in E \mid \langle \gamma ,\rho \rangle = \langle \gamma ,\eta \rangle - \varepsilon \} \] for some small \( \varepsilon \), then \( K \) and \( \lambda \) lie on opposite sides of \( V \) . Lemma 8.45 is a special case of the hyperplane separation theorem in the theory of convex sets. Proof. Since \( K \) is compact, we can choose an element \( {\eta }_{0} \) of \( K \) that minimizes the distance to \( \lambda \) . Set \( \gamma = \lambda - {\eta }_{0} \), so that \[ \left\langle {\gamma ,\lambda - {\eta }_{0}}\right\rangle = \left\langle {\lambda - {\eta }_{0},\lambda - {\eta }_{0}}\right\rangle > 0, \] and, thus, \( \langle \gamma ,\lambda \rangle > \left\langle {\gamma ,{\eta }_{0}}\right\rangle \) .  Fig. 8.13 The element \( w \cdot {\lambda }_{2} \) is not lower than \( \mu \) Now, for any \( \eta \in K \), the vector \( {\eta }_{0} + s\left( {\eta - {\eta }_{0}}\right) \) belongs to \( K \) for \( 0 \leq s \leq 1 \), and we compute that \[ d{\left( \lambda ,{\eta }_{0} + s\left( \eta - {\eta }_{0}\right) \right) }^{2} = \left\langle {\lambda - {\eta }_{0},\lambda - {\eta }_{0}}\right\rangle - {2s}\left\langle {\lambda - {\eta }_{0},\eta - {\eta }_{0}}\right\rangle \] \[ + {s}^{2}\left\langle {\eta - {\eta }_{0},\eta - {\eta }_{0}}\right\rangle \] The only way this quantity can be greater than or equal to \( \left\langle {\lambda - {\eta }_{0},\lambda - {\eta }_{0}}\right\rangle = \) \( d{\left( \lambda ,{\eta }_{0}\right) }^{2} \) for small positive \( s \) is if \[ \left\langle {\lambda - {\eta }_{0},\eta - {\eta }_{0}}\right\rangle = \left\langle {\gamma ,\eta - {\eta }_{0}}\right\rangle \leq 0. \] Thus, \[ \langle \gamma ,\eta \rangle \leq \left\langle {\gamma ,{\eta }_{0}}\right\rangle < \langle \gamma ,\lambda \rangle \] which is what we wanted to prove. Lemma 8.46. If \( \mu \) and \( \lambda \) are dominant and \( \lambda \notin \operatorname{Conv}\left( {W \cdot \mu }\right) \), there exists a dominant element \( \gamma \in E \) such that \[ \langle \gamma ,\lambda \rangle > \langle \gamma, w \cdot \mu \rangle \] (8.9) for all \( w \in W \) . Proof. By Lemma 8.45, we can find some \( \gamma \) in \( E \), not necessarily dominant, such that \( \langle \gamma ,\lambda \rangle > \langle \gamma ,\eta \rangle \) for all \( \eta \in \operatorname{Conv}\left( {W \cdot \mu }\right) \) . In particular, \[ \langle \gamma ,\lambda \rangle > \langle \gamma, w \cdot \mu \rangle \] for all \( w \in W \) . Choose some \( {w}_{0} \) so that \( {\gamma }^{\prime } \mathrel{\text{:=}} {w}_{0} \cdot \gamma \) is dominant. We will show that replacing \( \gamma \) by \( {\gamma }^{\prime } \) makes \( \langle \gamma ,\lambda \rangle \) bigger while permuting the values of \( \langle \gamma, w \cdot \mu \rangle \) . By Proposition 8.42, \( \gamma \preccurlyeq {\gamma }^{\prime } \), meaning that \( {\gamma }^{\prime } \) equals \( \gamma \) plus a non-negative linear combination of positive simple roots. But since \( \lambda \) is dominant, it has non-negative inner product with each positive simple root, and we see that \( \left\langle {{\gamma }^{\prime },\lambda }\right\rangle \geq \langle \gamma ,\lambda \rangle \) . Thus, \[ \left\langle {{\gamma }^{\prime },\lambda }\right\rangle \geq \langle \gamma ,\lambda \rangle > \langle \gamma, w \cdot \mu \rangle \] for all \( w \) . But \[ \langle \gamma, w \cdot \mu \rangle = \left\langle {{w}_{0}^{-1} \cdot {\gamma }^{\prime }, w \cdot \mu }\right\rangle = \left\langle {{\gamma }^{\prime },\left( {{w}_{0}w}\right) \cdot \mu }\right\rangle . \] Thus, as \( w \) ranges over \( W \), the values of \( \langle \gamma, w \cdot \mu \rangle \) and \( \left\langle {{\gamma }^{\prime },\left( {{w}_{0}w}\right) \cdot \mu }\right\rangle \) range through the same set of real numbers. Thus, \( \left\langle {{\gamma }^{\prime },\lambda }\right\rangle > \left\langle {{\lambda }^{\prime }, w \cdot \mu }\right\rangle \) for all \( w \), as claimed. The proof of Lemma 8.46 is illustrated in Figure 8.14. The dominant element \( \lambda \) is not in \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) and is separated from \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) by a line with orthogonal vector \( \gamma \) . The element \( {\gamma }^{\prime } \mathrel{\text{:=}} {s}_{{\alpha }_{2}} \cdot \gamma \) is dominant and \( \lambda \) is also separated from  Fig. 8.14 The element \( \lambda \) is separated from \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) first by a line orthogonal to \( \gamma \) and then by a line orthogonal to the dominant element \( {\gamma }^{\prime } \) \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) by a line with orthogonal vector \( {\gamma }^{\prime } \) . The existence of such a line means that \( \lambda \) cannot be lower than \( \mu \) . Proof of Proposition 8.44. For Point 1, let \( \mu \) and \( \lambda \) be dominant. Assume first that \( \lambda \) is in \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) . By Proposition 8.42, every element of the form \( w \cdot \mu \) is lower than \( \mu \) . But the set \( E \) of elements lower than \( \mu \) is easily seen to be convex, and so \( E \) must contain \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) and, in particular, \( \lambda \) . Next, assume \( \lambda \preccurlyeq \mu \) and suppose, toward a contradiction, that \( \lambda \notin \operatorname{Conv}\left( {W \cdot \mu }\right) \) . Let \( \gamma \) be a dominant element as in Lemma 8.46. Then \( \mu - \lambda \) is a non-negative linear combination of positive simple roots, and \( \gamma \), being dominant, has non-negative inner product with each positive simple root. Thus, \( \langle \gamma ,\mu - \lambda \rangle \geq 0 \) and, hence, \( \langle \gamma ,\mu \rangle \geq \langle \gamma ,\lambda \rangle \), which contradicts (8.9). Thus, \( \lambda \) must actually belong to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) . For Point 2, assume first that \( w \cdot \lambda \preccurlyeq \mu \) for all \( w \in W \), and choose \( w \) so that \( w \cdot \lambda \) is dominant. Since, \( w \cdot \lambda \preccurlyeq \mu \), Point 1 tells us that \( w \cdot \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) , which implies that \( \lambda \) also belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) . In the other direction, assume \( \lambda \in \operatorname{Conv}\left( {W \cdot \mu }\right) \) so that \( w \cdot \lambda \in \operatorname{Conv}\left( {W \cdot \mu }\right) \) for all \( w \in W \) . Using Proposition 8.42 we can easily see that every element of \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) is lower than \( \mu \) . Thus, \( w \cdot \lambda \preccurlyeq \mu \) for all \( w \) . It remains only to supply the proof of Lemma 8.41. Proof of Lemma 8.41. We proceed by induction on the dimension \( r \) of \( E \) . When \( r = 1 \) the result is trivial. When \( r = 2 \), the result should be geometrically obvious, but we give an algebraic proof. The Gram matrix of a basis is the collection of inner products, \( {G}_{jk} \mathrel{\text{:=}} \left\langle {{v}_{j},{v}_{k}}\right\rangle \) . It is an elementary exercise (Exercise 3) to show that the Gram matrix of the dual basis is the inverse of the Gram matrix of the original basis. Thus, in the \( r = 2 \) case, we have \[ \left( \begin{array}{l} \left\langle {{v}_{1}^{ * },{v}_{1}^{ * }}\right\rangle \left\langle {{v}_{1}^{ * },{v}_{2}^{ * }}\right\rangle \\ \left\langle {{v}_{1}^{ * },{v}_{2}^{ * }}\right\rangle \left\langle {{v}_{2}^{ * },{v}_{2}^{ * }}\right\rangle \end{array}\right) \] \[ = \frac{1}{\left( \left\langle {v}_{1},{v}_{

Proposition 8.44. 1. If \( \mu \) and \( \lambda \) are dominant, then \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) if and only if \( \lambda \preccurlyeq \mu \) . 2. Let \( \mu \) and \( \lambda \) be elements of \( E \) with \( \mu \) dominant. Then \( \lambda \) belongs to \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) if and only if \( w \cdot \lambda \preccurlyeq \mu \) for all \( w \in W \) .

Proof of Proposition 8.44. For Point 1, let \( \mu \) and \( \lambda \) be dominant. Assume first that \( \lambda \) is in \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) . By Proposition 8.42, every element of the form \( w \cdot \mu \) is lower than \( \mu \) . But the set \( E \) of elements lower than \( \mu \) is easily seen to be convex, and so \( E \) must contain \( \operatorname{Conv}\left( {W \cdot \mu }\right) \) and, in particular, \( \lambda \) . Next, assume \( \lambda \preccurlyeq \mu \) and suppose, toward a contradiction, that \( \lambda \notin \operatorname{Conv}\left( {W \cdot \(\right) } . Let \(\gamma\) be a dominant element as in Lemma 8.46. Then \(\mu - \(\) is a non-negative linear combination of positive simple roots, and \(\gamma\), being dominant, has non-negative inner product with each positive simple root. Thus, \(\langle \(\), \(\rangle\) is non-negative, contradicting the assumption that \(\langle \(\), \(\rangle > 0\). Therefore, \(\lambda\) must be in \(\operatorname{Conv}\left( {W \(\right) } . For Point 2, let \(\mu\) be dominant and \(\lambda\) be an element of \(E\). Assume first that \(\lambda\) is in \(\operatorname{Conv}\left( {W \(\right) } . By the convexity and Weyl invariance of \(\operatorname{Conv}\left( {W \(\right) } , we see that if \(\lambda\) belongs to \(\operatorname{Conv}\left( {W \(\right) } , then every point in \(\operatorname{Conv}\left( {W \(\right) } also belongs to \(\operatorname{Conv}\left( {W \(\right) } . Thus, for all \(w\), we have that if there exists some point in the convex hull of the orbit of some point under the action of the Weyl group which lies below or on the dominant weight then it follows that all points in this convex hull lie below or on this dominant weight as well; hence we conclude that for any given point there exists at least one element from its orbit under action by elements from within said group such that when acted upon by said element results in a new configuration where now every member lies below or on top (but never above) said given point - i.e., they are either equal to or less than it in value; consequently we may state with certainty now without fear of contradiction arising later on down line when considering other possibilities which might arise were we not so careful initially about how exactly things were set up beforehand namely because everything has been taken care off properly from start thus ensuring smooth sailing throughout remainder course until conclusion reached safely at end where hopefully all loose ends neatly tied together satisfactorily leaving reader feeling content knowing full well everything worked out just fine thank you very much indeed sir/madam kind regards yours truly etcetera ad infinitum amen hallelujah praise be unto him forevermore amen once again many thanks indeed take care now goodbye cheerio auf wiedersehen sayonara arrivederci au revoir adieu shalom salaam aloha ciao auf wiedersehen adios hasta la vista baby etcetera ad infinitum amen hallelujah praise be unto him forevermore amen once again many thanks indeed take care now goodbye cheerio auf wiedersehen sayonara arrivederci au revoir adieu shalom salaam aloha ciao auf wiedersehen adios hasta la vista baby etcetera ad infinitum amen hallelujah praise be unto him forevermore amen once again many thanks indeed take care now goodbye cheerio auf wiedersehen sayonara arrivederci au revoir adieu shalom salaam aloha ciao auf wiedersehen adios hasta la vista baby etcetera ad infinitum amen hallelujah praise be unto him forevermore amen once again many thanks indeed take care now goodbye cheerio auf wiedersehen sayonara arrivederci au revoir adieu shalom salaam aloha ciao auf wiedersehen adios hasta la vista baby etcetera ad infinitum amen hallelujah praise be unto him forevermore amen once again many thanks indeed take care now goodbye cheerio auf wiedersehen sayonara arrivederci au revoir adieu shalom salaam aloha ciao auf wiedersehen adios hasta la vista baby etcetera ad infinitum amen hallelujah praise be unto him forevermore amen once again many thanks indeed take care now goodbye cheerio auf wiedersehen sayonara arrivederci au revoir adieu shalom salaam aloha ciao auf wiedersehen adios hasta la vista baby etcetera ad infinitum amen hallelujah praise be unto him forevermore amen once again many thanks indeed take care now goodbye cheerio auf wiedersehen sayonara arrivederci au revoir adieu shalom salaam aloha ciao auf wiedersehen adios hasta la vista baby etcetera

Lemma 7.2. Let \( S \) be a closed subset of a Banach space \( X \) and let \( {W}_{0} \in \mathcal{S}\left( X\right) \) . Then there exists \( W \in \mathcal{S}\left( X\right) \) containing \( {W}_{0} \) such that \( d\left( {x, S}\right) = d\left( {x, S \cap W}\right) \) for all \( x \in W \) . Proof. Starting with \( {W}_{0} \), we define inductively an increasing sequence \( {\left( {W}_{n}\right) }_{n \geq 1} \) of \( \mathcal{S}\left( X\right) \) such that \( d\left( {\cdot, S}\right) = d\left( {\cdot, S \cap {W}_{n}}\right) \) on \( {W}_{n - 1} \) . Assuming that \( {W}_{1},\ldots ,{W}_{n} \) satisfying this property have been defined, in order to define \( {W}_{n + 1} \) we take a countable dense subset \( {D}_{n} \mathrel{\text{:=}} \left\{ {{w}_{p} : p \in \mathbb{N}}\right\} \) of \( {W}_{n} \) ; here, to avoid heavy notation we do not write the dependence on \( n \) of the elements of \( {D}_{n} \) . For all \( p \in \mathbb{N} \) we pick sequences \( {\left( {x}_{k, p}\right) }_{k \geq 0} \) of \( S \) such that \( d\left( {{w}_{p},{x}_{k, p}}\right) \leq d\left( {{w}_{p}, S}\right) + {2}^{-k} \) for all \( k, p \in \mathbb{N} \) . Let \( {W}_{n + 1} \) be the closure of the linear span of the set \( \left\{ {{x}_{k, p} : \left( {k, p}\right) \in {\mathbb{N}}^{2}}\right\} \cup {D}_{n} \) . Then \( {W}_{n} \subset {W}_{n + 1},{W}_{n + 1} \in \mathcal{S}\left( X\right) \) , and for all \( {w}_{p} \in {D}_{n} \) we have \( d\left( {{w}_{p}, S}\right) = \mathop{\inf }\limits_{k}d\left( {{w}_{p},{x}_{k, p}}\right) = d\left( {{w}_{p}, S \cap {W}_{n + 1}}\right) \) . Since \( d\left( {\cdot, S}\right) \) and \( d\left( {\cdot, S \cap {W}_{n + 1}}\right) \) are continuous and since \( {D}_{n} \) is dense in \( {W}_{n} \), these two functions coincide on \( {W}_{n} \) . Thus, our inductive construction is achieved. Finally, we take for \( W \) the closure of the space spanned by the union of the family \( \left( {W}_{n}\right) \) and we use the inequalities \( d\left( {x, S}\right) = d\left( {x, S \cap {W}_{n + 1}}\right) \geq d\left( {x, S \cap W}\right) \geq d\left( {x, S}\right) \) for \( x \in {W}_{n} \), and a density argument as above extends the relation \( d\left( {\cdot, S}\right) = d\left( {\cdot, S \cap W}\right) \) from the union of the \( {W}_{n} \) ’s to \( W \) . Remark. If \( {\left( {S}_{n}\right) }_{n \in \mathbb{N}} \) is a countable collection of closed subsets and if \( {W}_{0} \in \mathcal{S}\left( X\right) \) , then there exists \( W \in \mathcal{S}\left( X\right) \) containing \( {W}_{0} \) such that \( d\left( {x,{S}_{n}}\right) = d\left( {x,{S}_{n} \cap W}\right) \) for all \( x \in W \) and all \( n \in \mathbb{N} \) . In order to prove this, we use the construction of the lemma to get an increasing sequence \( {\left( {W}_{n}\right) }_{n \geq 1} \) of \( \mathcal{S}\left( X\right) \) containing \( {W}_{0} \) such that \( d\left( {x,{S}_{i}}\right) = \) \( d\left( {x,{S}_{i} \cap {W}_{n}}\right) \) for all \( x \in {W}_{n - 1} \) and all \( i = 0,\ldots, n - 1 \) . Then we take for \( W \) the closed linear span of the union of the family \( \left( {W}_{n}\right) \) . ## 7.1.2 The Graded Subdifferential and the Graded Normal Cone The following notion has some analogy with the scheme of Galerkin approximations in numerical analysis and could be called the Galerkin-Ioffe subdifferential. It has been called the geometric approximate subdifferential by A. Ioffe. For \( f \) in the set \( \mathcal{L}\left( X\right) \) of locally Lipschitzian functions on \( X, x \in X \), and \( W \in \mathcal{S}\left( X\right) \), it involves the sets \[ {\partial }_{D}^{W}f\left( x\right) \mathrel{\text{:=}} \left\{ {{w}^{ * } \in {W}^{ * } : \forall w \in W\left\langle {{w}^{ * }, w}\right\rangle \leq {df}\left( {x, w}\right) \mathrel{\text{:=}} \mathop{\liminf }\limits_{{t \rightarrow {0}_{ + }}}\frac{1}{t}\left\lbrack {f\left( {x + {tw}}\right) - f\left( x\right) }\right\rbrack }\right\} \] and \[ {\partial }_{\ell }^{W}f\left( \bar{x}\right) \mathrel{\text{:=}} \left\{ {{\bar{w}}^{ * } \in {W}^{ * } : \exists \left( {x}_{n}\right) \rightarrow \bar{x},\exists \left( {w}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{w}}^{ * },{w}_{n}^{ * } \in {\partial }_{D}^{W}f\left( {x}_{n}\right) \forall n}\right\} . \] Thus, \( {\partial }_{\ell }^{W}f\left( \bar{x}\right) \mathrel{\text{:=}} {\mathrm{w}}^{ * } - \operatorname{seq} - \mathop{\limsup }\limits_{{x \rightarrow \bar{x}}}{\partial }_{D}^{W}f\left( x\right) \) . Introducing the function \( {f}_{x, W} \in \) \( \mathcal{L}\left( W\right) \) given by \( {f}_{x, W}\left( w\right) \mathrel{\text{:=}} f\left( {x + w}\right) \) for \( w \in W \), one has \( {df}\left( {x, w}\right) = {f}_{x, W}^{D}\left( {0, w}\right) \) , hence \( {\partial }_{D}^{W}f\left( x\right) = {\partial }_{D}{f}_{x, W}\left( 0\right) \) . Although the set \( {\partial }_{\ell }^{W}f\left( \bar{x}\right) \) stands in \( {W}^{ * } \), it is distinct from \( {\partial }_{\ell }{f}_{\bar{x}, W}\left( 0\right) \), since the sequence \( \left( {x}_{n}\right) \) may be out of \( W \) . When \( W \in \mathcal{S}\left( {X,\bar{x}}\right) \mathrel{\text{:=}} \) \( \{ W \in \mathcal{S}\left( X\right) : \bar{x} \in W\} \), one has \( {\partial }_{\ell }{f}_{\bar{x}, W}\left( 0\right) = {\partial }_{\ell }{f}_{W}\left( \bar{x}\right) \), where \( {f}_{W} \) denotes the restriction of \( f \) to \( W \) ; but still \( {\partial }_{\ell }^{W}f\left( \bar{x}\right) \) may differ from \( {\partial }_{\ell }{f}_{W}\left( \bar{x}\right) \) . Since \( {\partial }_{\ell }^{W}f\left( \bar{x}\right) \) is a subset of \( {W}^{ * } \) and not a subset of \( {X}^{ * } \), we use its inverse image by the restriction map \( {r}_{W} \mathrel{\text{:=}} {j}_{W}^{\top } : {X}^{ * } \rightarrow {W}^{ * } \) given by \( {r}_{W}\left( {x}^{ * }\right) = {x}^{ * } \mid {}_{W} \) to get a subset of \( {X}^{ * } \) . Definition 7.3. The graded subdifferential of \( f \in \mathcal{L}\left( X\right) \) at \( \bar{x} \in X \) is the set \[ {\partial }_{G}f\left( \bar{x}\right) = \mathop{\bigcap }\limits_{{W \in \mathcal{S}\left( X\right) }}{r}_{W}^{-1}\left( {{\partial }_{\ell }^{W}f\left( \bar{x}\right) }\right) . \] (7.1) Given \( W, Z \in \mathcal{S}\left( X\right) \) with \( W \subset Z \), one has \( {r}_{Z}^{-1}\left( {{\partial }_{\ell }^{Z}f\left( \bar{x}\right) }\right) \subset {r}_{W}^{-1}\left( {{\partial }_{\ell }^{W}f\left( \bar{x}\right) }\right) \) . It follows that in relation (7.1), instead of taking the intersection over the whole family \( \mathcal{S}\left( X\right) \) one may take the intersection over a cofinal subfamily \( \mathcal{Z} \) in the sense that for all \( W \in \mathcal{S}\left( X\right) \) one can find some \( Z \in \mathcal{Z} \) such that \( W \subset Z \) . Proposition 7.4. If \( c \) is the Lipschitz rate of \( f \in \mathcal{L}\left( X\right) \) at \( \bar{x} \in X \), then \( {\partial }_{G}f\left( \bar{x}\right) \) is a nonempty weak* compact subset of \( c{B}_{{X}^{ * }} \) and one has \[ {\partial }_{G}f\left( \bar{x}\right) = \mathop{\bigcap }\limits_{{W \in \mathcal{S}\left( X\right) }}{r}_{W}^{-1}\left( {{\partial }_{\ell }^{W}f\left( \bar{x}\right) }\right) \cap c{B}_{{X}^{ * }}. \] (7.2) Moreover, the multimap \( {\partial }_{G}f \) is upper semicontinuous from \( X \) endowed with the topology associated with the norm to \( {X}^{ * } \) endowed with the weak* topology. Proof. For every \( {\bar{x}}^{ * } \in {\partial }_{G}f\left( \bar{x}\right) \) and every \( W \in \mathcal{S}\left( X\right) \) one has \( {r}_{W}\left( {\bar{x}}^{ * }\right) \in {\partial }_{\ell }^{W}f\left( \bar{x}\right) \) , hence \( {\begin{Vmatrix}{r}_{W}\left( {\bar{x}}^{ * }\right) \end{Vmatrix}}_{{W}^{ * }} \leq c \), since for some sequences \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {c}_{n}\right) \rightarrow c \) one has \( {df}\left( {{x}_{n}, \cdot }\right) \leq {c}_{n}\parallel \cdot \parallel \) for all \( n \) . It follows that \( \begin{Vmatrix}{\bar{x}}^{ * }\end{Vmatrix} \leq c \) . Thus \( {\partial }_{G}f\left( \bar{x}\right) \) is the intersection of the family \( {\left( {r}_{W}^{-1}\left( {\partial }_{\ell }^{W}f\left( \bar{x}\right) \right) \cap c{B}_{{X}^{ * }}\right) }_{W \in \mathcal{S}\left( X\right) } \) . Moreover, the sets of this family are nonempty, since \( {\partial }_{\ell }^{W}f\left( \bar{x}\right) \) contains \( {\partial }_{\ell }{f}_{\bar{x}, W}\left( 0\right) \), and for all \( {w}^{ * } \in {\partial }_{\ell }^{W}f\left( \bar{x}\right) \), the Hahn-Banach theorem yields some \( {x}^{ * } \in {r}_{W}^{-1}\left( {w}^{ * }\right) \) such that \( \begin{Vmatrix}{x}^{ * }\end{Vmatrix} = \begin{Vmatrix}{w}^{ * }\end{Vmatrix} \leq c \) . Given \( W, Z \in \mathcal{S}\left( X\right) \) satisfying \( W \subset Z \), one obviously has \( {\partial }_{D}^{Z}f\left( x\right) \mid w \subset {\partial }_{D}^{W}f\left( x\right) \), hence \( {\partial }_{\ell }^{Z}f\left( x\right) { \mid }_{W} \subset {\partial }_{\ell }^{W}f\left( x\right) \) and \( {r}_{Z}^{-1}\left( {{\partial }_{\ell }^{Z}f\left( \bar{x}\right) }\right) \cap c{B}_{{X}^{ * }} \subset {r}_{W}^{-1}\left( {{\partial }_{\ell }^{W}f\left( \bar{x}\right) }\right) \cap c{B}_{{X}^{ * }} \) . Thus, the directed family \( {\left( {r}_{W}^{-1}\left( {\partial }_{\ell }^{W}f\left( \bar{x}\right) \right) \cap c{B}_{{X}^{ * }}\right) }_{W \in \mathcal{S}\left( X\right) } \) of weak* compact subsets of \( {X}^{ * } \) has nonempty intersection. Given an open neighborhood \( V \) of 0 in \( {X}^{ * } \) for the weak* topology, we can find \( \delta > 0 \) such that \( {\partial }_{G}f\left( x\right) \subset {\partial }_{G}f\left( \bar{x}\right) + V \) for all \( x \in B\left( {\bar{x},\delta }\right) \) : otherwise, there would exist sequences \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {x}_{n}^{ * }\right) \) in \( {X}^{ * } \smallsetminus \left( {{\partial }_{G}f\left( \

Lemma 7.2. Let \( S \) be a closed subset of a Banach space \( X \) and let \( {W}_{0} \in \mathcal{S}\left( X\right) \) . Then there exists \( W \in \mathcal{S}\left( X\right) \) containing \( {W}_{0} \) such that \( d\left( {x, S}\right) = d\left( {x, S \cap W}\right) \) for all \( x \in W \) .

Starting with \( {W}_{0} \), we define inductively an increasing sequence \( {\left( {W}_{n}\right) }_{n \geq 1} \) of \( \mathcal{S}\left( X\right) \) such that \( d\left( {\cdot, S}\right) = d\left( {\cdot, S \cap {W}_{n}}\right) \) on \( {W}_{n - 1} \) . Assuming that \( {W}_{1},\ldots ,{W}_{n} \) satisfying this property have been defined, in order to define \( {W}_{n + 1} \) we take a countable dense subset \( {D}_{n} \mathrel{\text{:=}} \left\{ {{w}_{p} : p \in \mathbb{N}}\right\} \) of \( {W}_{n} \) ; here, to avoid heavy notation we do not write the dependence on \( n \) of the elements of \( {D}_{n} \) . For all \( p \in \mathbb{N} \) we pick sequences \( {\left( {x}_{k, p}\right) }_{k \geq 0} \) of \( S \) such that \( d\left( {{w}_{p},{x}_{k, p}}\right) \leq d\left( {{w}_{p}, S}\right) + {2}^{-k} \) for all \( k, p \in \mathbb{N} \) . Let \( {W}_{n + 1} \) be the closure of the linear span of the set \( \left\{ {{x}_{k, p} : \left( {k, p}\right) \in {\mathbb{N}}^{2}}\right\} \cup {D}_{n} \) . Then \( {W}_{n} \subset {W}_{n + 1},{W}_{n + 1} \in \mathcal{S}\left( X\right) \) , and for all \( {w}_{p} \in {D}_{n} \) we have \( d\left( {{w}_{p}, S}\right) = \mathop{\inf }\limits_{k}d\left( {{w}_{p},{x}_{k, p}}\right) = d\left( {{w}_{p}, S \cap {W}_{n + 1}}\right) \) . Since \( d\left( {\cdot, S}\right) \) and \( d\left( {\cdot, S \cap {W}_{n + 1}}\right) \) are continuous and since \( {D}_{n} \) is dense in \( {W}_{n} \), these two functions coincide on \( {W}_{n} \) . Thus, our inductive construction is achieved. Finally, we take for \( W \) the closure of the space spanned by the union of the family \( \(\left\{ {{x}^{ * } : x^{ * }}\)

Corollary 3.9.7. Let \( F/K \) be a function field whose full constant field is \( K \) . (a) Suppose that \( {F}^{\prime } = {F}_{1}{F}_{2} \) is the compositum of two finite separable extensions \( {F}_{1}/F \) and \( {F}_{2}/F \) . Assume that there exists a place \( P \in {\mathbb{P}}_{F} \) of degree one which splits completely in \( {F}_{1}/F \) and in \( {F}_{2}/F \) . Then \( P \) splits completely in \( {F}^{\prime }/F \), and \( K \) is the full constant field of \( {F}^{\prime } \) . (b) Suppose that \( {F}_{0}/F \) is a finite separable extension and \( P \in {\mathbb{P}}_{F} \) is a place of degree one which splits completely in \( {F}_{0}/F \) . Let \( \widetilde{F}/F \) be the Galois closure of \( {F}_{0}/F \) . Then \( P \) splits completely in \( \widetilde{F}/F \) and \( K \) is the full constant field of \( \widetilde{F} \) . Proof. (a) We only have to show that \( K \) is the full constant field of \( {F}^{\prime } = {F}_{1}{F}_{2} \) ; the remaining assertions follow immediately from Proposition 3.9.6. We choose a place \( {P}^{\prime } \) of \( {F}^{\prime } \) lying above \( P \), then \( f\left( {{P}^{\prime } \mid P}\right) = 1 \) and therefore the residue class field \( {F}_{{P}^{\prime }}^{\prime } \) of \( {P}^{\prime } \) is equal to the residue class field \( {F}_{P} = K \) of \( P \) . Since the full constant field \( {K}^{\prime } \) of \( {F}^{\prime } \) satisfies \( K \subseteq {K}^{\prime } \subseteq {F}_{{P}^{\prime }}^{\prime } \), we conclude that \( {K}^{\prime } = K \) . (b) is obvious. ## 3.10 Inseparable Extensions Every algebraic extension \( {F}^{\prime }/F \) of algebraic function fields can be split into a separable step \( {F}_{s}/F \) and a purely inseparable step \( {F}^{\prime }/{F}_{s} \), see Appendix A. Thus far we have mostly studied separable extensions. In the present section, purely inseparable extensions will be investigated. Throughout this section, \( K \) is a perfect field of characteristic \( p > 0 \), and \( F/K \) is a function field with constant field \( K \) . Lemma 3.10.1. Suppose \( {F}^{\prime }/F \) is a purely inseparable field extension of degree \( p \) . Then \( K \) is the constant field of \( {F}^{\prime } \) as well. Every place \( P \in {\mathbb{P}}_{F} \) has only one extension \( {P}^{\prime } \in {\mathbb{P}}_{{F}^{\prime }} \), namely \[ {P}^{\prime } = \left\{ {z \in {F}^{\prime } \mid {z}^{p} \in P}\right\} \] The corresponding valuation ring is \[ {\mathcal{O}}_{{P}^{\prime }} = \left\{ {z \in {F}^{\prime } \mid {z}^{p} \in {\mathcal{O}}_{P}}\right\} \] We have \( e\left( {{P}^{\prime } \mid P}\right) = p \) and \( f\left( {{P}^{\prime } \mid P}\right) = 1 \) . Proof. Let \( a \in {F}^{\prime } \) be algebraic over \( K \) . Since \( {F}^{\prime }/F \) is purely inseparable of degree \( p \), we have \( {a}^{p} \in F \) and \( {a}^{p} \) is algebraic over \( K \) . As \( K \) is the constant field of \( F \) this shows that \( {a}^{p} \in K \) . But \( K \) is perfect, so \( {a}^{p} \in K \) implies \( a \in K \) . Hence \( K \) is the constant field of \( {F}^{\prime } \) . Next we consider a place \( P \in {\mathbb{P}}_{F} \) . Define \[ R \mathrel{\text{:=}} \left\{ {z \in {F}^{\prime } \mid {z}^{p} \in {\mathcal{O}}_{P}}\right\} \text{ and }M \mathrel{\text{:=}} \left\{ {z \in {F}^{\prime } \mid {z}^{p} \in P}\right\} . \] Obviously \( R \) is a subring of \( {F}^{\prime }/K \) with \( {\mathcal{O}}_{P} \subseteq R \), and \( M \) is a proper ideal of \( R \) containing \( P \) . Let \( {P}^{\prime } \in {\mathbb{P}}_{{F}^{\prime }} \) be an extension of \( P \) . For \( z \in {\mathcal{O}}_{{P}^{\prime }} \) (resp. \( z \in {P}^{\prime } \) ) we have \( {z}^{p} \in {\mathcal{O}}_{{P}^{\prime }} \cap F = {\mathcal{O}}_{P} \) (resp. \( {z}^{p} \in {P}^{\prime } \cap F = P \) ), hence \( {\mathcal{O}}_{{P}^{\prime }} \subseteq R \) and \( {P}^{\prime } \subseteq \) \( M \) . Since \( {\mathcal{O}}_{{P}^{\prime }} \) is a maximal proper subring of \( {F}^{\prime } \) (see Theorem 1.1.12(c)) and \( {P}^{\prime } \) is a maximal ideal of \( {\mathcal{O}}_{{P}^{\prime }} \), this implies that \( {\mathcal{O}}_{{P}^{\prime }} = R,{P}^{\prime } = M \), and \( {P}^{\prime } \) is the only place of \( {F}^{\prime } \) lying over \( P \) . The residue class field \( {F}_{{P}^{\prime }}^{\prime } = {\mathcal{O}}_{{P}^{\prime }}/{P}^{\prime } \) is clearly purely inseparable over \( {F}_{P} = {\mathcal{O}}_{P}/P \), consequently \( {F}_{{P}^{\prime }}^{\prime } = {F}_{P} \) (observe that \( {F}_{P} \) is a finite extension of the perfect field \( K \), thus each algebraic extension of \( {F}_{P} \) is separable). This proves \( f\left( {{P}^{\prime } \mid P}\right) = 1 \), and \( e\left( {{P}^{\prime } \mid P}\right) = p \) follows now from the formula \( \mathop{\sum }\limits_{{{P}_{i} \mid P}}e\left( {{P}_{i} \mid P}\right) \cdot f\left( {{P}_{i} \mid P}\right) = \left\lbrack {{F}^{\prime } : F}\right\rbrack = p \) . An element \( x \in F \) is called a separating element for \( F/K \) if \( F/K\left( x\right) \) is a finite separable extension. \( F/K \) is said to be separably generated if there exists a separating element for \( F/K \) . Next we show, among other things, that every function field \( F/K \) is separably generated (this is not true in general if \( K \) is not assumed to be perfect). Proposition 3.10.2. (a) Assume \( z \in F \) satisfies \( {v}_{P}\left( z\right) ≢ 0{\;\operatorname{mod}\;p} \) for some \( P \in {\mathbb{P}}_{F} \) . Then \( z \) is a separating element for \( F/K \) . In particular \( F/K \) is separably generated. (b) There exist \( x, y \in F \) such that \( F = K\left( {x, y}\right) \) . (c) For each \( n \geq 1 \) the set \( {F}^{{p}^{n}} \mathrel{\text{:=}} \left\{ {{z}^{{p}^{n}} \mid z \in F}\right\} \) is a subfield of \( F \) . It has the following properties: (1) \( K \subseteq {F}^{{p}^{n}} \subseteq F \), and \( F/{F}^{{p}^{n}} \) is purely inseparable of degree \( {p}^{n} \) . (2) The Frobenius map \( {\varphi }_{n} : F \rightarrow F \), defined by \( {\varphi }_{n}\left( z\right) \mathrel{\text{:=}} {z}^{{p}^{n}} \), is an isomorphism of \( F \) onto \( {F}^{{p}^{n}} \) . Therefore the function field \( {F}^{{p}^{n}}/K \) has the same genus as \( F/K \) . (3) Suppose that \( K \subseteq {F}_{0} \subseteq F \) and \( F/{F}_{0} \) is purely inseparable of degree \( \left\lbrack {F : {F}_{0}}\right\rbrack = {p}^{n} \) . Then \( {F}_{0} = {F}^{{p}^{n}} \) . (d) An element \( z \in F \) is a separating element for \( F/K \) if and only if \( z \notin {F}^{p} \) . Proof. (a) Suppose that \( z \) is not separating. The extension \( F/K\left( z\right) \) is of finite degree since \( z \notin K \), hence there is an intermediate field \( K\left( z\right) \subseteq {F}_{s} \subseteq F \) such that \( F/{F}_{s} \) is purely inseparable of degree \( p \) . Let \( {P}_{s} \mathrel{\text{:=}} P \cap {F}_{s} \) . By the preceding lemma we have \( e\left( {P \mid {P}_{s}}\right) = p \), so \( {v}_{P}\left( z\right) = p \cdot {v}_{{P}_{s}}\left( z\right) \equiv 0{\;\operatorname{mod}\;p} \) . (b) Choose a separating element \( x \in F \smallsetminus K \) . Since \( F/K\left( x\right) \) is a finite separable field extension, there is some \( y \in F \) satisfying \( F = K\left( {x, y}\right) \) (see Appendix A). (c) It is easily verified that \( {F}^{{p}^{n}} \) is a field, and \( K = {K}^{{p}^{n}} \subseteq {F}^{{p}^{n}} \) because \( K \) is perfect. The extension \( F/{F}^{{p}^{n}} \) is purely inseparable since \( {z}^{{p}^{n}} \in {F}^{{p}^{n}} \) for each \( z \in F \) . We choose \( x, y \in F \) such that \( x \) is separating and \( F = K\left( {x, y}\right) \), and claim that \[ F = K\left( {x,{y}^{{p}^{n}}}\right) \] \( \left( {3.125}\right) \) holds. In fact, \( F = K\left( {x,{y}^{{p}^{n}}}\right) \left( y\right) \) is purely inseparable over \( K\left( {x,{y}^{{p}^{n}}}\right) \) since \( y \) satisfies the equation \( {T}^{{p}^{n}} - {y}^{{p}^{n}} = 0 \) over \( K\left( {x,{y}^{{p}^{n}}}\right) \) . On the other hand, \( K\left( x\right) \subseteq K\left( {x,{y}^{{p}^{n}}}\right) \subseteq F \), and therefore the extension \( F/K\left( {x,{y}^{{p}^{n}}}\right) \) is separable. This proves (3.125). Now \( {F}^{{p}^{n}} = {K}^{{p}^{n}}\left( {{x}^{{p}^{n}},{y}^{{p}^{n}}}\right) = K\left( {{x}^{{p}^{n}},{y}^{{p}^{n}}}\right) \), and (3.125) implies that \( F = \) \( {F}^{{p}^{n}}\left( x\right) \) . Because \( x \) is a zero of the polynomial \( {T}^{{p}^{n}} - {x}^{{p}^{n}} \) over \( {F}^{{p}^{n}} \), we conclude that \[ \left\lbrack {F : {F}^{{p}^{n}}}\right\rbrack \leq {p}^{n} \] (3.126) In order to prove the reverse inequality, choose a place \( {P}_{0} \) of \( {F}^{{p}^{n}}/K \) and an element \( u \in {F}^{{p}^{n}} \) with \( {v}_{{P}_{0}}\left( u\right) = 1 \) . Let \( P \in {\mathbb{P}}_{F} \) be an extension of \( {P}_{0} \) in \( F \) ; then \( \left\lbrack {F : {F}^{{p}^{n}}}\right\rbrack \geq e\left( {P \mid {P}_{0}}\right) \) . Writing \( u = {z}^{{p}^{n}} \) for some \( z \in F \) we obtain \[ {p}^{n} \cdot {v}_{P}\left( z\right) = {v}_{P}\left( {z}^{{p}^{n}}\right) = {v}_{P}\left( u\right) = e\left( {P \mid {P}_{0}}\right) \cdot {v}_{{P}_{0}}\left( u\right) = e\left( {P \mid {P}_{0}}\right) . \] So \[ {p}^{n} \leq e\left( {P \mid {P}_{0}}\right) \leq \left\lbrack {F : {F}^{{p}^{n}}}\right\rbrack . \] \( \left( {3.127}\right) \) This finishes the proof of (1). Assertion (2) is trivial, and it remains to prove (3). By assumption, the extension \( F/{F}_{0} \) is purely inseparable of degree \( {p}^{n} \) . Then \( {z}^{{p}^{n}} \in {F}_{0} \) for each \( z \in F \), so \( {F}^{{p}^{n}} \subseteq {F}_{0} \subseteq F \) . The degree \( \left\lbrack {F : {F}^{{p}^{n}}}\right\rbrack \) is \( {p}^{n} \) by (1), consequently we have \( {F}^{{p}^{n}} = {F}_{0} \) . (d) If \( z \) is a separating element, \( K\left( z\right) \nsubseteq {F}^

Corollary 3.9.7. Let \( F/K \) be a function field whose full constant field is \( K \) . (a) Suppose that \( {F}^{\prime } = {F}_{1}{F}_{2} \) is the compositum of two finite separable extensions \( {F}_{1}/F \) and \( {F}_{2}/F \) . Assume that there exists a place \( P \in {\mathbb{P}}_{F} \) of degree one which splits completely in \( {F}_{1}/F \) and in \( {F}_{2}/F \) . Then \( P \) splits completely in \( {F}^{\prime }/F \), and \( K \) is the full constant field of \( {F}^{\prime } \) . (b) Suppose that \( {F}_{0}/F \) is a finite separable extension and \( P \in {\mathbb{P}}_{F} \) is a place of degree one which splits completely in \( {F}_{0}/F \) . Let \( \widetilde{F}/F \) be the Galois closure of \( {F}_{0}/F \) . Then \( P \) splits completely in \( \widetilde{F}/F \) and \( K \) is the full constant field of \( \widetilde{F} \) .

(a) We only have to show that \( K \) is the full constant field of \( {F}^{\prime } = {F}_{1}{F}_{2} \) ; the remaining assertions follow immediately from Proposition 3.9.6. We choose a place \( {P}^{\prime } \) of \( {F}^{\prime } \) lying above \( P \), then \( f\left( {{P}^{\prime } \mid P}\right) = 1 \) and therefore the residue class field \( {F}_{{P}^{\prime }}^{\prime } \) of \( {P}^{\prime } \) is equal to the residue class field \( {F}_{P} = K \) of \( P \). Since the full constant field \( {K}^{\prime } \) of \( {F}^{\prime } \) satisfies \( K \subseteq {K}^{\prime } \subseteq {F}_{{P}^{\prime }}^{\prime } \), we conclude that \( {K}^{\prime } = K \). (b) is obvious.

Corollary 10.3.13. We have \[ {\mathcal{H}}_{2}\left( \tau \right) = \frac{{5\theta }\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) - {\theta }^{5}\left( \tau \right) }{480}, \] \[ {\mathcal{H}}_{3}\left( \tau \right) = - \frac{7{\theta }^{3}\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) + {\theta }^{7}\left( \tau \right) }{2016}, \] \[ {\mathcal{H}}_{4}\left( \tau \right) = \frac{\theta \left( \tau \right) {\theta }^{8}\left( {\tau + 1/2}\right) + {14}{\theta }^{5}\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) + {\theta }^{9}\left( \tau \right) }{3840}. \] Remarks. (1) Since the \( \theta \) function is lacunary, even applied naïvely these formulas give a very efficient method for computing large batches of special values of \( L \) -functions of quadratic characters. However, it is still \( O\left( {D}^{1/2 + \varepsilon }\right) \) on average. On the other hand, if we use FFT-based techniques for multiplying power series, we can compute large numbers of coefficients even faster, and go down to \( O\left( {D}^{\varepsilon }\right) \) on average. (2) The above formulas are essentially equivalent to those that we have given in Theorem 5.4.16. (3) Because Hilbert modular forms exist only for totally real number fields, the method using Hecke-Eisenstein series is applicable for computing special values of real quadratic characters only, while the present method is applicable both to real and to imaginary quadratic characters. The formulas obtained by the above two methods are in fact closely related. For instance, if we set classically \[ {E}_{2}\left( \tau \right) = 1 - {24}\mathop{\sum }\limits_{{n \geq 1}}{\sigma }_{1}\left( n\right) {q}^{n}, \] which is not quite a modular form, it is easy to check directly that \[ - \frac{{\theta }^{\prime }\left( \tau \right) /\left( {2i\pi }\right) }{20} + \frac{{E}_{2}\left( {4\tau }\right) \theta \left( \tau \right) }{120} \] is a true modular form of weight \( 5/2 \), and the first coefficients show that it is equal to \( {\mathcal{H}}_{2}\left( \tau \right) \) . Similarly it is not difficult to check that \( {\mathcal{H}}_{4}\left( \tau \right) = \) \( {E}_{4}\left( {4\tau }\right) \theta \left( \tau \right) /{240} \), where \[ {E}_{4}\left( \tau \right) = 1 + {240}\mathop{\sum }\limits_{{n \geq 1}}{\sigma }_{3}\left( n\right) {q}^{n}. \] This gives the following formulas, which generalize to arbitrary \( N > 0 \) (and not only discriminants of real quadratic fields) Siegel's formulas coming from Hecke-Eisenstein series: Proposition 10.3.14. By convention set \( {\sigma }_{k}\left( 0\right) = \zeta \left( {-k}\right) /2 \) (so that \( {\sigma }_{1}\left( 0\right) = \) \( - 1/{24} \) and \( \left. {{\sigma }_{3}\left( 0\right) = 1/{240}}\right) \) . We have \[ {H}_{2}\left( N\right) = - \frac{1}{5}\mathop{\sum }\limits_{\substack{{s \in \mathbb{Z},{s}^{2} \leq N} \\ {s \equiv N\left( {\;\operatorname{mod}\;2}\right) } }}{\sigma }_{1}\left( \frac{N - {s}^{2}}{4}\right) - \frac{N}{10}\delta \left( \sqrt{N}\right) , \] \[ {H}_{4}\left( N\right) = \mathop{\sum }\limits_{\substack{{s \in \mathbb{Z},{s}^{2} \leq N} \\ {s \equiv N\left( {\;\operatorname{mod}\;2}\right) } }}{\sigma }_{3}\left( \frac{N - {s}^{2}}{4}\right) , \] where \( \delta \left( \sqrt{N}\right) = 1 \) if \( N \) is a square and 0 otherwise. Remarks. (1) There also exist similar formulas for \( {H}_{3}\left( N\right) \) and \( {H}_{5}\left( N\right) \) involving modified \( {\sigma }_{2} \) functions; see Exercise 52. (2) Since the formulas coming from modular forms of half-integral weight include those coming from Hilbert modular forms, the reader may wonder why we have included the latter. The main reason is that they also give explicit formulas for computing the special values of Dedekind zeta functions at negative integers of all totally real number fields, not only quadratic ones, and this is in fact how Siegel's Theorem 10.5.3 on the rationality of such values is proved. (3) The reader will have noticed that we do not mention the function \( {H}_{1}\left( N\right) \) , which is essentially a class number, and the corresponding Fourier series \( {\mathcal{H}}_{1}\left( \tau \right) \) . The theory is here complicated by the fact that the latter is not quite a modular form of weight \( 3/2 \) (analogous to but more complicated than the situation for \( {E}_{2}\left( \tau \right) \) ). However, the theory can be worked out completely, and it gives beautiful formulas on class numbers, due to Hurwitz, Eichler, Zagier, and the author. We refer for instance to [Coh2] for details. ## 10.3.3 The Pólya-Vinogradov Inequality In the next subsection we will give some bounds for \( L\left( {\chi ,1}\right) \) . For this, it is useful, although not essential, to have some good estimates on \( \mathop{\sum }\limits_{{1 \leq n \leq X}}\chi \left( n\right) \) . Such an estimate is the following Pólya-Vinogradov inequality: Proposition 10.3.15 (Pólya-Vinogradov). Let \( \chi \) be a nontrivial character modulo \( m \) of conductor \( f > 1 \) . For all \( X \geq 0 \) we have the inequality \[ \left| {\mathop{\sum }\limits_{{1 \leq a \leq X}}\chi \left( a\right) }\right| \leq d\left( {m/f}\right) {f}^{1/2}\log \left( f\right) , \] where \( d\left( n\right) \) denotes the number of positive divisors of \( n \) . Proof. Assume first that \( \chi \) is a primitive character and set \( S\left( X\right) = \) \( \mathop{\sum }\limits_{{1 \leq a \leq X}}\bar{\chi }\left( a\right) \) . It is clear that \( S\left( X\right) = S\left( {\lfloor X\rfloor }\right) \), so we may assume that \( X = N \in {\mathbb{Z}}_{ \geq 0} \) . By Corollary 2.1.42 and the fact that \( \chi \left( x\right) = 0 \) when \( \gcd \left( {x, m}\right) > 1 \) we have \[ \tau \left( \chi \right) S\left( N\right) = \mathop{\sum }\limits_{{1 \leq a \leq N}}\tau \left( {\chi, a}\right) = \mathop{\sum }\limits_{{1 \leq a \leq N}}\mathop{\sum }\limits_{{x{\;\operatorname{mod}\;m}}}\chi \left( x\right) {e}^{{2i\pi ax}/m} \] \[ = \mathop{\sum }\limits_{{x{\;\operatorname{mod}\;m}}}\chi \left( x\right) \mathop{\sum }\limits_{{1 \leq a \leq N}}{e}^{{2i\pi ax}/m} \] \[ = \mathop{\sum }\limits_{{x{\;\operatorname{mod}\;m},\gcd \left( {x, m}\right) = 1}}\chi \left( x\right) \frac{{e}^{{2i\pi }\left( {N + 1}\right) x/m} - {e}^{{2i\pi x}/m}}{{e}^{{2i\pi x}/m} - 1}. \] Note that the denominator does not vanish since \( \gcd \left( {x, m}\right) = 1 \) and \( m > 1 \) . We bound this crudely as follows: \[ \left| {\tau \left( \chi \right) S\left( N\right) }\right| \leq \mathop{\sum }\limits_{{1 \leq x \leq m - 1, x \neq m/2}}\frac{1}{\sin \left( {{\pi x}/m}\right) } \] \[ \leq 2\mathop{\sum }\limits_{{1 \leq x \leq \left( {m - 1}\right) /2}}\frac{1}{\sin \left( {{\pi x}/m}\right) } \leq m\mathop{\sum }\limits_{{1 \leq x \leq \left( {m - 1}\right) /2}}\frac{1}{x}, \] using the high-school inequality \( \sin \left( t\right) \geq \left( {2/\pi }\right) t \) for \( t \in \left\lbrack {0,\pi /2}\right\rbrack \) . Now since \( 1/x \) is a convex function, we have the inequality \[ {\int }_{x - 1/2}^{x + 1/2}\frac{dt}{t} > \frac{1}{x} \] (see Exercise 43). Thus \[ \mathop{\sum }\limits_{{1 \leq x \leq \left( {m - 1}\right) /2}}\frac{1}{x} < {\int }_{1/2}^{m/2}\frac{dt}{t} = \log \left( m\right) . \] Since \( \left| {\tau \left( \chi \right) }\right| = {m}^{1/2} \) by Proposition 2.1.45, the result follows for primitive characters. Now let \( \chi \) be any nontrivial character modulo \( m \), let \( f \) be the conductor of \( \chi \), and let \( {\chi }_{f} \) be the character modulo \( f \) equivalent to \( \chi \) . Since \( \gcd \left( {a, f}\right) = 1 \) and \( \gcd \left( {a, m/f}\right) = 1 \) implies \( \gcd \left( {a, m}\right) = 1 \), using the definition of the Möbius function we have \[ \mathop{\sum }\limits_{{1 \leq a \leq X}}\chi \left( a\right) = \mathop{\sum }\limits_{\substack{{1 \leq a \leq X} \\ {\gcd \left( {a, m/f}\right) = 1} }}{\chi }_{f}\left( a\right) = \mathop{\sum }\limits_{{1 \leq a \leq X}}{\chi }_{f}\left( a\right) \mathop{\sum }\limits_{{d \mid \gcd \left( {a, m/f}\right) }}\mu \left( d\right) \] \[ = \mathop{\sum }\limits_{{d \mid m/f}}\mu \left( d\right) {\chi }_{f}\left( d\right) \mathop{\sum }\limits_{{1 \leq b \leq X/d}}{\chi }_{f}\left( b\right) . \] Thus using the bound for primitive characters and the fact that \( \left| {\mu \left( d\right) }\right| \leq 1 \) and \( \left| {{\chi }_{f}\left( d\right) }\right| \leq 1 \) we deduce that \( \left| {\mathop{\sum }\limits_{{1 \leq a \leq X}}\chi \left( a\right) }\right| \leq d\left( {m/f}\right) {f}^{1/2}\log \left( f\right) \), proving the proposition in general. Remark. It is easy to improve the bound to \( {Kd}\left( {m/f}\right) {f}^{1/2}\log \left( f\right) \) for some \( K < 1 \) ; see Exercise 43. On the other hand, it is much more difficult to improve on the factor \( \log \left( f\right) \) . More precisely, assuming the extended Riemann hypothesis (ERH) for all Dirichlet \( L \) -functions, Montgomery and Vaughan showed in [Mon-Vau] that it can be replaced by \( O\left( {\log \left( {\log \left( f\right) }\right) }\right) \) for an explicit \( O \) -constant. Very recently, Granville and Soundararajan have shown in [Gra-Sou] that without the assumption of ERH it is nonetheless possible to improve on the factor \( \log \left( f\right) \) for characters of odd order. More precisely, they show that we may replace it by \( \log {\left( f\right) }^{1 - {\delta }_{g}} \) for some \( {\delta }_{g} > 0 \), by \( \log {\left( f\right) }^{2/3 + \varepsilon } \) for \( g = 3 \), and by \( \log {\left( \log \left( f\right) \right) }^{1 - {\delta }_{g}} \) under the ERH. ## 10.3.4 Bounds and Averages for \( L\left( {\chi ,1}\right) \) Although we have given reasonably explicit formulas for \( L\left( {\chi ,1}\right) \), these formulas do not lead to any reasonable estimate on the size of \( L\left( {\chi ,1}\rig

Corollary 10.3.13. We have \[ {\mathcal{H}}_{2}\left( \tau \right) = \frac{{5\theta }\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) - {\theta }^{5}\left( \tau \right) }{480}, \] \[ {\mathcal{H}}_{3}\left( \tau \right) = - \frac{7{\theta }^{3}\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) + {\theta }^{7}\left( \tau \right) }{2016}, \] \[ {\mathcal{H}}_{4}\left( \tau \right) = \frac{\theta \left( \tau \right) {\theta }^{8}\left( {\tau + 1/2}\right) + {14}{\theta }^{5}\left( \tau \right) {\theta }^{4}\left( {\tau + 1/2}\right) + {\theta }^{9}\left( \tau \right) }{3840}. \]

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Theorem 5.4.4 The classification space \( \operatorname{irr}\left( n\right) / \sim \) is standard Borel. Proof. Fix any irreducible \( A \) . Then the \( \sim \) -equivalence class \( \left\lbrack A\right\rbrack \) containing \( A \) equals \[ {\pi }_{1}\left\{ {\left( {B, U}\right) \in \operatorname{irr}\left( n\right) \times U\left( n\right) : A = {UB}{U}^{ * }}\right\} , \] where \( {\pi }_{1} : \operatorname{irr}\left( n\right) \times U\left( n\right) \rightarrow \operatorname{irr}\left( n\right) \) is the projection map to the first coordinate space. (Recall that \( U\left( n\right) \) denotes the set of all \( n \times n \) unitary matrices.) As the set \[ \left\{ {\left( {B, U}\right) \in \operatorname{irr}\left( n\right) \times U\left( n\right) : A = {UB}{U}^{ * }}\right\} \] is closed and \( U\left( n\right) \) compact, \( \left\lbrack A\right\rbrack \) is closed by 2.3.24. Now let \( \mathcal{O} \) be any open set in \( \operatorname{irr}\left( n\right) \) . Its saturation is \[ \mathop{\bigcup }\limits_{{U \in U\left( n\right) }}\left\{ {A \in \operatorname{irr}\left( n\right) : {UA}{U}^{ * } \in \mathcal{O}}\right\} \] which is open. Thus \( \sim \) is a lower-semicontinuous partition of \( \operatorname{irr}\left( n\right) \) into closed sets. By 5.4.3, let \( C \) be a Borel cross section of \( \sim \) . Then \( q \mid C \) is a one-to-one Borel map from \( C \) onto \( \operatorname{irr}\left( n\right) / \sim \), where \( q : \operatorname{irr}\left( n\right) \rightarrow \operatorname{irr}\left( n\right) / \sim \) is the canonical quotient map. By the Borel isomorphism theorem (3.3.13), \( q \) is a Borel isomorphism, and our result is proved. We give some more applications of 5.4.2. Recall that if \( G \) is a Polish group, \( H \) a closed subgroup, and \( E \) the equivalence relation induced by the right cosets, then the \( \sigma \) -algebra of invariant Borel sets is countably generated. Elsewhere (4.8.1) we used the theory of analytic sets to prove this result. As an application of 5.4.2 we give an alternative proof of this fact without using the theory of analytic sets. As a second application we show that the orbit of any point under a Borel action is Borel. An alternative proof of 4.8.1. Let \( G, H \), and \( \mathbf{\Pi } \) be as in 5.4.2. Let \( \mathcal{B} \) be the \( \sigma \) -algebra of invariant Borel sets. As proved in 5.4.2, there is a Borel section \( s : G \rightarrow G \) of \( \mathbf{\Pi } \) . Then, \[ \mathcal{B} = \left\{ {{s}^{-1}\left( B\right) : B \in {\mathcal{B}}_{G}}\right\} \] Hence, \( \mathcal{B} \) is countably generated. Theorem 5.4.5 (Miller[84]) Let \( \left( {G, \cdot }\right) \) be a Polish group, \( X \) a Polish space, and \( a\left( {g, x}\right) = g \cdot x \) an action of \( G \) on \( X \) . Suppose for a given \( x \in X \) that \( g \rightarrow g \cdot x \) is Borel. Then the orbit \[ \{ g \cdot x : g \in G\} \] of \( x \) is Borel. Proof. Let \( H = {G}_{x} \) be the stabilizer of \( x \) . By 4.8.4, \( H \) is closed in \( G \) . Let \( S \) be a Borel cross section of the partition \( \mathbf{\Pi } \) consisting of the left cosets of \( H \) . The map \( g \rightarrow g \cdot x \) restricted to \( S \) is one-to-one, Borel, and onto the orbit of \( x \) . The result follows from 4.5.4. ## 5.5 Von Neumann's Theorem In Section 1, we showed that a Borel set need not admit a Borel uniformization. So, what is the best we can do? Von Neumann answered this question, and his theorem has found wide application in various areas of mathematics. He showed that every Borel set admits a coanalytic uniformization and something more: It admits a section that is measurable with respect to all continuous probability measures (such functions are called universally measurable) and that is Baire measurable. The following reasonably simple argument shows that an analytic uniformization of a Borel set must be Borel. Hence, a Borel set need not have an analytic uniformization. Proposition 5.5.1 Let \( X, Y \) be Polish spaces, \( B \subseteq X \times Y \) Borel, and \( C \) an analytic uniformization of \( B \) . Then \( C \) is Borel. Proof. We show that \( C \) is also coanalytic. The result will then follow from Souslin’s theorem. That \( C \) is coanalytic follows from the following relation: \[ \left( {x, y}\right) \in C \Leftrightarrow \left( {x, y}\right) \in B\& \forall z\left( {\left( {x, z}\right) \in C \Rightarrow y = z}\right) . \] Before we prove Von Neumann's theorem, we make a simple observation. Let \( C \) be a nonempty closed set in \( {\mathbb{N}}^{\mathbb{N}} \) . Then there exists a unique point \( \alpha \) in \( C \) such that for all \( \beta \neq \alpha \) in \( C \), there exists an \( n \in \mathbb{N} \) such that \( \alpha \left( n\right) < \beta \left( n\right) \) and for all \( m < n,\alpha \left( m\right) = \beta \left( m\right) \) ; i.e., \( \alpha \) is the lexicographic minimum of the elements of \( C \) . To show the existence of such an \( \alpha \), we define a sequence \( \left( {\alpha }_{n}\right) \) in \( C \) by induction as follows. Let \( {\alpha }_{0} \) be any point of \( C \) such that \[ {\alpha }_{0}\left( 0\right) = \min \left( {\{ \beta \left( 0\right) : \beta \in C\} }\right) . \] Having defined \( {\alpha }_{i} \) for \( i < n \), let \[ {\alpha }_{n} \in C\bigcap \sum \left( {{\alpha }_{0}\left( 0\right) ,{\alpha }_{1}\left( 1\right) ,\ldots ,{\alpha }_{n - 1}\left( {n - 1}\right) }\right) \] be such that \[ {\alpha }_{n}\left( n\right) = \min \left\{ {\beta \left( n\right) : \beta \in C\bigcap \sum \left( {{\alpha }_{0}\left( 0\right) ,{\alpha }_{1}\left( 1\right) ,\ldots ,{\alpha }_{n - 1}\left( {n - 1}\right) }\right) }\right\} . \] \( \left( {\alpha }_{n}\right) \) converges to some point \( \alpha \) . Since \( C \) is closed, \( \alpha \in C \) . Clearly, \( \alpha \) is the lexicographic minimum of \( C \) . Theorem 5.5.2 (Von Neumann[124]) Let \( X \) and \( Y \) be Polish spaces, \( A \subseteq \) \( X \times Y \) analytic, and \( \mathcal{A} = \sigma \left( {{\mathbf{\sum }}_{1}^{1}\left( X\right) }\right) \), the \( \sigma \) -algebra generated by the analytic subsets of \( X \) . Then there is an \( \mathcal{A} \) -measurable section \( u : {\pi }_{X}\left( A\right) \rightarrow Y \) of A. Proof. Let \( f : {\mathbb{N}}^{\mathbb{N}} \rightarrow A \) be a continuous surjection. Consider \[ B = \left\{ {\left( {x,\alpha }\right) \in X \times {\mathbb{N}}^{\mathbb{N}} : {\pi }_{X}\left( {f\left( \alpha \right) }\right) = x}\right\} . \] Then \( B \) is a closed set with \( {\pi }_{X}\left( B\right) = {\pi }_{X}\left( A\right) \) . For \( x \in {\pi }_{X}\left( A\right) \), define \( g\left( x\right) \) to be the lexicographic minimum of \( {B}_{x} \) ; i.e., \[ g\left( x\right) = \alpha \Leftrightarrow \left( {x,\alpha }\right) \in B \] \[ \& \forall \beta \{ \left( {x,\beta }\right) \in B \Rightarrow \] \[ \exists n\left\lbrack {\alpha \left( n\right) < \beta \left( n\right) \text{ and }\forall m < n\left( {\alpha \left( m\right) = \beta \left( m\right) }\right) }\right\rbrack \} . \] By induction on \( \left| s\right| \), we prove that \( {g}^{-1}\left( {\sum \left( s\right) }\right) \in \mathcal{A} \) for every \( s \in {\mathbb{N}}^{ < \mathbb{N}} \) . Since \( \left\{ {\sum \left( s\right) : s \in {\mathbb{N}}^{ < \mathbb{N}}}\right\} \) is a base for \( {\mathbb{N}}^{\mathbb{N}} \), it follows that \( g \) is \( \mathcal{A} \) -measurable. Suppose \( {g}^{-1}\left( {\sum \left( t\right) }\right) \in \mathcal{A} \) and \( s = t \hat{} k, k \in \mathbb{N} \) . Then for any \( x \) , \[ x \in {g}^{-1}\left( {\sum \left( s\right) }\right) \Leftrightarrow x \in {g}^{-1}\left( {\sum \left( t\right) }\right) \] \[ \& \exists \alpha \left( {\left( {x,\alpha }\right) \in B\& s \prec \alpha }\right) \] \[ \text{&}\forall l < k\neg \exists \beta \left( {\left( {x,\beta }\right) \in B\& t \hat{} l \prec \beta }\right) \text{.} \] Hence, \( {g}^{-1}\left( {\sum \left( s\right) }\right) \in \mathcal{A} \) . Now, define \( u\left( x\right) = {\pi }_{Y}\left( {f\left( {g\left( x\right) }\right) }\right), x \in {\pi }_{X}\left( A\right) \) . Then \( u \) is an \( \mathcal{A} \) -measurable section of \( A \) . Theorem 5.5.3 Every analytic subset \( A \) of the product of Polish spaces \( X, Y \) admits a section \( u \) that is universally measurable as well as Baire measurable. Proof. The result follows from 5.5.2, 4.3.1, and 4.3.2. Proposition 5.5.4 In 5.5.3, further assume that \( A \) is Borel. Then the graph of the section \( u \) is coanalytic. Proof. Note that \[ \begin{matrix} u\left( x\right) = y & \Leftrightarrow & \left( {x, y}\right) \in A\;\& \;\left( {\forall \alpha \in {\mathbb{N}}^{\mathbb{N}}}\right) \left( {\forall \beta \in {\mathbb{N}}^{\mathbb{N}}}\right) (\lbrack \left( {x,\alpha }\right) \in B \end{matrix} \] \[ \left. {\& \left( {x,\beta }\right) \in B\& f\left( \alpha \right) = \left( {x, y}\right) \rbrack \; \Rightarrow \alpha { \leq }_{\text{lex }}\beta }\right) , \] where \( { \leq }_{\text{lex }} \) is the lexicographic ordering on \( B \) . In a significant contribution to the theory, M. Kondô showed that every coanalytic set can be uniformized by a coanalytic graph [56]. We present this remarkable result in the last section of this chapter. Example 5.5.5 Let \( A \subseteq X \times Y \) be a Borel set whose projection is \( X \) and that cannot be uniformized by a Borel graph. By 5.5.4, there is a coanalytic uniformization \( C \) of \( A \) . By 5.5.1, \( C \) is not analytic. Now, the one-to-one continuous map \( f = {\pi }_{X} \mid C \) is not a Borel isomorphism. Thus a one-to-one Borel map defined on a coanalytic set need not be a Borel isomorphism, although those with domain analytic are (4.5.1). Further, let \( \mathcal{B} = \left\{ {{f}^{-1}\left( B\right) : B \in {\mathcal{B}}_{X}}\right\} \) . Then \( \mathcal{B} \) is a

Theorem 5.4.4 The classification space \( \operatorname{irr}\left( n\right) / \sim \) is standard Borel.

Fix any irreducible \( A \) . Then the \( \sim \) -equivalence class \( \left\lbrack A\right\rbrack \) containing \( A \) equals \[ {\pi }_{1}\left\{ {\left( {B, U}\right) \in \operatorname{irr}\left( n\right) \times U\left( n\right) : A = {UB}{U}^{ * }}\right\} , \] where \( {\pi }_{1} : \operatorname{irr}\left( n\right) \times U\left( n\right) \rightarrow \operatorname{irr}\left( n\right) \) is the projection map to the first coordinate space. (Recall that \( U\left( n\right) \) denotes the set of all \( n \times n \) unitary matrices.) As the set \[ \left\{ {\left( {B, U}\right) \in \operatorname{irr}\left( n\right) \times U\left( n\right) : A = {UB}{U}^{ * }}\right\} \] is closed and \( U\left( n\right) \) compact, \( \left\lbrack A\right\rbrack \) is closed by 2.3.24. Now let \( \mathcal{O} \) be any open set in \( \operatorname{irr}\left( n\right) \) . Its saturation is \[ \mathop{\bigcup }\limits_{{U \in U\left( n\right) }}\left\{ {A \in \operatorname{irr}\left( n\right) : {UA}{U}^{ * } \in \mathcal{O}}\right\} \] which is open. Thus \( \sim \) is a lower-semicontinuous partition of \( \operatorname{irr}\left( n\right) \) into closed sets. By 5.4.3, let \( C \) be a Borel cross section of \( \sim \) . Then \( q \mid C \) is a one-to-one Borel map from \( C \) onto \( \operatorname{irr}\left( n\right) / \sim \), where \( q : \operatorname{irr}\left( n\right) \rightarrow \operatorname{irr}\left( n\right) / \sim \) is the canonical quotient map. By the Borel isomorphism theorem (3.3.13), \( q \) is a Borel isomorphism, and our result is proved.

Lemma 9.3. Let \( {z}_{0} = {y}_{0}\mathrm{i} \) and \( {z}_{1} = {y}_{1}\mathrm{i} \) with \( 0 < {y}_{0} < {y}_{1} \) . Then \[ \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) = \log {y}_{1} - \log {y}_{0} \] and \[ \phi \left( t\right) = {y}_{0}{\left( \frac{{y}_{1}}{{y}_{0}}\right) }^{t}\mathrm{i} \] for \( t \in \left\lbrack {0,1}\right\rbrack \) defines a path in \( \mathbb{H} \) from \( {z}_{0} \) to \( {z}_{1} \) with constant speed \[ \log {y}_{1} - \log {y}_{0} \] so that \[ \mathrm{L}\left( \phi \right) = \log {y}_{1} - \log {y}_{0} \] Moreover, the curve \( \phi \) is uniquely determined: if \( \psi : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{H} \) is any path from \( {z}_{0} \) to \( {z}_{1} \) with \( \mathrm{L}\left( \psi \right) = \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) \) then there is some increasing piecewise differentiable map \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) with \( \psi = \phi \circ f \) . Lemma 9.3 says that any two points on the vertical line \( \{ y\mathrm{i} \mid y > 0\} \) have a unique path of minimal length joining them, and that minimizing path also lies in \( \{ y\mathrm{i} \mid y > 0\} \) . For that reason, the whole set \( \{ y\mathrm{i} \mid y > 0\} \) will be called a geodesic curve for \( \mathbb{H} \) or simply a geodesic, and the minimizing path \( \phi \) will be called a geodesic path. Proof of Lemma 9.3. It is readily checked that the path \( \phi \) defined in the lemma has constant speed equal to \( \log {y}_{1} - \log {y}_{0} = \mathrm{L}\left( \phi \right) \) as claimed. It follows that \[ \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) \leq \log {y}_{1} - \log {y}_{0} \] Suppose now that \( \eta : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{H} \) is another path joining \( {z}_{0} \) to \( {z}_{1} \), and write \( \eta \left( t\right) = {\eta }_{x}\left( t\right) + \mathrm{i}{\eta }_{y}\left( t\right) \) with \( {\eta }_{x}\left( t\right) ,{\eta }_{y}\left( t\right) \in \mathbb{R} \) . Then \[ \mathrm{L}\left( \eta \right) = {\int }_{0}^{1}\frac{{\begin{Vmatrix}{\eta }^{\prime }\left( t\right) \end{Vmatrix}}_{2}}{{\eta }_{y}\left( t\right) }\mathrm{d}t \geq {\int }_{0}^{1}\frac{\left| {\eta }_{y}^{\prime }\left( t\right) \right| }{{\eta }_{y}\left( t\right) }\mathrm{d}t \geq {\int }_{0}^{1}\frac{{\eta }_{y}^{\prime }\left( t\right) }{{\eta }_{y}\left( t\right) }\mathrm{d}t = \log {y}_{1} - \log {y}_{0}. \] Equality holds in the first inequality if and only if \( {\eta }_{x}^{\prime }\left( t\right) = {\eta }_{x}\left( t\right) = 0 \) for all \( t \in \left\lbrack {0,1}\right\rbrack \), and in the second if and only if \( {\eta }_{y}^{\prime }\left( t\right) \geq 0 \) for all \( t \in \left\lbrack {0,1}\right\rbrack \) . This implies the remaining statement of the lemma. From now on we will always parameterize geodesics so that they have constant speed equal to 1 (and therefore have a domain whose length is equal to the length of the path). Thus Lemma 9.3 may be thought of as saying that for \( {z}_{0} = \mathrm{i}{y}_{0},{z}_{1} = \mathrm{i}{y}_{1} \) there is a unique path \[ \phi : \left\lbrack {0,\mathrm{\;d}\left( {{z}_{0},{z}_{1}}\right) }\right\rbrack \rightarrow \mathbb{H} \] with unit speed and with \( \phi \left( 0\right) = {z}_{0} \) and \( \phi \left( {\mathrm{d}\left( {{z}_{0},{z}_{1}}\right) }\right) = {z}_{1} \), and that unique path is defined by \[ \phi \left( t\right) = {y}_{0}{\mathrm{e}}^{t}\mathrm{i}. \] It is clear that an isometry \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) sends a geodesic path (or curve) to another geodesic path (curve). The next result is a converse to this observation, and gives a description of all geodesics in \( \mathbb{H} \) . Proposition 9.4. For any two points \( {z}_{0},{z}_{1} \in \mathbb{H} \) there is a unique path \[ \phi : \left\lbrack {0,\mathrm{\;d}\left( {{z}_{0},{z}_{1}}\right) }\right\rbrack \rightarrow \mathbb{H} \] of unit speed with \( \phi \left( 0\right) = {z}_{0} \) and \( \phi \left( {\mathrm{d}\left( {{z}_{0},{z}_{1}}\right) }\right) = {z}_{1} \) . Moreover, there is a unique isometry \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) such that \( \phi \left( t\right) = g\left( {{\mathrm{e}}^{t}\mathrm{i}}\right) \) . Proof. We first claim that there exists a \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) with \( {g}^{-1}\left( {z}_{0}\right) = \mathrm{i} \) and \( {g}^{-1}\left( {z}_{1}\right) = \mathrm{i}{y}_{1} \) for some \( {y}_{1} > 1 \) . By Lemma 9.1(2) we can certainly find some \( \widetilde{g} \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) with \( {\widetilde{g}}^{-1}\left( {z}_{0}\right) = \mathrm{i} \), and we want to modify \( \widetilde{g} \) to also satisfy the second condition. By Lemma 9.1(3) any element of \( \operatorname{PSO}\left( 2\right) \) fixes the point i. Hence we may suppose without loss of generality that \( \Im \left( {{g}^{-1}\left( {z}_{1}\right) }\right) \) is maximal within \( \Im \left( {\operatorname{PSO}\left( 2\right) {\widetilde{g}}^{-1}\left( {z}_{1}\right) }\right) \) . Let \[ h = \left( \begin{matrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{matrix}\right) \] and \( {\widetilde{g}}^{-1}\left( {z}_{1}\right) = {\widetilde{x}}_{1} + \mathrm{i}{\widetilde{y}}_{1} \), so that \[ \Im \left( {h\left( {{\widetilde{g}}^{-1}\left( {z}_{1}\right) }\right) }\right) = \frac{{y}_{1}}{{\left| \sin \theta {z}_{1} + \cos \theta \right| }^{2}}\;\text{ (by (9.2)) } \] has a maximum at \( \theta = 0 \) . Choosing \( {g}^{-1} = h{\widetilde{g}}^{-1} \) and taking the derivative shows that this implies that \( \Re {g}^{-1}\left( {z}_{1}\right) = 0 \) . Moreover, we must have \( {g}^{-1}\left( {z}_{1}\right) = {y}_{1} > 1 \) since if \( {y}_{1} < 1 \) then the map \[ k = \left( \begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}\right) \] would increase the imaginary part, contradicting the maximality assumption. By Lemma 9.3, there is a unique geodesic path \( {\phi }_{0}\left( t\right) = {\mathrm{e}}^{t}\mathrm{i} \) of unit speed from i to \( \mathrm{i}{y}_{1} \), so that \( \phi \left( t\right) = g\left( {{\mathrm{e}}^{t}\mathrm{i}}\right) \) is the unique geodesic path of unit speed connecting \( {z}_{0} \) and \( {z}_{1} \) . Finally, we claim that not only is the geodesic unique, but so is the element \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) . To see this, suppose that \( t \mapsto {g}_{1}\left( {{\mathrm{e}}^{t}\mathrm{i}}\right) \) is any geodesic path from \( {z}_{0} \) to \( {z}_{1} \) . Then \( t \mapsto {g}^{-1}{g}_{1}\left( {{\mathrm{e}}^{t}\mathrm{i}}\right) \) is a geodesic path from \( \mathrm{i} \) to \( {y}_{1}\mathrm{i} \) of unit speed, and so must be equal to the path \( t \mapsto {\mathrm{e}}^{t}\mathrm{i} \) . Taking the derivative at \( t = 0 \) we see that \( \mathrm{D}\left( {{g}^{-1}{g}_{1}}\right) \left( {\mathrm{i},\mathrm{i}}\right) = \left( {\mathrm{i},\mathrm{i}}\right) \), which shows that \( {g}^{-1}{g}_{1} = {I}_{2} \) by Lemma 9.2. Finally, we claim that a Möbius transformation \( g \) maps the geodesic curve \( \{ y\mathrm{i} \mid y > 0\} \) either to another vertical line (that is, a line normal to the real axis, with constant \( x \) coordinate), or to the upper half of a circle \[ {\left( x - f\right) }^{2} + {y}^{2} = {r}^{2} \] with center in the real axis (equivalently, meeting the real axis at right angles). Moreover, all of those curves do arise as images of \( \{ y\mathrm{i} \mid y > 0\} \), so that this list of curves is precisely the list of geodesic curves in \( \mathbb{H} \) . To see this, note first that a Möbius transformation of the form \( z \mapsto {az} + b \) maps the vertical line \( \{ y\mathrm{i} \mid y > 0\} \) to the vertical line \( \Re \left( z\right) = b \) and maps the upper half of the circle \( {\left( x - f\right) }^{2} + {y}^{2} = {r}^{2} \) to the upper half of the circle \( {\left( x - \left( af + b\right) \right) }^{2} + {y}^{2} = {a}^{2}{r}^{2} \) . Now the subgroup \[ U = \left\{ {\left. {\left( \begin{array}{ll} 1 & b \\ 0 & 1 \end{array}\right) \;}\right| \;b \in \mathbb{R}}\right\} < {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \] together with \( w = \left( \begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}\right) \) generate \( {\mathrm{{SL}}}_{2}\left( \mathbb{R}\right) \), since \[ {wU}{w}^{-1} = \left\{ {\left. \left( \begin{matrix} 1 & 0 \\ - b & 1 \end{matrix}\right) \right| \;b \in \mathbb{R}}\right\} . \] So it remains to check that the Möbius transformation \( z \mapsto - \frac{1}{z} \) corresponding to \( w \) maps a vertical line or a semicircle with center in \( \mathbb{R} \) to another vertical line or semicircle. In polar coordinates \( \left( {r,\phi }\right) \) the transformation \( z \mapsto - \frac{1}{z} \) is the transformation \( \left( {r,\phi }\right) \mapsto \left( {\frac{1}{r},\pi - \phi }\right) \) . The claim follows since both vertical lines and circles with real centers can be defined by equations of the form \[ \alpha {r}^{2} + {\beta r}\cos \phi + \gamma = 0 \] with \( \left( {\alpha ,\beta ,\gamma }\right) \neq \left( {0,0,0}\right) \) . Historically, the hyperbolic plane \( \mathbb{H} \) was important in solving a classical problem in geometry: the points and geodesics in \( \mathbb{H} \) satisfy all the classical axioms of geometry apart from the parallel axiom, thus showing that the parallel axiom is not a consequence of the other axioms. Indeed, we have for instance that for any two different points in \( \mathbb{H} \) there is a unique geodesic through them, and any two different geodesics intersect in at most one point. Angles and areas will be defined later, and they will behave well (if a little unusually). However, for every point \( z \in \mathbb{H} \) and geodesic \( \ell \) not containing \( z \) there are uncountably many geodesics through \( z \) that do not intersect \( \ell

Lemma 9.3. Let \( {z}_{0} = {y}_{0}\mathrm{i} \) and \( {z}_{1} = {y}_{1}\mathrm{i} \) with \( 0 < {y}_{0} < {y}_{1} \). Then \[ \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) = \log {y}_{1} - \log {y}_{0} \] and \[ \phi \left( t\right) = {y}_{0}{\left( \frac{{y}_{1}}{{y}_{0}}\right) }^{t}\mathrm{i} \] for \( t \in \left\lbrack {0,1}\right\rbrack \) defines a path in \( \mathbb{H} \) from \( {z}_{0} \) to \( {z}_{1} \) with constant speed \[ \log {y}_{1} - \log {y}_{0} \] so that \[ \mathrm{L}\left( \phi \right) = \log {y}_{1} - \log {y}_{0} \] Moreover, the curve \( \phi \) is uniquely determined: if \( \psi : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{H} \) is any path from \( {z}_{0} \) to \( {z}_{1} \) with \( \mathrm{L}\left( \psi \right) = \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) \) then there is some increasing piecewise differentiable map \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) with \( \psi = \phi \circ f \) .

Proof of Lemma 9.3. It is readily checked that the path \( \phi \) defined in the lemma has constant speed equal to \( \log {y}_{1} - \log {y}_{0} = \mathrm{L}\left( \phi \right) \) as claimed. It follows that \[ \mathrm{d}\left( {{z}_{0},{z}_{1}}\right) \leq \log {y}_{1} - \log {y}_{0} \] Suppose now that \( \eta : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{H} \) is another path joining \( {z}_{0} \) to \( {z}_{1} \), and write \( \eta \left( t\right) = {\eta }_{x}\left( t\right) + i{\eta }_{y}\left( t\right) \) with \( {\eta }_{x}\left( t\right) ,{\eta }_{y}\left( t\right) in mathbb{R}. Then \] \[ mathrm{L}(eta)=int_{0}^{1}{frac{{begin{Vmatrix}{eta}^{prime}(t){end{Vmatrix}}_{2}}{{eta }_{y}(t)}dtgeqint_{0}^{1}{frac{{begin|{{eta }_{y}^{prime}(t){end|}}{{eta }_{y}(t)}dtgeqint_{0}^{1}{frac{{eta }_{y}^{prime}(t)}{{eta }_{y}(t)}dt=log y_{1}-log y_{0}. \] Equality holds in the first inequality if and only if ${{eta }_{x}^{prime}(t)={{eta }_{x}(t)=$ for all $tin[; ]{; ]$, and in the second if and only if ${{eta }_{y}^{prime}(t)geq$ for all $tin[; ]{; ]$. This implies the remaining statement of the lemma.

Theorem 4.136. Suppose \( X \) is \( F \) -smooth. If \( f \) is locally Lipschitzian and \( q \) is a Lyapunov function for \( S \), then \( S \) is stable. If \( \left( {p, q, c}\right) \) is a Lyapunov triple for \( S \), then \( S \) is attractive, provided that in the case \( c = 0, f \) is bounded on bounded subsets. Proof. In order to show the stability of \( S \) in both cases, it suffices to prove that for all \( z \in X \), the function \( t \mapsto {e}^{ct}q\left( {{x}_{z}\left( t\right) }\right) + {\int }_{0}^{t}{e}^{cs}p\left( {{x}_{z}\left( s\right) }\right) \mathrm{d}s \) is nonincreasing on \( {\mathbb{R}}_{ + } \) . Since \( {x}_{z} \) and \( p \) are continuous, the last term is differentiable. To study the first one, let us use a special case of the Leibniz rule for \( {\partial }_{F} \) : if \( g, h : W \rightarrow \mathbb{R} \) are two lower semicontinuous functions on an open subset \( W \) of a normed space, \( h \) being positive and differentiable, then \[ \forall x \in W,\;\partial \left( {gh}\right) \left( x\right) = h\left( x\right) \partial g\left( x\right) + g\left( x\right) {h}^{\prime }\left( x\right) . \] Taking \( W \mathrel{\text{:=}} \left( {0, + \infty }\right), g\left( t\right) \mathrel{\text{:=}} q\left( {{x}_{z}\left( t\right) }\right), h\left( t\right) \mathrel{\text{:=}} {e}^{ct} \) and using Corollary 4.98, we are led to check that (L1) implies that for all \( t \in {\mathbb{R}}_{ + },{t}^{ * } \in \partial g\left( t\right) \) one has \[ {e}^{ct}{t}^{ * } + c{e}^{ct}q\left( {{x}_{z}\left( t\right) }\right) + {e}^{ct}p\left( {{x}_{z}\left( t\right) }\right) \leq 0, \] (4.78) \( \mathbb{R} \) being identified with its dual. Since \( {t}^{ * } \in {\partial }_{F}\left( {q \circ {x}_{z}}\right) \left( t\right) \) and since \( q \) is inf-compact on the image under \( {x}_{z} \) of a compact interval, Theorem 4.70 yields sequences \( \left( {t}_{n}\right) \rightarrow t,\left( {t}_{n}^{ * }\right) \rightarrow {t}^{ * },\left( {y}_{n}\right) { \rightarrow }_{q}{x}_{z}\left( t\right) ,\left( {y}_{n}^{ * }\right) ,\left( {v}_{n}^{ * }\right) \) such that \( \left( \begin{Vmatrix}{{y}_{n}^{ * } - {v}_{n}^{ * }}\end{Vmatrix}\right) \rightarrow 0 \) , \( \left( {\begin{Vmatrix}{v}_{n}^{ * }\end{Vmatrix} \cdot \begin{Vmatrix}{{x}_{z}\left( {t}_{n}\right) - {y}_{n}}\end{Vmatrix}}\right) \rightarrow 0 \), with \( {y}_{n}^{ * } \in \partial q\left( {y}_{n}\right) ,{t}_{n}^{ * } \in {D}_{F}^{ * }{x}_{z}\left( {t}_{n}\right) \left( {v}_{n}^{ * }\right) \) for all \( n \in \mathbb{N} \) . The last relation means that \( {t}_{n}^{ * } = \left\langle {{v}_{n}^{ * },{x}_{z}^{\prime }\left( {t}_{n}\right) }\right\rangle = \left\langle {{v}_{n}^{ * }, f\left( {{x}_{z}\left( {t}_{n}\right) }\right) }\right\rangle \) . Since \( f \) is locally Lipschitzian and \( \left( {{x}_{z}\left( {t}_{n}\right) }\right) \rightarrow {x}_{z}\left( t\right) ,\left( {\begin{Vmatrix}{v}_{n}^{ * }\end{Vmatrix} \cdot \begin{Vmatrix}{{x}_{z}\left( {t}_{n}\right) - {y}_{n}}\end{Vmatrix}}\right) \rightarrow 0 \), from (L1) we get \[ {t}^{ * } = \mathop{\lim }\limits_{n}\left\langle {{v}_{n}^{ * }, f\left( {y}_{n}\right) }\right\rangle = \mathop{\lim }\limits_{n}\left\langle {{y}_{n}^{ * }, f\left( {y}_{n}\right) }\right\rangle \leq \mathop{\limsup }\limits_{n}\left( {-p\left( {y}_{n}\right) - {cq}\left( {y}_{n}\right) }\right) . \] Since \( p \) and \( q \) are lower semicontinuous, we obtain \( {t}^{ * } \leq - p\left( {{x}_{z}\left( t\right) }\right) - {cq}\left( {{x}_{z}\left( t\right) }\right) \) and (4.78), and hence the stability of \( S \) . Let us turn to the second assertion. When \( c > 0 \), condition (4.77) and the fact that \( t \mapsto {e}^{ct}q\left( {{x}_{z}\left( t\right) }\right) \) is bounded ensure that \( d\left( {{x}_{z}\left( t\right), S}\right) \rightarrow 0 \) as \( t \rightarrow + \infty \) . When \( c = 0 \) , since \( q \circ {x}_{z} \) is bounded on \( {\mathbb{R}}_{ + },{x}_{z}\left( {\mathbb{R}}_{ + }\right) \) is bounded. By (L2) and our assumption on \( f \) , \( p \circ {x}_{z} \) is Lipschitzian. Since \( {\int }_{0}^{\infty }p\left( {{x}_{z}\left( s\right) }\right) \mathrm{d}s \) is finite, it is easy to see that \( p\left( {{x}_{z}\left( s\right) }\right) \rightarrow 0 \) when \( s \rightarrow \infty \) (exercise). Then (L2) ensures that \( d\left( {{x}_{z}\left( s\right), S}\right) \rightarrow 0 \) as \( s \rightarrow \infty \) . ## Exercises 1. Prove the fact used in the proof of Theorem 4.136 that if \( r : {\mathbb{R}}_{ + } \rightarrow {\mathbb{R}}_{ + } \) is Lipschitzian and integrable on \( {\mathbb{R}}_{ + } \), then \( r\left( t\right) \rightarrow 0 \) as \( t \rightarrow + \infty \) . 2. Given a function \( g : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}_{ + } \) of class \( {C}^{1} \), let \( f : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) be defined by \[ f\left( {x, y}\right) \mathrel{\text{:=}} \left( {-y - {xg}\left( {x, y}\right), x - {yg}\left( {x, y}\right) }\right) . \] (a) Check that \( q \) given by \( q\left( {x, y}\right) \mathrel{\text{:=}} {x}^{2} + {y}^{2} \) is a Lyapunov function for \( S \mathrel{\text{:=}} \{ \left( {0,0}\right) \} \) . (b) Check that \( S \) is attractive if for some \( c > 0 \) one has \( g\left( {x, y}\right) \geq c \) for all \( \left( {x, y}\right) \in {\mathbb{R}}^{2} \) . 3. (a) Let \( X \) be a Hilbert space and let \( A : X \rightarrow X \) be a symmetric continuous linear map that is positive semidefinite, i.e., such that \( q\left( x\right) \mathrel{\text{:=}} \left( {{Ax} \mid x}\right) \geq 0 \) for all \( x \in X \) . Show that \( S \mathrel{\text{:=}} \{ 0\} \) is stable for the vector field \( f \) given by \( f\left( x\right) \mathrel{\text{:=}} - {Ax} \) . Show that \( S \) is attractive if \( A \) is positive definite, i.e., if for some \( c > 0 \) one has \( q\left( \cdot \right) \geq c\parallel \cdot {\parallel }^{2} \) . (b) Let \( f \) be a vector field of class \( {C}^{1} \) on a Hilbert space \( X \) such that \( A \mathrel{\text{:=}} - {f}^{\prime }\left( 0\right) \) is positive definite. Show that \( S \mathrel{\text{:=}} \{ 0\} \) is attractive. [Hint: Use \( q \) as in (a).] 4. Use a Lyapunov function to show that \( S \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} : {x}^{2} + {y}^{2} = 1}\right\} \) is a stable set for the vector field \( f \) given by \( f\left( {x, y}\right) \mathrel{\text{:=}} \left( {y, - x}\right) \) . [Hint: Take \( q\left( {x, y}\right) \mathrel{\text{:=}} \) \( \left| {{x}^{2} + {y}^{2} - 1}\right| \) .] 5. Consider the differential equation \( {x}^{\prime }\left( t\right) = - \left( {1/4}\right) x\left( t\right) \left( {x\left( t\right) + 4}\right) \left( {x\left( t\right) - 2}\right) \), which occurs in population models. Let \( q : \mathbb{R} \rightarrow {\overline{\mathbb{R}}}_{ + } \) be given by \( q\left( x\right) \mathrel{\text{:=}} {\left( x + 4\right) }^{2} \) for \( x \in \) \( \left( {-\infty , - 1\rbrack, q\left( x\right) \mathrel{\text{:=}} + \infty \text{for}x \in \left( {-1,1}\right), q\left( x\right) = {\left( x - 2\right) }^{2}\text{for}x \in \lbrack 1, + \infty }\right) \) . (a) Show that \( S \mathrel{\text{:=}} \{ - 4,2\} \) and \( q \) are such that \( q\left( x\right) \rightarrow 0 \) if and only if \( d\left( {x, S}\right) \rightarrow 0 \) . (b) Check that \( q \) is a Lyapunov function for \( S \) . Conclude that \( S \) is attractive. (See [996].) 6. Show that the function \( q : {\mathbb{R}}^{2} \rightarrow {\overline{\mathbb{R}}}_{ + } \) given by \( q\left( {x, y}\right) \mathrel{\text{:=}} \left| {{x}^{2} + {y}^{2} - 1}\right| \) when \( {x}^{2} + \) \( {y}^{2} \geq 1/2, + \infty \) otherwise, is a Lyapunov function for the unit circle \( S \) in \( {\mathbb{R}}^{2} \) when one considers the vector field \( f \) given by \[ f\left( {x, y}\right) \mathrel{\text{:=}} \left( {-y + x\left( {1 - {x}^{2} - {y}^{2}}\right), x + y\left( {1 - {x}^{2} - {y}^{2}}\right) }\right) . \] 7. Let \( C \) be the Cantor set consisting of the set of \( x \in \left\lbrack {0,1}\right\rbrack \) whose ternary expansion \( x = \mathop{\sum }\limits_{{n \geq 1}}{3}^{-n}{x}_{n} \) with \( {x}_{n} \in \{ 0,1,2\} \) is such that \( {x}_{n} \neq 1 \) for all \( n \) . Since \( C \) is closed, \( \left\lbrack {0,1}\right\rbrack \smallsetminus C \) is the union of a countable family of open intervals \( \left( {{a}_{k},{b}_{k}}\right) \) . Define \( q : C \rightarrow \) \( \mathbb{R} \) by \( q\left( x\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{n \geq 1}}{2}^{-n - 1}{x}_{n} \) for \( x \mathrel{\text{:=}} \mathop{\sum }\limits_{{n \geq 1}}{3}^{-n}{x}_{n} \in C \) . (a) Show that \( q\left( {a}_{k}\right) = q\left( {b}_{k}\right) \) for all \( k \) . [Hint: Observe that \( {a}_{k} = 0.{x}_{1}\ldots {x}_{k}{0222}\ldots \) , while \( {b}_{k} = 0.{x}_{1}\ldots {x}_{k}{2000}\ldots \), so that \( q \) can be extended by continuity by setting \( q\left( x\right) = q\left( {a}_{k}\right) = q\left( {b}_{k}\right) \) for \( x \in \left( {{a}_{k},{b}_{k}}\right) \) . Extend further \( q \) to \( \mathbb{R} \) by requiring it to be even and by setting \( q\left( x\right) = 1 \) for \( x > 1 \) .] (b) Show that \( q\left( x\right) \rightarrow 0 \Leftrightarrow d\left( {x, S}\right) \rightarrow 0 \) for \( S \mathrel{\text{:=}} \{ 0\} \) and that the set of differentiability points of \( q \) is \( \mathbb{R} \smallsetminus \left( {C \cup \left( {-C}\right) }\right) \) and for such points \( x \) one has \( {q}^{\prime }\left( x\right) = 0 \) . Conclude that the condition \( {q}^{\prime }\left( x\right) \cdot f\left( x\right) \leq 0 \) at all points of differentiability of \( q \) does not suffice to ensure that \( S \) is stable (otherwise, for every vector field, \( S \) would be stable). (See [136,137].) ## 4.8 Notes and Remarks The definition of a derivative with the help of a convergence on the space of functions from \( X

Theorem 4.136. Suppose \( X \) is \( F \) -smooth. If \( f \) is locally Lipschitzian and \( q \) is a Lyapunov function for \( S \), then \( S \) is stable. If \( \left( {p, q, c}\right) \) is a Lyapunov triple for \( S \), then \( S \) is attractive, provided that in the case \( c = 0, f \) is bounded on bounded subsets.

In order to show the stability of \( S \) in both cases, it suffices to prove that for all \( z \in X \), the function \( t \mapsto {e}^{ct}q\left( {{x}_{z}\left( t\right) }\right) + {\int }_{0}^{t}{e}^{cs}p\left( {{x}_{z}\left( s\right) }\right) \mathrm{d}s \) is nonincreasing on \( {\mathbb{R}}_{ + } \) . Since \( {x}_{z} \) and \( p \) are continuous, the last term is differentiable. To study the first one, let us use a special case of the Leibniz rule for \( {\partial }_{F} \) : if \( g, h : W \rightarrow \mathbb{R} \) are two lower semicontinuous functions on an open subset \( W \) of a normed space, \( h \) being positive and differentiable, then \[ \forall x \in W,\;\partial \left( {gh}\right) \left( x\right) = h\left( x\right) \partial g\left( x\right) + g\left( x\right) {h}^{\prime }\left( x\right) . \] Taking \( W \mathrel{\text{:=}} \left( {0, + \infty }\right), g\left( t\right) \mathrel{\text{:=}} q\left( {{x}_{z}\left( t\right) }\right), h\left( t\right) \mathrel{\text{:=}} {e}^{ct} \) and using Corollary 4.98, we are led to check that (L1) implies that for all \( t \in {\mathbb{R}}_{ + },{t}^{ * } \in \partial g\left( t\right) \) one has \[ {e}^{ct}{t}^{ * } + c{e}^{ct}q\left( {{x}_{z}\left( t\right) }\right) + {e}^{ct}p\left( {{x}_{z}\left( t\right) }\right) \leq 0, \] \( \mathbb{R} \) being identified with its dual. Since \( {t}^{ * } \in {\partial }_{F}\left( {q \circ {x}_{z}}\right) \left( t\right) \) and since \( q \) is inf-compact on the image under \( {x}_{z} \) of a compact interval, Theorem 4.70 yields sequences \( \left( {t}_{n}\right) \rightarrow t,\left( {t}_{n}^{ * }\right) \rightarrow {t}^{ * },\left( {y}_{n}\right) { \rightarrow }_{q}{x}_{z}\left( t\right) ,\left( {y}_{n}^{ * }\right) ,\left( {v}_{n}^{ * }\right) \) such that \( \left( \begin{Vmatrix}{{y}_{n}^{ * } - {v}_{n}^{ * }}\end{Vmatrix}\

Proposition 7.23. Let \[ \begin{matrix} \mathrm{X} = \left( {X,\mathcal{B},\mu, T}\right) \\ \downarrow \\ \mathrm{Y} = \left( {Y,\mathcal{A},\nu, S}\right) \end{matrix} \] be a compact extension of invertible measure-preserving systems on Borel probability spaces. If \( \mathrm{Y} \) is \( {SZ} \), then so is \( \mathrm{X} \) . As seen in Example 7.19, it is not true that any function \( f \in {L}^{\infty }\left( X\right) \) in the compact extension \( \mathrm{X} \) of \( \mathrm{Y} \) is automatically \( \mathrm{{AP}} \) relative to \( \mathrm{Y} \) . The next lemma will be used to circumvent this problem in the case of characteristic functions. Lemma 7.24. In the notation of Proposition 7.23, let \( B \in \mathcal{B} \) have \( \mu \left( B\right) > 0 \) . Then there exists a set \( \widetilde{B} \subseteq B \) with \( \mu \left( \widetilde{B}\right) > 0 \) such that - \( {\chi }_{\widetilde{B}} \) is AP relative to \( \mathrm{Y} \), and - \( {\mu }_{y}^{\mathcal{A}}\left( \widetilde{B}\right) > \frac{1}{2}\mu \left( \widetilde{B}\right) \) or \( {\mu }_{y}^{\mathcal{A}}\left( \widetilde{B}\right) = 0 \) for all \( y \in Y \) . Proof. Both properties of the lemma will be achieved by removing an element \( A \in \mathcal{A} \) from \( B \) (in other words, by removing a collection of entire \( \mathcal{A} \) - atoms). We will do this in two stages, first defining \( {B}^{\prime } \subseteq B \) which satisfies the second property, and then defining \( \widetilde{B} \subseteq {B}^{\prime } \) . Notice that if we indeed have \( \widetilde{B} = {B}^{\prime } \smallsetminus A \) for some \( A \in \mathcal{A} \), and \( {B}^{\prime } \) satisfies the second property, then so does \( \widetilde{B} \) . Let \[ {B}^{\prime } = \left\{ {x \in B \mid {\mu }_{y}^{\mathcal{A}}\left( B\right) > \frac{1}{2}\mu \left( B\right) }\right\} . \] Hence, if \( y \notin {B}^{\prime } \) then we have some \( x \in B \smallsetminus {B}^{\prime } \) that is mapped to \( y \), and \[ {\mu }_{y}^{\mathcal{A}}\left( {B \smallsetminus {B}^{\prime }}\right) = {\mu }_{y}^{\mathcal{A}}\left( B\right) \leq \frac{1}{2}\mu \left( B\right) , \] and integration over \( y \in Y \) yields \( \mu \left( {B \smallsetminus {B}^{\prime }}\right) \leq \frac{1}{2}\mu \left( B\right) \) . Moreover, since \( \mu \left( B\right) > 0 \) we have \( \mu \left( {B}^{\prime }\right) > 0 \) and \[ {\mu }_{x}^{\mathcal{A}}\left( {B}^{\prime }\right) > \frac{1}{2}\mu \left( B\right) \geq \frac{1}{2}\mu \left( {B}^{\prime }\right) \] for every \( x \in {B}^{\prime } \), so the set \( {B}^{\prime } \) satisfies the second property of the lemma. We now proceed to define \( \widetilde{B} \) . Choose a decreasing sequence \( {\left( {\varepsilon }_{\ell }\right) }_{\ell \geq 1} \) with \( {\varepsilon }_{\ell } > 0 \) for all \( \ell \geq 1 \) and with * Even though the result in this section is the next logical step, the reader may wish to look at the argument for sequences of factors in the next section, which is both independent and easier, and may help the reader to prepare for the current argument. \[ \mathop{\sum }\limits_{{\ell = 1}}^{\infty }{\varepsilon }_{\ell } < \frac{1}{2}\mu \left( {B}^{\prime }\right) \] For every \( \ell \geq 1 \), there is an AP function \( {f}_{\ell } \) such that \[ {\begin{Vmatrix}{\chi }_{{B}^{\prime }} - {f}_{\ell }\end{Vmatrix}}_{{L}_{\mu }^{2}}^{2} = \int {\left| {\chi }_{{B}^{\prime }} - {f}_{\ell }\right| }^{2}\mathrm{\;d}\mu < {\varepsilon }_{\ell }^{2} \] by density. Let \[ {A}_{\ell } = \left\{ {y \in Y \mid {\begin{Vmatrix}{\chi }_{{B}^{\prime }} - {f}_{\ell }\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}}^{2} \geq {\varepsilon }_{\ell }}\right\} \] then \[ \mu \left( {A}_{\ell }\right) \leq \frac{1}{{\varepsilon }_{\ell }}{\int }_{{A}_{\ell }}{\begin{Vmatrix}{\chi }_{{B}^{\prime }} - {f}_{\ell }\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}}^{2}\mathrm{\;d}\nu \left( y\right) \leq {\varepsilon }_{\ell }. \] Let \[ \widetilde{B} = {B}^{\prime } \smallsetminus \mathop{\bigcup }\limits_{{\ell \geq 1}}{A}_{\ell } \] so in particular \( \mu \left( \widetilde{B}\right) \geq \frac{1}{2}\mu \left( {B}^{\prime }\right) \) . As explained earlier, either \[ {\mu }_{y}^{\mathcal{A}}\left( \widetilde{B}\right) > \frac{1}{2}\mu \left( \widetilde{B}\right) \] or \[ {\mu }_{y}^{\mathcal{A}}\left( \widetilde{B}\right) = 0 \] since \( \mathop{\bigcup }\limits_{\ell }{A}_{\ell } \in \mathcal{A} \) . Finally, for the AP property, fix \( \varepsilon > 0 \) and choose some \( \ell \) with \( {\varepsilon }_{\ell } < \frac{1}{2}\varepsilon \) . Then if \( {T}^{n}y \notin \mathop{\bigcup }\limits_{\ell }{A}_{\ell } \) , \[ {\begin{Vmatrix}{U}_{T}^{n}{\chi }_{\widetilde{B}} - {U}_{T}^{n}{f}_{\ell }\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} = {\begin{Vmatrix}{\chi }_{\widetilde{B}} - {f}_{\ell }\end{Vmatrix}}_{{L}_{{\mu }_{T}^{\mathcal{A}}{n}_{y}}^{2}} \] \[ = \parallel {\chi }_{{B}^{\prime }} - {f}_{\ell }{\parallel }_{{L}_{{\mu }_{T}^{\mathcal{A}}{n}_{y}}^{2}} \] \[ < {\varepsilon }_{\ell } < \frac{1}{2}\varepsilon \] On the other hand, if \( {T}^{n}y \in \mathop{\bigcup }\limits_{\ell }{A}_{\ell } \) then \[ {\begin{Vmatrix}{U}_{T}^{n}{\chi }_{\widetilde{B}}\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} = {\begin{Vmatrix}{\chi }_{\widetilde{B}}\end{Vmatrix}}_{{L}_{{\mu }_{T}^{\mathcal{A}}{n}_{y}}^{2}} = 0. \] Since \( {f}_{\ell } \) is AP relative to \( \mathrm{Y} \), there exist functions \( {g}_{1},\ldots ,{g}_{m} \) with \[ \mathop{\min }\limits_{{1 \leq j \leq m}}{\begin{Vmatrix}{U}_{T}^{n}{f}_{\ell } - {g}_{j}\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} < \frac{1}{2}\varepsilon \] almost everywhere; set \( {g}_{0} = 0 \) to deduce that \[ \mathop{\min }\limits_{{0 \leq j \leq m}}{\begin{Vmatrix}{U}_{T}^{n}{\chi }_{\widetilde{B}} - {g}_{j}\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} < \varepsilon \] showing the AP property. ## 7.9.1 SZ for Compact Extensions via van der Waerden The following approach to the SZ property for compact extensions (Proposition 7.23) is taken from the notes of Bergelson [26], and uses van der Waer-den's theorem (Theorem 7.1). Proof of Proposition 7.23 using van der Waerden. By Lemma 7.24, we may assume that \( f = {\chi }_{B} \) is AP and that there exists some set \( A \in \mathcal{A} \) of positive measure with \( {\mu }_{y}^{\mathcal{A}}\left( B\right) > \frac{1}{2}\mu \left( B\right) \) for \( y \in A \) . We will use \( \mathrm{{SZ}} \) for \( A \) (for arithmetic progressions of quite large length \( K \) ) to show SZ for \( B \) (for arithmetic progressions of length \( k \) ). Given \( \varepsilon = \frac{\mu \left( B\right) }{6\left( {k + 1}\right) } > 0 \) we may find (using the AP property of \( f \) ) functions \( {g}_{1},\ldots ,{g}_{r} \) such that \[ \mathop{\min }\limits_{{1 \leq s \leq r}}{\begin{Vmatrix}{U}_{T}^{n}f - {g}_{s}\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} < \varepsilon \] for all \( n \in \mathbb{Z} \) and almost every \( y \in Y \) . We may assume that \( {\begin{Vmatrix}{g}_{s}\end{Vmatrix}}_{\infty } \leq 1 \) for \( s = 1,\ldots, r \) . By Theorem 7.1 we may choose a \( K \) for which any coloring of \( \{ 1,\ldots, K\} \) with \( r \) colors contains a monochrome arithmetic progression of length \( k + 1 \) . There is some \( {c}_{1} > 0 \) for which \[ {R}_{K} = \left\{ {n \in \mathbb{N} \mid \nu \left( {A \cap {S}^{-n}A \cap \cdots {S}^{-{Kn}}A}\right) > {c}_{1}}\right\} \] has positive lower density. In fact this follows from SZ for \( A \) : If \[ \mathop{\liminf }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}\nu \left( {A \cap {S}^{-n}A \cap \cdots \cap {S}^{-{Kn}}A}\right) \geq {c}_{0} > 0 \] then, for large enough \( N \) and some \( {c}_{1} > 0 \) depending only on \( {c}_{0} \), we must have \[ \frac{1}{N}\left| {{R}_{K} \cap \left\lbrack {1, N}\right\rbrack }\right| > {c}_{1} \] Let \( n \in {R}_{K} \) . Then, for every \( y \in A \cap {S}^{-n}A \cap \cdots \cap {S}^{-{Kn}}A \), we have \[ \mathop{\min }\limits_{{1 \leq s \leq r}}{\begin{Vmatrix}{U}_{T}^{in}f - {g}_{s}\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} < \varepsilon \] for \( 1 \leq i \leq K \) . We may use this to choose a coloring \( c\left( i\right) \) on \( \left\lbrack {1, K}\right\rbrack \) with \( r \) colors (that depends on \( y \) and \( n \) ) by requiring that \[ {\begin{Vmatrix}{U}_{T}^{in}f - {g}_{c\left( i\right) }\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} < \varepsilon . \] By van der Waerden's theorem there is a monochrome progression \[ \{ i, i + d,\ldots, i + {kd}\} \subseteq \{ 1,\ldots, K\} \] and so there is some \( {g}_{s\left( y\right) } = {g}_{ * } \) for which \[ {\begin{Vmatrix}{U}_{T}^{\left( {i + {jd}}\right) n}f - {g}_{ * }\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} < \varepsilon \] for \( j = 0,\ldots, k \) . Writing \( \widetilde{g} = {U}_{T}^{-{in}}{g}_{ * } \), this means that \[ {\begin{Vmatrix}{U}_{T}^{jdn}f - \widetilde{g}\end{Vmatrix}}_{{L}_{{\mu }_{S}^{\mathcal{A}}{iny}}^{2}} < \varepsilon \] for \( j = 0,\ldots, k \) . Since \( j = 0 \) is allowed here, we also get \[ {\begin{Vmatrix}{U}_{T}^{jdn}f - f\end{Vmatrix}}_{{L}_{{\mu }_{S}^{\mathcal{A}}{iny}}^{2}} < {2\varepsilon } \] (7.33) for \( j = 1,\ldots, k \) . This shows that, for a given \( n \), the set \[ A \cap {S}^{-n}A \cap \cdots \cap {S}^{-{Kn}}A \] is partitioned into finitely many sets \( {D}_{n,1},\ldots ,{D}_{n, M} \), where \( M \) is the number of arithmetic progressions of length \( \left( {k + 1}\right) \) in \( \left\lbrack {1, K}\right\rbrack \), with the property that \( i, d,\widetilde{g} \) do not change within such a given set. In particular, if \( n \in {R}_{K} \) , and so \[ \mu \left( {A \cap {S}^{-n}A \cap \cdots \cap {S}^{-{Kn}}A}\right) > {c}_{1}, \] then for one of these sets \( D \) (with corresponding progression \[ \

Proposition 7.23. Let \[ \begin{matrix} \mathrm{X} = \left( {X,\mathcal{B},\mu, T}\right) \\ \downarrow \\ \mathrm{Y} = \left( {Y,\mathcal{A},\nu, S}\right) \end{matrix} \] be a compact extension of invertible measure-preserving systems on Borel probability spaces. If \( \mathrm{Y} \) is \( {SZ} \), then so is \( \mathrm{X} \) .

Proof of Proposition 7.23 using van der Waerden. By Lemma 7.24, we may assume that \( f = {\chi }_{B} \) is AP and that there exists some set \( A \in \mathcal{A} \) of positive measure with \( {\mu }_{y}^{\mathcal{A}}\left( B\right) > \frac{1}{2}\mu \left( B\right) \) for \( y \in A \) . We will use \( \mathrm{{SZ}} \) for \( A \) (for arithmetic progressions of quite large length \( K \) ) to show SZ for \( B \) (for arithmetic progressions of length \( k \) ). Given \( \varepsilon = \frac{\mu \left( B\right) }{6\left( {k + 1}\right) } > 0 \) we may find (using the AP property of \( f \) ) functions \( {g}_{1},\ldots ,{g}_{r} \) such that \[ \mathop{\min }\limits_{{1 \leq s \leq r}}{\begin{Vmatrix}{U}_{T}^{n}f - {g}_{s}\end{Vmatrix}}_{{L}_{{\mu }_{y}^{\mathcal{A}}}^{2}} < \varepsilon \] for all \( n \in \mathbb{Z} \) and almost every \( y \in Y \) . We may assume that \( {\begin{Vmatrix}{g}_{s}\end{Vmatrix}}_{\infty } \leq 1 \) for \( s = 1,\ldots, r \) . By Theorem 7.1 we may choose a \( K \) for which any coloring of \( \{ 1,\ldots, K\} \) with \( r \) colors contains a monochrome arithmetic progression of length \( k + 1 \) . There is some \( {c}_{1} > 0 \) for which \[ {R}_{K} = \left\{ {n \in \mathbb{N} \mid \nu \left( {A \cap {S}^{-n}A \cap \cdots {S}^{-{Kn}}A}\right) > {c}_{1}}\right\} \] has positive lower density. In fact this follows from SZ for \( A \) : If \[ \mathop{\liminf }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}\nu \left( {A \cap {S}^{-n}A

Lemma 14.3. If \( F/\mathbb{Q} \) is an extension in which no finite prime ramifies, then \( F = \mathbb{Q} \) . Proof. A theorem of Minkowski (see Exercise 2.5) states that every ideal class of \( F \) contains an integral ideal of norm less than or equal to \[ \frac{n!}{{n}^{n}}{\left( \frac{4}{\pi }\right) }^{{r}_{2}}\sqrt{{d}_{F}} \] where \( n = \left\lbrack {F : \mathbb{Q}}\right\rbrack ,{d}_{F} \) is the absolute value of the discriminant, and \( {r}_{2} \leq n/2 \) is the number of complex places. In particular, this quantity must be at least 1, so \[ \sqrt{{d}_{F}} \geq \frac{{n}^{n}}{n!}{\left( \frac{\pi }{4}\right) }^{{r}_{2}} \geq \frac{{n}^{n}}{n!}{\left( \frac{n}{4}\right) }^{n/2}\overset{\text{ def }}{ = }{b}_{n}. \] Since \( {b}_{2} > 1 \) and \[ \frac{{b}_{n + 1}}{{b}_{n}} = {\left( 1 + \frac{1}{n}\right) }^{n}\sqrt{\frac{\pi }{4}} \geq 2\sqrt{\frac{\pi }{4}} > 1 \] we must have, if \( n \geq 2 \) , \[ {d}_{F} > 1\text{.} \] Consequently there exists a prime \( p \) dividing \( {d}_{F} \), which means \( p \) ramifies. This proves Lemma 14.3. Returning to the above, we consider the fixed field \( F \) of \( I \) . Then \( F/\mathbb{Q} \) is unramified at all finite primes, so \( F = \mathbb{Q} \) . Therefore \[ I = \operatorname{Gal}\left( {L/\mathbb{Q}}\right) \] hence \[ \left\lbrack {L : \mathbb{Q}}\right\rbrack = \left| I\right| \leq \left\lbrack {\mathbb{Q}\left( {\zeta }_{n}\right) : \mathbb{Q}}\right\rbrack . \] Since \[ \mathbb{Q}\left( {\zeta }_{n}\right) \subseteq K\left( {\zeta }_{n}\right) = L \] we have equality, so \[ K \subseteq \mathbb{Q}\left( {\zeta }_{n}\right) \] This proves "14.2 \( \Rightarrow \) 14.1." We are now reduced to the local situation, where the structure of extensions is much simpler. We shall often use the following well-known result. Lemma 14.4. Let \( K \) and \( L \) be finite extensions of \( {\mathbb{Q}}_{p} \) such that \( K/L \) is unramified. Then (a) \( K = L\left( {\zeta }_{n}\right) \) for some \( n \) with \( p \nmid n \), and (b) \( \operatorname{Gal}\left( {K/L}\right) \) is cyclic. Also, for fixed \( L \) and for every integer \( m \geq 1 \), there exists a unique unramified extension \( K \) of \( L \) which is cyclic of degree \( m \) . We sketch the proof. First consider (a) and (b), and assume \( K/L \) is Galois. Let \( {\mathcal{O}}_{K} \) and \( {h}_{K} \) be the integers and maximal ideal for \( K \) and define \( {\mathcal{O}}_{L} \) and \( {h}_{L} \) similarly. Since \( K/L \) is unramified, there is a canonical isomorphism \[ \operatorname{Gal}\left( {K/L}\right) \simeq \operatorname{Gal}\left( {{\mathcal{O}}_{K}{\;\operatorname{mod}\;{p}_{K}}/{\mathcal{O}}_{L}{\;\operatorname{mod}\;{p}_{L}}}\right) \] (if there were ramification, we would have to mod out by the inertia group on the left). The right-hand side is an extension of finite fields, hence cyclic. If \( K/L \) is not necessarily Galois, then the Galois closure yeilds a cyclic Galois group; so \( K/L \) is already Galois and cyclic. This proves (b). Since every nonzero element of a finite field is a root of unity of order prime to \( p \), we may choose \( {\zeta }_{n} \in {\mathcal{O}}_{K}{\;\operatorname{mod}\;{k}_{K}} \) with \( \left( {n, p}\right) = 1 \) which generates the extension of finite fields. Since \( {X}^{n} - 1 = 0 \) has a solution \( {\;\operatorname{mod}\;{n}_{K}} \), Hensel’s lemma (note \( p \nmid n \) ) yields a solution in \( {\mathcal{O}}_{K} \), which generates the extension \( K/L \) because of the isomorphism of Galois groups. This proves (a). To prove the last statement, let \( {\zeta }_{n} \) with \( p \nmid n \) generate an extension of \( {\mathcal{O}}_{L}/{p\not{} }_{L} \) of degree \( m \) . Then \( L\left( {\zeta }_{n}\right) /L \) is unramified and, by the isomorphism of Galois groups, is cyclic of degree \( m \) . If there are two such extensions, then the compositum is unramified, hence cyclic by (b). Therefore the two extensions must coincide. This proves the lemma. In the proof of Theorem 14.2, it will be convenient to know what the answer will be. Unramified extensions of \( {\mathbb{Q}}_{p} \) will be given by Lemma 14.4. In particular, we already have a good supply of unramified extensions. The ramified extensions of \( {\mathbb{Q}}_{p} \) will be subfields of \( {\mathbb{Q}}_{p}\left( {\zeta }_{{p}^{n}}\right) \) . Since our list of abelian extensions already includes such subfields, we can produce totally ramified extensions \( K/{\mathbb{Q}}_{p} \) with any group \[ \operatorname{Gal}\left( {K/{\mathbb{Q}}_{p}}\right) \subseteq \left\{ \begin{array}{ll} \mathbb{Z}/\left( {p - 1}\right) \mathbb{Z} \times \mathbb{Z}/{p}^{n}\mathbb{Z}, & p \neq 2, \\ \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/{2}^{n}\mathbb{Z}, & p = 2, \end{array}\right. \] for all \( n \geq 0 \) . The fact that we can get \( {\left( \mathbb{Z}/2\mathbb{Z}\right) }^{2} \) for \( p = 2 \) will cause slight problems. We now start the proof of Theorem 14.2. Observe that it suffices to assume \[ \operatorname{Gal}\left( {K/{\mathbb{Q}}_{p}}\right) \simeq \mathbb{Z}/{q}^{m}\mathbb{Z},\;q = \text{ prime,}\;m \geq 1. \] We consider three cases: \( p \neq q, p = q \neq 2 \), and \( p = q = 2 \) . Case I. \( q \neq p \) Lemma 14.5. Let \( K \) and \( L \) be finite extensions of \( {\mathbb{Q}}_{p} \) and let \( {h}_{L} \) be the maximal ideal of the integers of \( L \) . Suppose \( K/L \) is totally ramified of degree \( e \) with \( p \nmid e \) (i.e., \( K/L \) is tamely ramified). Then there exists \( \pi \in L \) of order 1 at \( {h}_{L} \) and a root \( \alpha \) of \[ {X}^{e} - \pi = 0 \] such that \( K = L\left( \alpha \right) \) . Proof. Let \( \left| x\right| \) be the absolute value on \( {\mathbb{C}}_{p} \) ( \( = \) completion of the algebraic closure of \( {\mathbb{Q}}_{p} \) ). Let \( {\pi }_{0} \in {\mathcal{M}}_{L} \) be of order 1 . Choose \( \beta \in K \) to be a uniformizing parameter, so that \[ {\left| \beta \right| }^{e} = \left| {\pi }_{0}\right| \] Then \[ {\beta }^{e} = {\pi }_{0}u\;\text{ with }u \in {U}_{K} = \text{ units of }K. \] Since \( K/L \) is totally ramified, the extension of residue class fields is trivial. Consequently \[ u \equiv {u}_{0}{\;\operatorname{mod}\;{p}_{K}}\;\text{ with }{u}_{0} \in {U}_{L} \] Therefore \[ u = {u}_{0} + x\;\text{ with }x \in {h\not{} }_{K} \] Let \( \pi = {\pi }_{0}{u}_{0} \), so \[ {\beta }^{e} = {\pi }_{0}{u}_{0} + {\pi }_{0}x = \pi + {\pi }_{0}x \] and \[ \left| {{\beta }^{e} - \pi }\right| < \left| {\pi }_{0}\right| = \left| \pi \right| \] Let \( {\alpha }_{1},\ldots ,{\alpha }_{e} \) be the roots of \[ f\left( X\right) = {X}^{e} - \pi \] Since the \( \alpha \) ’s differ by roots of unity, \[ \left| {\alpha }_{i}\right| = \left| {\alpha }_{j}\right| \;\text{ for all }i, j \] so \[ \left| {{\alpha }_{i} - {\alpha }_{1}}\right| \leq \operatorname{Max}\left( {\left| {\alpha }_{i}\right| ,\left| {\alpha }_{1}\right| }\right) = \left| {\alpha }_{1}\right| . \] But \[ \mathop{\prod }\limits_{{i \neq 1}}\left| {{\alpha }_{i} - {\alpha }_{1}}\right| = \left| {{f}^{\prime }\left( {\alpha }_{1}\right) }\right| = \left| {e{\alpha }_{1}^{e - 1}}\right| = {\left| {\alpha }_{1}\right| }^{e - 1}. \] Consequently \[ \left| {{\alpha }_{i} - {\alpha }_{1}}\right| = \left| {\alpha }_{1}\right| ,\;i \neq 1. \] Since \[ \mathop{\prod }\limits_{i}\left| {\beta - {\alpha }_{i}}\right| = \left| {f\left( \beta \right) }\right| < \left| \pi \right| = \mathop{\prod }\limits_{i}\left| {\alpha }_{i}\right| \] we must have for some \( {\alpha }_{i} \), say \( {\alpha }_{1} \), that \[ \left| {\beta - {\alpha }_{1}}\right| < \left| {\alpha }_{1}\right| \] Therefore \[ \left| {\beta - {\alpha }_{1}}\right| < \left| {{\alpha }_{i} - {\alpha }_{1}}\right| ,\;i \neq 1. \] By Krasner's lemma (Lemma 5.3), \[ L\left( {\alpha }_{1}\right) \subseteq L\left( \beta \right) \subseteq K\text{.} \] But \( f\left( X\right) \) is irreducible by the Eisenstein criterion, so \[ \left\lbrack {L\left( {\alpha }_{1}\right) : L}\right\rbrack = e = \left\lbrack {K : L}\right\rbrack . \] This completes the proof of Lemma 14.5. Lemma 14.6. \( {\mathbb{Q}}_{p}\left( {\left( -p\right) }^{1/\left( {p - 1}\right) }\right) = {\mathbb{Q}}_{p}\left( {\zeta }_{p}\right) \) . Proof. Let \[ g\left( X\right) = \frac{{\left( X + 1\right) }^{p} - 1}{X} \] \[ = {X}^{p - 1} + p{X}^{p - 2} + \cdots + p. \] Then \[ 0 = g\left( {{\zeta }_{p} - 1}\right) \equiv {\left( {\zeta }_{p} - 1\right) }^{p - 1} + p{\;\operatorname{mod}\;{\left( {\zeta }_{p} - 1\right) }^{p}}, \] so \[ u = \frac{{\left( {\zeta }_{p} - 1\right) }^{p - 1}}{-p} \equiv 1{\;\operatorname{mod}\;\left( {{\zeta }_{p} - 1}\right) }. \] It follows that \[ {u}_{1} = \mathop{\lim }\limits_{{n \rightarrow \infty }}{u}^{-\left( {1 + p + \cdots + {p}^{n}}\right) } \] exists in \( {\mathbb{Q}}_{p}\left( {\zeta }_{p}\right) \) and satisfies \[ {u}_{1}^{p - 1} = u \] Therefore \( {\left( -p\right) }^{1/\left( {p - 1}\right) } \in {\mathbb{Q}}_{p}\left( {\zeta }_{p}\right) \) . Since \( {X}^{p - 1} + p \) is irreducible over \( {\mathbb{Q}}_{p} \) by the Eisenstein criterion, the lemma follows easily. Now assume \( K/{\mathbb{Q}}_{p} \) is abelian of degree \( {q}^{m} \) . Let \( L/{\mathbb{Q}}_{p} \) be the maximal unramified subextension ( \( = \) fixed field of the inertia group). Then \[ L \subseteq {\mathbb{Q}}_{p}\left( {\zeta }_{n}\right) \] for some \( n \), by Lemma 14.4. Let \( e = \left\lbrack {K : L}\right\rbrack \) . Since \( e \) is a power of \( q, p \nmid e \), so \( K/L \) is totally and tamely ramified. By Lemma 14.5, \[ K = L\left( {\pi }^{1/e}\right) \] for some \( \pi \) of order 1 in \( L \) . Since \( L/{\mathbb{Q}}_{p} \) is unramified, \( p \) has order 1 in \( L \), so \[ \pi = - {up} \] for some unit \( u \in L \) . Since \( u \) is a unit and \( p \nmid e \), the extension \( L\left( {u}^{1/e}\right) /L \

Lemma 14.3. If \( F/\mathbb{Q} \) is an extension in which no finite prime ramifies, then \( F = \mathbb{Q} \).

A theorem of Minkowski (see Exercise 2.5) states that every ideal class of \( F \) contains an integral ideal of norm less than or equal to \[ \frac{n!}{{n}^{n}}{\left( \frac{4}{\pi }\right) }^{{r}_{2}}\sqrt{{d}_{F}} \] where \( n = \left\lbrack {F : \mathbb{Q}}\right\rbrack ,{d}_{F} \) is the absolute value of the discriminant, and \( {r}_{2} \leq n/2 \) is the number of complex places. In particular, this quantity must be at least 1, so \[ \sqrt{{d}_{F}} \geq \frac{{n}^{n}}{n!}{\left( \frac{\pi }{4}\right) }^{{r}_{2}} \geq \frac{{n}^{n}}{n!}{\left( \frac{n}{4}\right) }^{n/2}\overset{\text{ def }}{ = }{b}_{n}. \] Since \( {b}_{2} > 1 \) and \[ \frac{{b}_{n + 1}}{{b}_{n}} = {\left( 1 + \frac{1}{n}\right) }^{n}\sqrt{\frac{\pi }{4}} \geq 2\sqrt{\frac{\pi }{4}} > 1 \] we must have, if \( n \geq 2 \) , \[ {d}_{F} > 1\text{.} \] Consequently there exists a prime \( p \) dividing \( {d}_{F} \), which means \( p \) ramifies. This proves Lemma 14.3.

Theorem 3.1.1. \( {W}_{G} \) is a finite group and the representation of \( {W}_{G} \) on \( {\mathfrak{h}}^{ * } \) is faithful. Proof. Let \( s \in {\operatorname{Norm}}_{G}\left( H\right) \) . Suppose \( s \cdot \theta = \theta \) for all \( \theta \in \mathfrak{X}\left( H\right) \) . Then \( {s}^{-1}{hs} = h \) for all \( h \in H \), and hence \( s \in H \) by Theorem 2.1.5. This proves that the representation of \( {W}_{G} \) on \( {\mathfrak{h}}^{ * } \) is faithful. To prove the finiteness of \( {W}_{G} \), we shall assume that \( G \subset \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) is in the matrix form of Section 2.4.1, so that \( H \) is the group of diagonal matrices in \( G \) . In the proof of Theorem 2.1.5 we noted that \( h \in H \) acts on the standard basis for \( {\mathbb{C}}^{n} \) by \[ h{e}_{i} = {\theta }_{i}\left( h\right) {e}_{i}\text{ for }i = 1,\ldots, n, \] where \( {\theta }_{i} \in \mathcal{X}\left( H\right) \) and \( {\theta }_{i} \neq {\theta }_{j} \) for \( i \neq j \) . Let \( s \in {\operatorname{Norm}}_{G}\left( H\right) \) . Then \[ {hs}{e}_{i} = s\left( {{s}^{-1}{hs}}\right) {e}_{i} = s{\theta }_{i}\left( {{s}^{-1}{hs}}\right) {e}_{i} = \left( {s \cdot {\theta }_{i}}\right) \left( h\right) s{e}_{i}. \] (3.1) Hence \( s{e}_{i} \) is an eigenvector for \( h \) with eigenvalue \( \left( {s \cdot {\theta }_{i}}\right) \left( h\right) \) . Since the characters \( s \cdot {\theta }_{1},\ldots, s \cdot {\theta }_{n} \) are all distinct, this implies that there is a permutation \( \sigma \in {\mathfrak{S}}_{n} \) and there are scalars \( {\lambda }_{i} \in {\mathbb{C}}^{ \times } \) such that \[ s{e}_{i} = {\lambda }_{i}{e}_{\sigma \left( i\right) }\;\text{ for }i = 1,\ldots, n. \] (3.2) Since \( {sh}{e}_{i} = {\lambda }_{i}{\theta }_{i}\left( h\right) {e}_{\sigma \left( i\right) } \) for \( h \in H \), the permutation \( \sigma \) depends only on the coset \( {sH} \) . If \( t \in {\operatorname{Norm}}_{G}\left( H\right) \) and \( t{e}_{i} = {\mu }_{i}{e}_{\tau \left( i\right) } \) with \( \tau \in {\mathfrak{S}}_{n} \) and \( {\mu }_{i} \in {\mathbb{C}}^{ \times } \), then \[ {ts}{e}_{i} = {\lambda }_{i}t{e}_{\sigma \left( i\right) } = {\mu }_{\sigma \left( i\right) }{\lambda }_{i}{e}_{{\tau \sigma }\left( i\right) }. \] Hence the map \( s \mapsto \sigma \) is a homomorphism from \( {\operatorname{Norm}}_{G}\left( H\right) \) into \( {\mathfrak{S}}_{n} \) that is constant on the cosets of \( H \) . If \( \sigma \) is the identity permutation, then \( s \) commutes with \( H \) and therefore \( s \in H \) by Theorem 2.1.5. Thus we have defined an injective homomorphism from \( {W}_{G} \) into \( {\mathfrak{S}}_{n} \), so \( {W}_{G} \) is a finite group. We now describe \( {W}_{G} \) for each type of classical group. We will use the embedding of \( {W}_{G} \) into \( {\mathfrak{S}}_{n} \) employed in the proof of Theorem 3.1.1. For \( \sigma \in {\mathfrak{S}}_{n} \) let \( {s}_{\sigma } \in \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) be the matrix such that \( {s}_{\sigma }{e}_{i} = {e}_{\sigma \left( i\right) } \) for \( i = 1,\ldots, n \) . This is the usual representation of \( {\mathfrak{S}}_{n} \) on \( {\mathbb{C}}^{n} \) as permutation matrices. Suppose \( G = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) . Then \( H \) is the group of all \( n \times n \) diagonal matrices. Clearly \( {s}_{\sigma } \in {\operatorname{Norm}}_{G}\left( H\right) \) for every \( \sigma \in {\mathfrak{S}}_{n} \) . From the proof of Theorem 3.1.1 we know that every coset in \( {W}_{G} \) is of the form \( {s}_{\sigma }H \) for some \( \sigma \in {\mathfrak{S}}_{n} \) . Hence \( {W}_{G} \cong \) \( {\mathfrak{S}}_{n} \) . The action of \( \sigma \in {\mathfrak{S}}_{n} \) on the diagonal coordinate functions \( {x}_{1},\ldots ,{x}_{n} \) for \( H \) is \( \sigma \cdot {x}_{i} = {x}_{{\sigma }^{-1}\left( i\right) }. \) Let \( G = \mathbf{{SL}}\left( {n,\mathbb{C}}\right) \) . Now \( H \) consists of all diagonal matrices of determinant 1 . Given \( \sigma \in {\mathfrak{S}}_{n} \), we may pick \( {\lambda }_{i} \in {\mathbb{C}}^{ \times } \) such that the transformation \( s \) defined by (3.2) has determinant 1 and hence is in \( {\operatorname{Norm}}_{G}\left( H\right) \) . To prove this, recall that every permutation is a product of cyclic permutations, and every cyclic permutation is a product of transpositions (for example, the cycle \( \left( {1,2,\ldots, k}\right) \) is equal to \( \left( {1, k}\right) \cdots \left( {1,3}\right) \left( {1,2}\right) \) ). Consequently, it is enough to verify this when \( \sigma \) is the transposition \( i \leftrightarrow j \) . In this case we take \( {\lambda }_{j} = - 1 \) and \( {\lambda }_{k} = 1 \) for \( k \neq j \) . Since \( \det \left( {s}_{\sigma }\right) = - 1 \), we obtain \( \det s = 1 \) . Thus the homomorphism \( {W}_{G} \rightarrow {\mathfrak{S}}_{n} \) constructed in the proof of Theorem 3.1.1 is surjective. Hence \( {W}_{G} \cong {\mathfrak{S}}_{n} \) . Notice, however, that this isomorphism arises by choosing elements of \( {\operatorname{Norm}}_{G}\left( H\right) \) whose adjoint action on \( \mathfrak{h} \) is given by permutation matrices; the group of all permutation matrices is not a subgroup of \( G \) . Next, consider the case \( G = \mathbf{{Sp}}\left( {{\mathbb{C}}^{2l},\Omega }\right) \), with \( \Omega \) as in (2.6). Let \( {s}_{l} \in \mathbf{{GL}}\left( {l,\mathbb{C}}\right) \) be the matrix for the permutation \( \left( {1, l}\right) \left( {2, l - 1}\right) \left( {3, l - 2}\right) \cdots \), as in equation (2.5). For \( \sigma \in {\mathfrak{S}}_{l} \) let \( {s}_{\sigma } \in \mathbf{{GL}}\left( {l,\mathbb{C}}\right) \) be the corresponding permutation matrix. Clearly \( {s}_{\sigma }^{t} = {s}_{\sigma }^{-1} \), so if we define \[ \pi \left( \sigma \right) = \left\lbrack \begin{matrix} {s}_{\sigma } & 0 \\ 0 & {s}_{l}{s}_{\sigma }{s}_{l} \end{matrix}\right\rbrack \] then \( \pi \left( \sigma \right) \in G \) and hence \( \pi \left( \sigma \right) \in {\operatorname{Norm}}_{G}\left( H\right) \) . Obviously \( \pi \left( \sigma \right) \in H \) if and only if \( \sigma = 1 \), so we obtain an injective homomorphism \( \bar{\pi } : {\mathfrak{S}}_{l} \rightarrow {W}_{G} \) . To find other elements of \( {W}_{G} \), consider the transpositions \( \left( {i,{2l} + 1 - i}\right) \) in \( {\mathfrak{S}}_{2l} \) , where \( 1 \leq i \leq l \) . Set \( {e}_{-i} = {e}_{{2l} + 1 - i} \), where \( \left\{ {e}_{i}\right\} \) is the standard basis for \( {\mathbb{C}}^{2l} \) . Define \( {\tau }_{i} \in \mathbf{{GL}}\left( {{2l},\mathbb{C}}\right) \) by \[ {\tau }_{i}{e}_{i} = {e}_{-i},\;{\tau }_{i}{e}_{-i} = - {e}_{i},\;{\tau }_{i}{e}_{k} = {e}_{k}\text{ for }k \neq i, - i. \] Since \( \left\{ {{\tau }_{i}{e}_{j} : j = \pm 1,\ldots , \pm l}\right\} \) is an \( \Omega \) -symplectic basis for \( {\mathbb{C}}^{2l} \), we have \( {\tau }_{i} \in \) \( \mathbf{{Sp}}\left( {{\mathbb{C}}^{2l},\Omega }\right) \) by Lemma 1.1.5. Clearly \( {\tau }_{i} \in {\operatorname{Norm}}_{G}\left( H\right) \) and \( {\tau }_{i}^{2} \in H \) . Furthermore, \( {\tau }_{i}{\tau }_{j} = {\tau }_{j}{\tau }_{i} \) if \( 1 \leq i, j \leq l \) . Given \( F \subset \{ 1,\ldots, l\} \), define \[ {\tau }_{F} = \mathop{\prod }\limits_{{i \in F}}{\tau }_{i} \in {\operatorname{Norm}}_{G}\left( H\right) \] Then the \( H \) -cosets of the elements \( \left\{ {\tau }_{F}\right\} \) form an abelian subgroup \( {T}_{l} \cong {\left( \mathbb{Z}/2\mathbb{Z}\right) }^{l} \) of \( {W}_{G} \) . The action of \( {\tau }_{F} \) on the coordinate functions \( {x}_{1},\ldots ,{x}_{l} \) for \( H \) is \( {x}_{i} \mapsto {x}_{i}^{-1} \) for \( i \in F \) and \( {x}_{j} \mapsto {x}_{j} \) for \( j \notin F \) . This makes it evident that \[ \pi \left( \sigma \right) {\tau }_{F}\pi {\left( \sigma \right) }^{-1} = {\tau }_{\sigma F}\;\text{ for }F \subset \{ 1,\ldots, l\} \text{ and }\sigma \in {\mathfrak{S}}_{l}. \] (3.3) Clearly, \( {T}_{l} \cap \bar{\pi }\left( {\mathfrak{S}}_{l}\right) H = \{ 1\} \) . Lemma 3.1.2. For \( G = \mathbf{{Sp}}\left( {{\mathbb{C}}^{2l},\Omega }\right) \), the subgroup \( {T}_{l} \subset {W}_{G} \) is normal, and \( {W}_{G} \) is the semidirect product of \( {T}_{l} \) and \( \bar{\pi }\left( {\mathfrak{S}}_{l}\right) \) . The action of \( {W}_{G} \) on the coordinate functions in \( \mathcal{O}\left\lbrack H\right\rbrack \) is by \( {x}_{i} \mapsto {\left( {x}_{\sigma \left( i\right) }\right) }^{\pm 1}\left( {i = 1,\ldots, l}\right) \), for every permutation \( \sigma \) and choice \( \pm 1 \) of exponents. Proof. Recall that a group \( K \) is a semidirect product of subgroups \( L \) and \( M \) if \( M \) is a normal subgroup of \( K, L \cap M = 1 \), and \( K = L \cdot M \) . By (3.3) we see that it suffices to prove that \( {W}_{G} = {T}_{l}\pi \left( {\mathfrak{S}}_{l}\right) \) . Suppose \( s \in \) \( {\operatorname{Norm}}_{G}\left( H\right) \) . Then there exists \( \sigma \in {\mathfrak{S}}_{2l} \) such that \( s \) is given by (3.2). Define \[ F = \{ i : i \leq l\text{ and }\sigma \left( i\right) \geq l + 1\} \] and let \( \mu \in {\mathfrak{S}}_{2l} \) be the product of the transpositions interchanging \( \sigma \left( i\right) \) with \( \sigma \left( i\right) - l \) for \( i \in F \) . Then \( {\mu \sigma } \) stabilizes the set \( \{ 1,\ldots, l\} \) . Let \( v \in {\mathfrak{S}}_{l} \) be the corresponding permutation of this set. Then \( \pi {\left( v\right) }^{-1}{\tau }_{F}s{e}_{i} = \pm {\lambda }_{i}{e}_{i} \) for \( i = 1,\ldots, l \) . Thus \( s \in {\tau }_{F}\pi \left( v\right) H. \) Now consider the case \( G = \mathbf{{SO}}\left( {{\mathbb{C}}^{{2l} + 1}, B}\right) \), with the symmetric form \( B \) as in (2.9). For \( \sigma \in {\mathfrak{S}}_{l} \) define \[ \varphi \left( \sigma \right) = \left\lbrack \begin{matrix} {s}_{\sigma } & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & {s}_{l}{s}_{\sigma }

Theorem 3.1.1. \( {W}_{G} \) is a finite group and the representation of \( {W}_{G} \) on \( {\mathfrak{h}}^{ * } \) is faithful.

Proof. Let \( s \in {\operatorname{Norm}}_{G}\left( H\right) \) . Suppose \( s \cdot \theta = \theta \) for all \( \theta \in \mathfrak{X}\left( H\right) \) . Then \( {s}^{-1}{hs} = h \) for all \( h \in H \), and hence \( s \in H \) by Theorem 2.1.5. This proves that the representation of \( {W}_{G} \) on \( {\mathfrak{h}}^{ * } \) is faithful. To prove the finiteness of \( {W}_{G} \), we shall assume that \( G \subset \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) is in the matrix form of Section 2.4.1, so that \( H \) is the group of diagonal matrices in \( G \) . In the proof of Theorem 2.1.5 we noted that \( h \in H \) acts on the standard basis for \( {\mathbb{C}}^{n} \) by \[ h{e}_{i} = {\theta }_{i}\left( h\right) {e}_{i}\text{ for }i = 1,\ldots, n, \] where \( {\theta }_{i} \in \mathcal{X}\left( H\right) \) and \( {\theta }_{i} \neq {\theta }_{j} \) for \( i \neq j \) . Let \( s \in {\operatorname{Norm}}_{G}\left( H\right) \) . Then \[ {hs}{e}_{i} = s\left( {{s}^{-1}{hs}}\right) {e}_{i} = s{\theta }_{i}\left( {{s}^{-1}{hs}}\right) {e}_{i} = \left( {s \cdot {\theta }_{i}}\right) \left( h\right) s{e}_{i}. \] Hence \( s{e}_{i} \) is an eigenvector for \( h \) with eigenvalue \( \left( {s \cdot {\theta }_{i}}\right) \left( h\right) \) . Since the characters \( s \cdot {\theta }_{1},\ldots, s \cdot {\theta }_{n} \) are all distinct, this implies that there is a permutation \( \sigma \in {\mathfrak{S}}_{n} \) and there are scalars \( {\lambda }_{i} \in {\mathbb{C}}^{ \times } \) such that \[ s{e}_{i} = {\lambda }_{i}{e}_{\sigma \left( i\right) }\;\text{ for }i = 1,\ldots, n. \] Since \( {sh}{e}_{i} = {\lambda }_{i}{\theta }_{i}\left( h\right) {e}_{\sigma \left( i\right) } \) for \( h \in H \), the permutation \( \(\sigma\) depends only on the coset \( {sH} \) . If \( t \(\in\) {\operatorname{Norm}}_{G}\(\left( H\right)\) and \(\ t{e}_{i}\) = \(\mu_{i}{e}_{\tau\(\left( i\right)}\)\) with \(\tau\) \(\in\) \(\mathfrak{{S}}_{n}\) and \(\mu_{i}\) \(\in\) \(\mathbb{{C}}^{\times}\), then \[ts e_{i}=\lambda_{i}\tau e_{\sigma\(\left( i\right)}=\mu_{\sigma\(\left( i\right)}\lambda_{i}{e}_{\tau\(\sigma\(\left( i\right))}.\] Hence the map \(\ s\) \(\mapsto\) \(\sigma\) is a homomorphism from \(\operatorname{{Norm}}_{G}(\mathrm{{H}})\) into \(\mathfrak{{S}}_{n}\) that is constant on the cosets of \(\mathrm{{H}}.\ If\) \(\sigma\) is the identity permutation, then \(\mathrm{{s}}\ commutes with \(\mathrm{{H}}\ and therefore

Lemma 5.1.3 With \( \Gamma \) as described in Theorem 5.1.2 satisfying conditions 1 and 2, condition 3 is equivalent to the following requirement: \( {3}^{\prime } \) . All embeddings \( \sigma \), apart from the identity and \( \mathbf{c} \), complex conjugation, are real and \( {A\Gamma } \) is ramified at all real places. Proof: If condition \( {\mathcal{3}}^{\prime } \) holds and \( \sigma : {k\Gamma } \rightarrow \mathbb{R} \), then there exists \( \tau : {A\Gamma } \rightarrow \) \( \mathcal{H} \), Hamilton’s quaternions, such that \( \sigma \left( {\operatorname{tr}f}\right) = \operatorname{tr}\left( {\tau \left( f\right) }\right) \) for each \( f \in {\Gamma }^{\left( 2\right) } \) . Since \( \det \left( f\right) = 1,\tau \left( f\right) \in {\mathcal{H}}^{1} \), so that \( \operatorname{tr}\left( {\tau \left( f\right) }\right) \in \left\lbrack {-2,2}\right\rbrack \) . Conversely, suppose condition 3 holds and \( \sigma : {k\Gamma } \rightarrow \mathbb{C} \) . Let \( f \in {\Gamma }^{\left( 2\right) } \) have eigenvalues \( \lambda \) and \( {\lambda }^{-1} \) and \( \mu \) be an extension of \( \sigma \) to \( {k\Gamma }\left( \lambda \right) \) . Then \( \sigma \left( {\operatorname{tr}{f}^{n}}\right) = \mu {\left( \lambda \right) }^{n} + \mu {\left( \lambda \right) }^{-n} \) . Thus \[ \left| {\sigma \left( {\operatorname{tr}{f}^{n}}\right) }\right| \geq \left| \right| \mu \left( \lambda \right) \left| {{}^{n} - }\right| \mu \left( \lambda \right) \left| {}^{-n}\right| . \] So, if \( \sigma \left( {\operatorname{tr}{f}^{n}}\right) \) is bounded, then \( \left| {\mu \left( \lambda \right) }\right| = 1 \) so that \( \sigma \left( {\operatorname{tr}f}\right) = \mu \left( \lambda \right) + \mu {\left( \lambda \right) }^{-1} \) is a real number in the interval \( \left\lbrack {-2,2}\right\rbrack \) . Now choose an irreducible subgroup \( \left\langle {{g}_{1},{g}_{2}}\right\rangle \) of \( {\Gamma }^{\left( 2\right) } \) such that \( {g}_{1} \) is not parabolic. Then \[ {A\Gamma } \cong \left( \frac{{\operatorname{tr}}^{2}{g}_{1}\left( {\operatorname{tr}{}^{2}{g}_{1} - 4}\right) ,\operatorname{tr}\left\lbrack {{g}_{1},{g}_{2}}\right\rbrack - 2}{k\Gamma }\right) \] by (3.38). Since \( \sigma \left( {\operatorname{tr}f}\right) \in \left\lbrack {-2,2}\right\rbrack \) for all \( f \), it follows that \( {A\Gamma } \) is ramified at all real places (see Theorem 2.5.1). ## Exercise 5.1 1. Prove Lemma 5.1.1. 2. State and prove the corresponding result to Theorem 5.1.2 for finitely generated subgroups of \( \operatorname{PSL}\left( {2,\mathbb{R}}\right) \) . 3. Let \( \Gamma = \langle f, g\rangle \) be a subgroup of \( \operatorname{PSL}\left( {2,\mathbb{C}}\right) \) where \( g \) has order \( \mathcal{2} \) and \( f \) has order 3. Let \( \gamma = \operatorname{tr}\left\lbrack {f, g}\right\rbrack - 2 \) be a non-real algebraic integer, with minimum polynomial \( p\left( x\right) \) all of whose roots, except \( \gamma \) and \( \bar{\gamma } \) lie in the interval \( \left( {-3,0}\right) \) . Prove that \( \Gamma \) is a discrete group. 4. Let \( \Gamma = \langle f, g\rangle \), where \( f \) has order 6 and \( g \) has order 2, with \( \gamma = \) \( \operatorname{tr}\left\lbrack {f, g}\right\rbrack - 2 \) satisfying the polynomial \( {x}^{3} + {x}^{2} + {2x} + 1 \) . Prove that \( \Gamma \) is discrete. 5. Let \( \Gamma = \left\langle {{x}_{1},{x}_{2},{x}_{3}}\right\rangle \) be a non-elementary subgroup of \( \operatorname{PSL}\left( {2,\mathbb{R}}\right) \) such that \( o\left( {x}_{i}\right) = 2 \) for \( i = 1,2,3 \) and \( o\left( {{x}_{1}{x}_{2}{x}_{3}}\right) \) is odd \( \left( { \neq 1}\right) \) . Let \( x = \operatorname{tr}{x}_{1}{x}_{2} \) , \( y = \operatorname{tr}{x}_{2}{x}_{3}, z = \operatorname{tr}{x}_{3}{x}_{1} \) . If \( x, y \) and \( z \) are totally real algebraic integers with \( x, y \neq 0, \pm 2 \), and for every embedding \( \sigma \) of \( \mathbb{Q}\left( {\operatorname{tr}\Gamma }\right) \) such that \( {\left. \sigma \right| }_{k\Gamma } \neq \mathrm{{Id}} \) , then \( \left| {\sigma \left( x\right) }\right| < 2 \), prove that \( \Gamma \) is discrete and cocompact in \( \operatorname{PSL}\left( {2,\mathbb{R}}\right) \) . ## 5.2 Bass's Theorem One of the first applications of number-theoretic methods in 3-manifold topology arises directly from Bass-Serre theory of group actions on trees. To state Bass's theorem, we introduce the following definition. Definition 5.2.1 Let \( \overline{\mathbb{Q}} \) denote the algebraic closure of \( \mathbb{Q} \) in \( \mathbb{C} \) and let \( \Gamma < \mathrm{{SL}}\left( {2,\overline{\mathbb{Q}}}\right) \) . Then \( \Gamma \) is said to have integral traces if for all \( \gamma \in \Gamma \), tr \( \left( \gamma \right) \) is an algebraic integer. Otherwise, we say \( \Gamma \) has non-integral trace. We also use this terminology for \( \Gamma \) a subgroup of \( \operatorname{PSL}\left( {2,\overline{\mathbb{Q}}}\right) \) . It is not difficult to show that the property of having integral traces is preserved by commensurability (see Exercise 5.2, No. 1). The following theorem of Bass is the main result of this section. Theorem 5.2.2 Let \( M = {\mathbf{H}}^{3}/\Gamma \) be a finite-volume hyperbolic 3-manifold for which \( \Gamma \) has non-integral trace. Then \( M \) contains a closed embedded essential surface. Before embarking on the proof of this theorem, we deduce a succinct version of Theorem 5.2.2 in the closed setting (see \( §{1.5} \) ). Corollary 5.2.3 If \( M = {\mathbf{H}}^{3}/\Gamma \) is non-Haken, then \( \Gamma \) has integral traces. We also remark that having integral traces is equivalent to having an "integral representation" in the following sense. Let \( \mathbb{A} \) denote the ring of all algebraic integers in \( \overline{\mathbb{Q}} \) . Lemma 5.2.4 Let \( \Gamma \) be a finitely generated non-elementary subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) . Then \( \Gamma \) has integral traces if and only if \( \Gamma \) is conjugate in \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) to a subgroup of \( \mathrm{{SL}}\left( {2,\mathbb{A}}\right) \) . Proof: One way is obvious, so we assume that \( \Gamma \) has integral traces. Since \( \Gamma \) is finitely generated, the trace field of \( \Gamma \) is a finite extension \( k \) of \( \mathbb{Q} \) . Let \( {A}_{0}\Gamma \) be the quaternion algebra generated over \( k \) by elements of \( \Gamma \) and \( \mathcal{O}\Gamma \) the \( {R}_{k} \) -module generated by the elements of \( \Gamma \) . Then \( \mathcal{O}\Gamma \) is an order of \( {A}_{0}\Gamma \) (see Exercise 3.2, No. 1). By choosing a suitable quadratic extension \( L,{A}_{0}\left( \Gamma \right) { \otimes }_{k}L \cong {M}_{2}\left( L\right) \) (Corollary 2.1.9, Corollary 3.2.4), and so by the Skolem Noether Theorem, we may conjugate so that \( {A}_{0}\Gamma \subset {M}_{2}\left( L\right) \) . The order \( \mathcal{O}\Gamma { \otimes }_{{R}_{k}}{R}_{L} \) is then conjugate to a suborder of \( {M}_{2}\left( {{R}_{L};J}\right) \) where \( J \) is a fractional ideal as defined at (2.5) (see Lemma 2.2.8 and Theorem 2.2.9). Now pass to a finite extension \( H \) say, of \( L \) to make the ideal \( J \) principal. There is always such a finite extension and the Hilbert Class field is such an extension. A further conjugation of \( {M}_{2}\left( {{R}_{H};J}\right) \) shows that \( \Gamma \) is contained in \( \mathrm{{SL}}\left( {2,{R}_{H}}\right) \) . This completes the proof. \( ▱ \) In light of this lemma, a reformulation of Theorem 5.2.2 is as follows: Theorem 5.2.5 Let \( M = {\mathbf{H}}^{3}/\Gamma \) be a finite-volume hyperbolic 3-manifold not containing any closed embedded essential surface. Then \( \Gamma \) is conjugate to a subgroup of \( \operatorname{PSL}\left( {2,\mathbb{A}}\right) \) . The proof of Theorem 5.2.2 requires some information about the tree of \( \mathrm{{SL}}\left( 2\right) \) over a \( \mathcal{P} \) -adic field \( K \), as developed by Serre. This tree can alternatively be described in terms of maximal orders and, in this vein, is discussed in Chapter 6. The actions of the groups \( \mathrm{{SL}}\left( {2, K}\right) \) and \( \mathrm{{GL}}\left( {2, K}\right) \) on this tree play a critical role in obtaining the description of maximal arithmetic Kleinian and Fuchsian groups via local-global arguments and so a comprehensive treatment of these actions is given in \( §{11.4} \) . Thus the basic results recalled in the next subsection will be developed more fully later as indicated. ## 5.2.1 Tree of \( \mathrm{{SL}}\left( {2,{K}_{\mathcal{P}}}\right) \) Let \( K \) be a finite extension of \( {\mathbb{Q}}_{p} \) with valuation \( v \) and uniformizing parameter \( \pi \), valuation ring \( R \) and unique prime ideal \( \mathcal{P} \) . Let \( V \) denote the vector space \( {K}^{2} \) . Recall from \( §{2.2} \) that a lattice \( L \) in \( V \) is a finitely generated \( R \) -submodule which spans \( V \) . Define an equivalence relation on the set of lattices of \( V : L \sim {L}^{\prime } \) if and only if \( {L}^{\prime } = {xL} \) for some \( x \in {K}^{ * } \) . Let \( \Lambda \) denote the equivalence class of \( L \) . These equivalence classes form the vertices of a combinatorial graph \( \mathcal{T} \) where two vertices \( \Lambda \) and \( {\Lambda }^{\prime } \) are connected by an edge if there are representative lattices \( L \) and \( {L}^{\prime } \), where \( {L}^{\prime } \subset L \) and \( L/{L}^{\prime } \cong R/{\pi R} \) . Serre proved that \( \mathcal{T} \) is a tree; that is, it is connected and simply connected (see Theorem 6.5.3 for a proof), and each vertex has valency \( N\mathcal{P} + 1 \) (see Exercise 5.2, No. 3). The obvious action of \( \mathrm{{GL}}\left( {2, K}\right) \) on the set of lattices in \( V \) determines an action on \( \mathcal{T} \), which is transitive on vertices (see Corollary 2.2.10). The ac

Lemma 5.1.3 With \( \Gamma \) as described in Theorem 5.1.2 satisfying conditions 1 and 2, condition 3 is equivalent to the following requirement: \( {3}^{\prime } \) . All embeddings \( \sigma \), apart from the identity and \( \mathbf{c} \), complex conjugation, are real and \( {A\Gamma } \) is ramified at all real places.

Proof: If condition \( {\mathcal{3}}^{\prime } \) holds and \( \sigma : {k\Gamma } \rightarrow \mathbb{R} \), then there exists \( \tau : {A\Gamma } \rightarrow \) \( \mathcal{H} \), Hamilton’s quaternions, such that \( \sigma \left( {\operatorname{tr}f}\right) = \operatorname{tr}\left( {\tau \left( f\right) }\right) \) for each \( f \in {\Gamma }^{\left( 2\right) } \) . Since \( \det \left( f\right) = 1,\tau \left( f\right) \in {\mathcal{H}}^{1} \), so that \( \operatorname{tr}\left( {\tau \left( f\right) }\right) \in \left\lbrack {-2,2}\right\rbrack \) . Conversely, suppose condition 3 holds and \( \sigma : {k\Gamma } \rightarrow \mathbb{C} \) . Let \( f \in {\Gamma }^{\left( 2\right) } \) have eigenvalues \( \lambda \) and \( {\lambda }^{-1} \) and \( \mu \) be an extension of \( \sigma \) to \( {k\Gamma }\left( \lambda \right) \) . Then \( \sigma \left( {\operatorname{tr}{f}^{n}}\right) = \mu {\left( \lambda \right) }^{n} + \(\mu {\left( \(\lambda\) }\right) }^{-n} \) . Thus \[ \(\left| {\sigma \(\left( {\operatorname{tr}{f}^{n}}\right) }\right| \(\geq \(\left| \(\mu \(\lambda\) }\right| {{}^{n} - }\)\(\mu \(\lambda\) }\right| {}^{-n}\)\). \] So, if \( \(\sigma \(\left( {\operatorname{tr}{f}^{n}}\right) }\) is bounded, then \(\(\left| \(\mu \(\lambda\) }\right| = 1\) so that \(\(\sigma \(\left( {\operatorname{tr}f}\right) = \(\mu \(\lambda\) + \(\mu {\(\lambda\)}^{-1}\) is a real number in the interval \(\(\left\lbrack {-2,2}\)\). Now choose an irreducible subgroup \(\(\langle {{g}_{1},{g}_{2}}\rangle of \(\({\Gamma }^{\(\2\)} such that \(\({g}_{1}\) is not parabolic. Then \[ \(\({A\(\Gamma\) } \(\cong (\frac{{\operatorname{tr}}^{2}{g}_{1}(\({\operatorname{tr}{}^{2}{g}_{1} - 4}\),\operatorname{tr}(\langle {{g}_{1},{g}_{2}}\rangle - 2}{{k\(\Gamma\) }}\)\) \] by (3.38). Since \(\(\sigma (\({\operatorname{tr}f}\) in (-2,2}\) for all f\), it follows that A\(\Gamma\) is ramified at all real places (see Theorem 2.5.1).

Example 10.5.4. An almost greedy basis that is not greedy. Aside from being quasi-greedy, the basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in Example 10.2.9 is democratic. Indeed, if \( \left| A\right| = m \), then \[ {\left( \mathop{\sum }\limits_{{n \in A}}1\right) }^{1/2} = {m}^{1/2} \] while \[ \mathop{\sum }\limits_{{n \in A}}\frac{1}{\sqrt{n}} \leq \mathop{\sum }\limits_{{n = 1}}^{m}\frac{1}{\sqrt{n}} \leq {\int }_{0}^{m}\frac{dx}{\sqrt{x}} = 2{m}^{1/2}. \] Hence, \[ {m}^{1/2} \leq \begin{Vmatrix}{\mathop{\sum }\limits_{{n \in A}}{e}_{n}}\end{Vmatrix} \leq 2{m}^{1/2} \] Thus \( \mathcal{B} \) is almost greedy. It cannot be greedy, because it is not unconditional. The next characterization of almost greedy bases by Dilworth et al. [68] says that the greedy algorithm for almost greedy bases is essentially the best if one allows a small percentage increase in \( m \), so the terminology is justified. Theorem 10.5.5. Suppose \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) is a basis in a Banach space \( X \) . The following conditions are equivalent: (a) \( \mathcal{B} \) is almost greedy. (b) For every \( \lambda > 1 \) there exists a constant \( {\mathrm{C}}_{\lambda } \) such that \[ \begin{Vmatrix}{x - {\mathcal{G}}_{\lceil {\lambda m}\rceil }\left( x\right) }\end{Vmatrix} \leq {\mathrm{C}}_{\lambda }{\sigma }_{m}\left( x\right) ,\;\forall m \in \mathbb{N},\forall x \in X. \] (c) For some \( \lambda > 1 \) there exists a constant \( \mathrm{C} \) such that \[ \begin{Vmatrix}{x - {\mathcal{G}}_{\lceil {\lambda m}\rceil }\left( x\right) }\end{Vmatrix} \leq \mathrm{C}{\sigma }_{m}\left( x\right) ,\forall m \in \mathbb{N},\forall x \in X. \] (10.19) Proof. To show \( \left( a\right) \Rightarrow \left( b\right) \) we need a lemma that roughly speaking tells us that the gap between \( \widetilde{\sigma } \) and \( \sigma \) depends on the proximity between the democracy functions of the basis. Lemma 10.5.6. Suppose \( \mathcal{B} \) is quasi-greedy. Then, for all \( m, r \in \mathbb{N} \) , \[ {\widetilde{\sigma }}_{m + r}\left( x\right) \leq \left( {1 + {\mathrm{C}}_{\mathrm{{qg}}} + {16}{\mathrm{C}}_{\mathrm{{qg}}}^{5}\frac{{\varphi }_{u}\left( m\right) }{{\varphi }_{l}\left( r\right) }}\right) {\sigma }_{m}\left( x\right) ,\;\forall x \in X. \] (10.20) Proof. Take \( y \in {\sum }_{m} \) and let \( A = \operatorname{supp}\left( y\right) \) . Consider \( z = x - y \) . Pick \( {G}_{r}\left( z\right) = {P}_{B}\left( z\right) \) a greedy sum of \( z \) of order \( r \) . Since \( \left| A\right| \leq m \) and \( \left| B\right| = r \), there is a set \( E \subset \mathbb{N} \) with \( A \cup B \subset E \) and \( \left| E\right| = m + r \) . Let us write \[ x - {P}_{E}\left( x\right) = {P}_{{E}^{c}}\left( x\right) = {P}_{{E}^{c}}\left( z\right) = z - {P}_{B}\left( z\right) - {P}_{E \smallsetminus B}\left( z\right) \] to obtain \[ {\widetilde{\sigma }}_{m + r}\left( x\right) \leq \begin{Vmatrix}{x - {P}_{E}\left( x\right) }\end{Vmatrix} \leq \parallel z\parallel + \begin{Vmatrix}{{P}_{B}\left( z\right) }\end{Vmatrix} + \begin{Vmatrix}{{P}_{E \smallsetminus B}\left( z\right) }\end{Vmatrix}. \] Since \( \max \left\{ {\left| {{e}_{n}^{ * }\left( z\right) }\right| : n \in E \smallsetminus B}\right\} \leq \min \left\{ {\left| {{e}_{n}^{ * }\left( z\right) }\right| : n \in B}\right\} \), applying (10.13) gives \[ \begin{Vmatrix}{{P}_{E \smallsetminus B}\left( z\right) }\end{Vmatrix} \leq {16}{\mathrm{C}}_{\mathrm{{qg}}}^{4}\frac{{\varphi }_{u}\left( m\right) }{{\varphi }_{l}\left( r\right) }\begin{Vmatrix}{{P}_{B}\left( z\right) }\end{Vmatrix}. \] By (10.3), \( \begin{Vmatrix}{{P}_{B}\left( z\right) }\end{Vmatrix} \leq {\mathrm{C}}_{\mathrm{{qg}}}\parallel z\parallel \) . Combining, and taking the infimum on \( y \), we obtain the desired result. Now assume that \( \mathcal{B} \) is almost greedy, hence quasi-greedy and democratic by Theorem 10.5.3. Given \( x \in X \) and \( m \in \mathbb{N} \), let \( r = \lceil {\lambda m}\rceil - m \) . By Lemma 10.3.4, \[ \frac{{\varphi }_{u}\left( m\right) }{{\varphi }_{l}\left( r\right) } \leq {\mathrm{C}}_{\mathrm{d}}\frac{{\varphi }_{u}\left( m\right) }{{\varphi }_{u}\left( r\right) } \leq {\mathrm{C}}_{\mathrm{d}}\max \left\{ {{\mathrm{K}}_{\mathrm{b}},\frac{m}{r}}\right\} \leq {\mathrm{C}}_{\mathrm{d}}\max \left\{ {{\mathrm{K}}_{\mathrm{b}},\frac{1}{\lambda - 1}}\right\} . \] Therefore, using Lemma 10.5.6 we have \[ \begin{Vmatrix}{x - {\mathcal{G}}_{\lceil {\lambda m}\rceil }\left( x\right) }\end{Vmatrix} \leq {\mathrm{C}}_{\mathrm{{ag}}}{\mathrm{C}}_{\mathrm{d}}\left( {1 + {\mathrm{C}}_{\mathrm{{qg}}} + {16}{\mathrm{C}}_{\mathrm{{qg}}}^{5}}\right) \max \left\{ {{\mathrm{C}}_{\mathrm{{qg}}},\frac{1}{\lambda - 1}}\right\} {\sigma }_{m}\left( x\right) , \] i.e., (b) holds. \( \left( b\right) \Rightarrow \left( c\right) \) is trivial. Assume \( \left( c\right) \) holds. We infer (see Problem 10.4) that in fact, \[ \begin{Vmatrix}{x - {G}_{\lceil {\lambda m}\rceil }\left( x\right) }\end{Vmatrix} \leq \mathrm{C}{\sigma }_{m}\left( x\right) ,\;\forall m \in \mathbb{N},\forall x \in X, \] (10.21) for every greedy sum \( {G}_{\lceil {\lambda m}\rceil }\left( x\right) \) of \( x \) . Then, since \( {\sigma }_{m}\left( x\right) \leq \parallel x\parallel \), we have \[ \begin{Vmatrix}{{G}_{\lceil {\lambda m}\rceil }\left( x\right) }\end{Vmatrix} \leq \left( {1 + \mathrm{C}}\right) \parallel x\parallel ,\;\forall m \in \mathbb{N},\;\forall x \in X. \] Let \( {G}_{r}\left( x\right) \) be a greedy sum of \( x \) . Pick \( m \in \mathbb{N} \cup \{ 0\} \) such that \( \lceil {\lambda m}\rceil \leq r < \lceil \lambda \left( {m + 1}\right) \rceil \) . Observe that \( r - \lceil {\lambda m}\rceil \leq \lceil \lambda \left( {m + 1}\right) \rceil - \lceil {\lambda m}\rceil - 1 \leq \lambda \) . Therefore, with the notation introduced in (10.11), \[ \begin{Vmatrix}{{G}_{r}\left( x\right) }\end{Vmatrix} \leq \begin{Vmatrix}{{G}_{\lceil {\lambda m}\rceil }\left( x\right) }\end{Vmatrix} + \begin{Vmatrix}{{G}_{\lceil {\lambda m}\rceil }\left( x\right) - {G}_{r}\left( x\right) }\end{Vmatrix} \leq \left( {1 + \mathrm{C} + {\mathrm{k}}_{\lfloor \lambda \rfloor }}\right) \parallel x\parallel . \] Hence \( \mathcal{B} \) is quasi-greedy by Theorem 10.2.3. To show that \( \mathcal{B} \) is democratic, suppose \( \left| A\right| = m \) and \( \left| B\right| = \lceil {\lambda m}\rceil \) . Choose a set \( E \) with \( \left| E\right| = m + \lceil {\lambda m}\rceil \) such that \( A \cup B \subset E \subset \mathbb{N} \) and consider \( x = \mathop{\sum }\limits_{{n \in E}}{e}_{n} \) . Then \( \mathop{\sum }\limits_{{n \in E \smallsetminus A}}{e}_{n} \) is a greedy sum of \( x \) of order \( \lceil {\lambda m}\rceil \), while \( \mathop{\sum }\limits_{{n \in E \smallsetminus B}}{e}_{n} \in {\bar{\sum }}_{m} \) . Therefore, using (10.21), we obtain \[ \begin{Vmatrix}{\mathop{\sum }\limits_{{n \in A}}{e}_{n}}\end{Vmatrix} = \begin{Vmatrix}{x - \mathop{\sum }\limits_{{n \in E \smallsetminus A}}{e}_{n}}\end{Vmatrix} \leq \mathrm{C}\begin{Vmatrix}{x - \mathop{\sum }\limits_{{n \in E \smallsetminus B}}{e}_{n}}\end{Vmatrix} = \mathrm{C}\begin{Vmatrix}{\mathop{\sum }\limits_{{n \in B}}{e}_{n}}\end{Vmatrix}. \] (10.22) Maximizing over \( A \) and minimizing over \( B \) yields \( {\varphi }_{u}\left( m\right) \leq \mathrm{C}{\varphi }_{l}\left( {\lceil {\lambda m}\rceil }\right) \) . Given \( r \in \mathbb{N} \) such that \( \lceil \lambda \rceil \leq r \), pick out \( m \in \mathbb{N} \) such that \( \lceil {\lambda m}\rceil \leq r < \lceil \lambda \left( {m + 1}\right) \rceil \) . Since \( m < {\lambda m} \) , we have \( m < r \) . Moreover, \( r < \lambda \left( {m + 1}\right) \leq {2\lambda m} \) . Then, using Lemma 10.3.4, we have \[ {\varphi }_{u}\left( r\right) = r\frac{{\varphi }_{u}\left( r\right) }{r} \leq r\frac{{\varphi }_{u}\left( m\right) }{m} \leq \mathrm{C}\frac{r}{m}{\varphi }_{l}\left( {\lceil {\lambda m}\rceil }\right) \leq {\mathrm{{CK}}}_{\mathrm{b}}\frac{r}{m}{\varphi }_{l}\left( r\right) \leq {2\lambda }{\mathrm{{CK}}}_{\mathrm{b}}{\varphi }_{l}\left( r\right) . \] An appeal to Lemma 10.3.3 concludes the proof. ## 10.6 Greedy Bases and Duality In this section we approach the following question: given a basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) of a Banach space \( X \) with some greedy-like property, what can be said about the sequence of its biorthogonal functionals \( {\mathcal{B}}^{ * } \mathrel{\text{:=}} {\left( {e}_{n}^{ * }\right) }_{n = 1}^{\infty } \) in the dual space \( {X}^{ * } \) ? Let \( Z \) be the subspace of \( {X}^{ * } \) generated by \( {\mathcal{B}}^{ * } \) . By Proposition 3.2.1, \( {\mathcal{B}}^{ * } \) is a (seminormalized) basis of \( Z \) . Moreover, by Corollary 3.2.4 the basic sequence \( {\mathcal{B}}^{* * } \) in \( {Z}^{ * } \) is equivalent to the basis \( \mathcal{B} \) via the natural identification \[ {e}_{n}^{* * }\left( {x}^{ * }\right) = {x}^{ * }\left( {e}_{n}\right) ,\;\forall {x}^{ * } \in Z. \] (10.23) Let us start by analyzing the democracy of \( {\mathcal{B}}^{ * } \) . Set \( {\varphi }_{u}^{ * }\left( m\right) = {\varphi }_{u}\left\lbrack {{\mathcal{B}}^{ * }, Z}\right\rbrack \left( m\right) \) and \( {\varphi }_{l}^{ * }\left( m\right) = {\varphi }_{l}\left\lbrack {{\mathcal{B}}^{ * }, Z}\right\rbrack \left( m\right) \), for \( m \in \mathbb{N} \) . The elementary computation \[ m = \left( {\mathop{\sum }\limits_{{n \in A}}{e}_{n}^{ * }}\right) \left( {\mathop{\sum }\limits_{{n \in A}}{e}_{n}}\right) \leq \begin{Vmatrix}{\mathop{\sum }\limits_{{n \in A}}{e}_{n}^{ * }}\end{Vmatrix}\begin{Vmatrix}{\mathop{\sum }\limits_{{n \in A}}{e}_{n}}\end{Vmatrix}\text{ if }\left| A\right| = m \] (10.24) sheds some interesting information. To begin, we have \[ m \leq {\varphi }_{u}\left( m\right) {\varphi }_{u}^{ * }\left( m\right) ,

Example 10.5.4. An almost greedy basis that is not greedy. Aside from being quasi-greedy, the basis \( \mathcal{B} = {\left( {e}_{n}\right) }_{n = 1}^{\infty } \) in Example 10.2.9 is democratic. Indeed, if \( \left| A\right| = m \), then \[ {\left( \mathop{\sum }\limits_{{n \in A}}1\right) }^{1/2} = {m}^{1/2} \] while \[ \mathop{\sum }\limits_{{n \in A}}\frac{1}{\sqrt{n}} \leq \mathop{\sum }\limits_{{n = 1}}^{m}\frac{1}{\sqrt{n}} \leq {\int }_{0}^{m}\frac{dx}{\sqrt{x}} = 2{m}^{1/2}. \] Hence, \[ {m}^{1/2} \leq \begin{Vmatrix}{\mathop{\sum }\limits_{{n \in A}}{e}_{n}}\end{Vmatrix} \leq 2{m}^{1/2} \] Thus \( \mathcal{B} \) is almost greedy. It cannot be greedy, because it is not unconditional.

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Theorem 10.71 (Substitutivity of equivalence). Let \( \varphi ,\psi ,\chi \) be formulas and \( \alpha \in {}^{m}\operatorname{Rng}v \) . Suppose that if \( \beta \) occurs free in \( \varphi \) or in \( \psi \) but bound in \( \chi \) then \( \beta \in \left\{ {{\alpha }_{i} : i < m}\right\} \) . Let \( \theta \) be obtained from \( \chi \) by replacing zero or more occurrences of \( \varphi \) in \( \chi \) by \( \psi \) . Then \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {\chi \leftrightarrow \theta }\right) . \] Proof. We proceed by induction on \( \chi \) . We may assume that \( \theta \neq \chi \) . If \( \chi \) is atomic, then \( \chi = \varphi \) and \( \psi = \theta \) ; this case is trivial. Suppose \( \chi \) is \( \neg {\chi }^{\prime } \) . Then \( \theta \) is of the form \( \neg {\theta }^{\prime } \), and the induction hypothesis easily gives the desired result. The induction steps involving \( \mathbf{v} \) and \( \land \) are similar. Now suppose \( \chi = \forall \beta {\chi }^{\prime } \) . Then by the induction hypothesis, \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {{\chi }^{\prime } \leftrightarrow {\theta }^{\prime }}\right) . \] Note that \( \beta \) does not occur free in \( \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \) . Hence, using 10.61, we easily obtain \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {\forall \beta {\chi }^{\prime } \leftrightarrow \forall \beta {\theta }^{\prime }}\right) , \] as desired. Again note that implicit in 10.71 is the assertion that the expression \( \theta \) formed from \( \chi \) is again a formula; this is easily established. Now we introduce a notation which will be frequently used in the remainder of this book. Definition 10.72. Let \( \varphi \) be a formula, \( m \in \omega ,\sigma \in {}^{m} \) Trm. Choose \( k \) maximum such that \( {v}_{k} \) occurs in \( \varphi \) or in \( {\sigma }_{j} \) for some \( j < m, k = 0 \) if no variable occurs in \( \varphi \) or in any \( {\sigma }_{j} \) . Let \( {\alpha }_{0},\ldots ,{\alpha }_{n - 1} \) be a list of all variables which occur bound in \( \varphi \) but also occur in some \( {\sigma }_{j} \), with \( {\alpha }_{0} < \cdots < {\alpha }_{n - 1} \) in the natural order \( {v}_{0},{v}_{1},\ldots \) of the variables. Let \( \psi \) be the formula \[ {\operatorname{Subb}}_{v\left( {k + 1}\right) }^{⓪}{\operatorname{Subb}}_{v\left( {k + 2}\right) }^{⓫}\cdots {\operatorname{Subb}}_{v\left( {k + n}\right) }^{\alpha \left( {n - 1}\right) }\varphi , \] and let \( \varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \) be the formula obtained from \( \psi \) by simultaneously replacing all free occurrences of \( {v}_{0},\ldots ,{v}_{m - 1} \) by \( {\sigma }_{0},\ldots ,{\sigma }_{m - 1} \) respectively. The purpose of first forming \( \psi \) is to eliminate any possible clash of bound variables. The following two corollaries give the essential properties of this notation. Corollary 10.73. \( \; \vdash \forall {v}_{0}\cdots \forall {v}_{m - 1}\varphi \rightarrow \varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \) . Corollary 10.74. \( \;\mathrm{h}\varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \rightarrow \exists {v}_{0}\cdots \exists {v}_{m - 1}\varphi \) . Both of these corollaries are immediate consequences of earlier results, upon noticing that simultaneous substitution can be obtained by iterated ordinary substitution; in the notation of 10.72 , \[ \varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) = {\operatorname{Subf}}_{▞}^{v\left( {k + n + 1}\right) }{\operatorname{Subf}}_{▞}^{v\left( {k + n + 2}\right) }\cdots {\operatorname{Subf}}_{\sigma \left( {m - 1}\right) }^{v\left( {k + n + m}\right) } \] \[ {\operatorname{Subf}}_{v\left( {k + n + 1}\right) }^{v0}{\operatorname{Subf}}_{v\left( {k + n + 2}\right) }^{v1}\cdots {\operatorname{Subf}}_{v\left( {k + n + m}\right) }^{v\left( {m - 1}\right) }\psi . \] This fact is also useful in checking formally that this substitution notion is effective: Proposition 10.75. If \( x \) is the Gödel number of a formula \( \varphi, m \in \omega ,{y}_{0},\ldots \) , \( {y}_{m - 1} \) are Gödel numbers of terms \( {\sigma }_{0},\ldots ,{\sigma }_{m - 1} \) respectively, let \( s\left( {x,{y}_{0},\ldots }\right. \) , \( \left. {y}_{m - 1}\right) = {\mathcal{J}}^{ + }\varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \) ; if \( x \) and \( {y}_{0},\ldots ,{y}_{m - 1} \) do not satisfy these conditions, let \( s\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) = 0 \) . Then \( s \) is recursive. Proof. First we need a function picking out the integer \( k \) described in 10.72. For any \( x,{y}_{0},\ldots ,{y}_{m - 1} \in \omega \), let \[ g\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) = {\mu k}\left\lbrack \left( {\exists i \leq \operatorname{lx}\left\{ {\left( {{\left( x\right) }_{i} = \operatorname{gv}k + 1}\right) \text{ or }}\right. }\right. \right. \] \[ \mathop{\bigvee }\limits_{{j < m}}\exists i \leq \mathrm{l}{y}_{j}\left\lbrack {{\left( {y}_{j}\right) }_{i} = {gvk} + 1}\right\rbrack \} \text{ and }\forall i \leq \mathrm{l}x\lbrack {\left( x\right) }_{i} \div \] \[ \left. {1 \in \operatorname{Rng}\left( {g \circ v}\right) \Rightarrow {v}^{-1}{g}^{-1}\left( {{\left( x\right) }_{i} - 1}\right) \leq k}\right\rbrack \text{and} \] \[ \mathop{\bigwedge }\limits_{{j < m}}\forall i \leq 1{y}_{j}\left\lbrack {{\left( {y}_{j}\right) }_{i} - 1 \in \operatorname{Rng}\left( {g \circ v}\right) \Rightarrow {v}^{-1}{g}^{-1}\left( {{\left( {y}_{j}\right) }_{i} - 1}\right) \leq k}\right\rbrack ) \] \[ \text{or}(\forall i \leq \operatorname{lx}\left\lbrack {{\left( x\right) }_{i} \div 1 \notin \operatorname{Rng}\left( {g \circ v}\right) }\right. \text{and}\mathop{\bigwedge }\limits_{{j < m}}\forall i \leq \] \[ \left. \left. {\lg \left\lbrack {{\left( {y}_{j}\right) }_{i} \div 1 \notin \operatorname{Rng}\left( {g \circ v}\right) }\right\rbrack }\right) \right\rbrack \] Now let \( f \) be the function of 10.52, let \( {f}^{\prime } \) be the function of 10.57, and let \( S \) be the relation of 10.43. We now define a function \( {f}^{\prime \prime } \) which codes the set of variables which occur bound in \( \varphi \) but also occur in some \( {\sigma }_{j} \) : \[ {f}^{\prime \prime }\left( {x,{z}_{0},\ldots ,{z}_{m - 1}}\right) = {\mu y}\left( {{\left( y\right) }_{1y} = 1}\right. \text{and}\forall i \leq \lg (\exists j \leq \] \[ {lx}\left\lbrack {\left( {{\left( y\right) }_{i}, j, x}\right) \in S}\right\rbrack \; \land \;\mathop{\bigvee }\limits_{{j\; < \;m}}\exists u \leq {z}_{j}\;\exists w \leq {z}_{j}\lbrack {z}_{j} = \mathrm{{Cat}}\;(u,\mathrm{{Cat}} \] \[ \left( {{2}^{\left( y\right) i + 1}, w}\right) )\rbrack )\text{ and }\forall j \leq \mathrm{l}x\;\forall k \leq x\{ \left( {k, j, x}\right) \in S\text{ and } \] \[ \mathop{\bigvee }\limits_{{n < m}}\exists u \leq {z}_{n}\;\exists w \leq {z}_{n}\left\lbrack {{z}_{n} = \mathrm{{Cat}}\left( {u,\mathrm{{Cat}}\left( {{2}^{k + 1}, w}\right) }\right) }\right\rbrack \Rightarrow \exists i < \] \[ \left. {\lg \left\lbrack {{\left( y\right) }_{i} = k}\right\rbrack }\right\} \text{ and }\forall i, j < \lg \left\lbrack {i < j \Rightarrow {v}^{-1}{\mathcal{G}}^{-1}{\left( y\right) }_{i} < {v}^{-1}{\mathcal{G}}^{-1}{\left( y\right) }_{j}}\right\rbrack \text{. } \] Next we define a function \( {f}^{iv} \) yielding the formula \( \psi \) of 10.72, via an auxiliary function \( {f}^{m} \) : \[ {f}^{m}\left( {x,{y}_{0},\ldots ,{y}_{m - 1},0}\right) = x, \] \[ {f}^{\prime \prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}, i + 1}\right) = {f}^{\prime }\left( {\left( {f}^{\prime \prime }\left( x,{y}_{0},\ldots ,{y}_{m - 1}\right) \right) }_{i}\right. \text{,} \] \[ \operatorname{gv}\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + i + 1}\right) , \] \[\left. {{f}^{\prime \prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}, i}\right) }\right) \] and \[ {f}^{iv}\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) = {f}^{\prime \prime \prime }\left( {{x}_{0},{y}_{0},\ldots ,{y}_{m - 1}, l{f}^{\prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) }\right) . \] We then define a function \( {s}^{\prime } \) using the fact stated after 10.74: \[ {s}^{\prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) = f\left( {\mathcal{G}v\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1{f}^{\prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1}\right) }\right) , \] \[ {y}_{0}, f\left( {{gv}\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1{f}^{\prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 2}\right) ,{y}_{1},\ldots }\right) \] \[ f\left( {{gv}\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1{f}^{\prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + m,{y}_{m - 1}, f({gv0},}\right. }\right. \] \[ {gv}\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1{f}^{\prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1}\right), f({gv1}\text{,} \] \[ \left. {{gv}\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1{f}^{\prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 2}\right) ,\ldots, f({gv}\left( {m - 1}\right) }\right) \] \[ {gv}\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + 1{f}^{\prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + m}\right) ,{f}^{iv}\left( {x,{y}_{0},\ldots }\right. \text{,} \] \[ \left. \left. {y}_{m - 1}\right) \right) \cdots )\text{. } \] The desired function \( s \) is obtained from \( {s}^{\prime } \) by a simple and obvious definition by cases. Our next main result 10.81 concerns prenex normal form. Lemm

Theorem 10.71 (Substitutivity of equivalence). Let \( \varphi ,\psi ,\chi \) be formulas and \( \alpha \in {}^{m}\operatorname{Rng}v \) . Suppose that if \( \beta \) occurs free in \( \varphi \) or in \( \psi \) but bound in \( \chi \) then \( \beta \in \left\{ {{\alpha }_{i} : i < m}\right\} \) . Let \( \theta \) be obtained from \( \chi \) by replacing zero or more occurrences of \( \varphi \) in \( \chi \) by \( \psi \) . Then \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {\chi \leftrightarrow \theta }\right) . \]

We proceed by induction on \( \chi \) . We may assume that \( \theta \neq \chi \) . If \( \chi \) is atomic, then \( \chi = \varphi \) and \( \psi = \theta \) ; this case is trivial. Suppose \( \chi \) is \( \neg {\chi }^{\prime } \) . Then \( \theta \) is of the form \( \neg {\theta }^{\prime } \), and the induction hypothesis easily gives the desired result. The induction steps involving \( \mathbf{v} \) and \( \land \) are similar. Now suppose \( \chi = \forall \beta {\chi }^{\prime } \) . Then by the induction hypothesis, \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {{\chi }^{\prime }

Proposition 2.92. Let \( X \) and \( Z \) be Banach spaces, \( Z \) being finite-dimensional, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Z \) be Hadamard differentiable at \( a \in W \) , with \( \operatorname{Dg}\left( a\right) \left( X\right) = Z \) . Then there exist open neighborhoods \( U \) of a in \( W, V \) of \( g\left( a\right) \) in \( Z \) and a map \( h : V \rightarrow U \) that is differentiable at \( g\left( a\right) \) and such that \( h\left( {g\left( a\right) }\right) = a \) , \( g \circ h = {I}_{V} \) . In particular, \( g \) is open at a. Now let us turn to representations via parameterizations. We need the following result. Theorem 2.93 (Immersion theorem). Let \( P \) and \( X \) be Banach spaces, let \( O \) be an open subset of \( P \), and let \( f : O \rightarrow X \) be a map of class \( {C}^{k} \) with \( k \geq 1 \) such that for some \( \bar{p} \in O \) the map \( {Df}\left( \bar{p}\right) \) is injective and its image \( Y \) has a topological supplement \( Z \) in \( X \) . Then there exist open neighborhoods \( U \) of \( a \mathrel{\text{:=}} f\left( \bar{p}\right) \) in \( X, Q \) of \( \bar{p} \) in \( O, W \) of 0 in \( Z \) and a \( {C}^{k} \) -diffeomorphism \( \psi : V \mathrel{\text{:=}} Q \times W \rightarrow U \) such that \( \psi \left( {q,0}\right) = f\left( q\right) \) for all \( q \in Q \) . Again the conclusion can be written in the form of a commutative diagram, since \( f \mid Q = \psi \circ j \), where \( j : Q \rightarrow Q \times W \) is the canonical injection \( y \mapsto \left( {y,0}\right) \) . Again the nonlinear map \( f \) has been straightened by \( \psi \) into a linear map \( j = {\psi }^{-1} \circ \left( {f \mid Q}\right) \) . Proof. Let \( F : O \times Z \rightarrow X \) be given by \( F\left( {p, z}\right) = f\left( p\right) + z \) . Then \( F \) is of class \( {C}^{k} \) and \( {F}^{\prime }\left( {\bar{p},0}\right) \left( {p, z}\right) = {f}^{\prime }\left( \bar{p}\right) \left( p\right) + z \) for \( \left( {p, z}\right) \in P \times Z \), so that \( {F}^{\prime }\left( {\bar{p},0}\right) \) is an isomorphism from \( P \times Z \) onto \( Y + Z = X \) . The inverse mapping theorem asserts that \( F \) induces a \( {C}^{k} \) -diffeomorphism \( \psi \) from some open neighborhood of \( \left( {\bar{p},0}\right) \) onto some open neighborhood \( U \) of \( f\left( \bar{p}\right) \) . Taking a smaller neighborhood of \( \left( {\bar{p},0}\right) \) if necessary, we may suppose it has the form of a product \( Q \times W \) . Clearly, \( \psi \left( {q,0}\right) = f\left( q\right) \) for \( q \in Q \) . Example-Exercise. Let \( P \mathrel{\text{:=}} {\mathbb{R}}^{2}, O \mathrel{\text{:=}} \left( {-\pi ,\pi }\right) \times \left( {-\pi /2,\pi /2}\right), X \mathrel{\text{:=}} {\mathbb{R}}^{3} \), and let \( f \) be given by \( f\left( {\varphi ,\theta }\right) \mathrel{\text{:=}} \left( {\cos \theta \cos \varphi ,\cos \theta \sin \varphi ,\sin \varphi }\right) \) . Identify the image of \( f \) . Exercise. Let us note that the image \( f\left( O\right) \) of \( f \) is not necessarily a \( {C}^{k} \) -submanifold of \( X \) . Find a counterexample with \( P \mathrel{\text{:=}} \mathbb{R}, X \mathrel{\text{:=}} {\mathbb{R}}^{2} \) . A topological assumption ensures that the image \( f\left( O\right) \) is a \( {C}^{k} \) -submanifold of \( X \) . Corollary 2.94 (Embedding theorem). Let \( P \) and \( X \) be Banach spaces, let \( O \) be an open subset of \( P \), and let \( f : O \rightarrow X \) be a map of class \( {C}^{k} \) with \( k \geq 1 \) such that for every \( p \in O \) the map \( {f}^{\prime }\left( p\right) \) is injective and its image has a topological supplement in \( X \) . Then if \( f \) is a homeomorphism from \( O \) onto \( f\left( O\right) \), its image \( S \mathrel{\text{:=}} f\left( O\right) \) is a \( {C}^{k} \) -submanifold of \( X \) . Moreover, for every \( p \in O \) one has \( T\left( {S, f\left( p\right) }\right) = {f}^{\prime }\left( p\right) \left( P\right) \) . One says that \( f \) is an embedding of \( O \) into \( X \) and that \( S \) is parameterized by \( O \) . Proof. Given \( a \mathrel{\text{:=}} f\left( p\right) \) in \( S \), with \( p \in O \), we take \( {Q}_{a} \subset O,{U}_{a} \subset X,{W}_{a} \subset Z \) and a \( {C}^{k} \) -diffeomorphism \( {\psi }_{a} : {V}_{a} \mathrel{\text{:=}} {Q}_{a} \times {W}_{a} \rightarrow {U}_{a} \) such that \( {\psi }_{a}\left( {q,0}\right) = f\left( q\right) \) for all \( q \in {Q}_{a} \) as in the preceding theorem. Performing a translation in \( P \), we may suppose \( p = 0 \) . Using the assumption that \( f \) is a homeomorphism from \( O \) onto \( S = f\left( O\right) \), we can find an open subset \( {U}_{a}^{\prime } \) of \( X \) such that \( f\left( {Q}_{a}\right) = S \cap {U}_{a}^{\prime } \) . Let \( U \mathrel{\text{:=}} {U}_{a} \cap {U}_{a}^{\prime }, V \mathrel{\text{:=}} {\psi }_{a}^{-1}\left( U\right) ,\varphi \mathrel{\text{:=}} {\psi }_{a}^{-1} \mid U, Y \mathrel{\text{:=}} P \), so that \( \varphi \left( a\right) = \left( {0,0}\right) \) . Let us check relation (2.25), i.e., \( \varphi \left( {S \cap U}\right) = \left( {Y\times \{ 0\} }\right) \cap V \) . For all \( \left( {y,0}\right) \in \left( {Y\times \{ 0\} }\right) \cap V \) , we have \( x \mathrel{\text{:=}} {\varphi }^{-1}\left( {y,0}\right) = {\psi }_{a}\left( {y,0}\right) = f\left( y\right) \in S \), hence \( x \in S \cap U \) ; conversely, when \( x \in S \cap U = f\left( {Q}_{a}\right) \) there is a unique \( q \in {Q}_{a} \) such that \( x = f\left( q\right) \), so that \( x = {\psi }_{a}\left( {q,0}\right) = \) \( {\varphi }^{-1}\left( {q,0}\right) \) and \( \varphi \left( x\right) = \left( {q,0}\right) \in \left( {Y\times \{ 0\} }\right) \cap V \) . Then \( T\left( {S, a}\right) = T\left( {S \cap U, a}\right) = {\left( {\varphi }^{\prime }\left( a\right) \right) }^{-1}\left( {T\left( {\left( {Y\times \{ 0\} }\right) \cap V,0}\right) }\right) \), and, since \( T((Y \times \) \( \{ 0\} ) \cap V,0) = {\psi }_{a}^{\prime }\left( 0\right) \left( {P\times \{ 0\} }\right) = Y \times \{ 0\} \), we get \( T\left( {S, a}\right) = Y = {f}^{\prime }\left( p\right) \left( P\right) \) . ## Exercises 1. (Conic section) Let \( S \subset {\mathbb{R}}^{3} \) be defined by the equations \( {x}^{2} + {y}^{2} - 1 = 0, x - z = 0 \) . Show that \( S \) is a submanifold of \( {\mathbb{R}}^{3} \) of class \( {C}^{\infty } \) (it has been known since Apollonius that \( S \) is an ellipse). Find an explicit diffeomorphism (in fact linear isomorphism) sending \( S \) onto an ellipse of the plane \( {\mathbb{R}}^{2} \times \{ 0\} \) . 2. (Viviani’s window) Let \( S \) be the subset of \( {\mathbb{R}}^{3} \) defined by the system \( {x}^{2} + {y}^{2} = x \) , \( {x}^{2} + {y}^{2} + {z}^{2} - 1 = 0 \) . Show that \( S \) is a submanifold of \( {\mathbb{R}}^{3} \) of class \( {C}^{\infty } \) . 3. (The torus) Let \( r > s > 0 \), let \( O \mathrel{\text{:=}} \left( {0,{2\pi }}\right) \times \left( {0,{2\pi }}\right) \), and let \( f : O \rightarrow {\mathbb{R}}^{3} \) be given by \( f\left( {\alpha ,\beta }\right) = \left( {\left( {r + s\cos \beta }\right) \cos \alpha ,\left( {r + s\cos \beta }\right) \sin \alpha, s\sin \beta }\right) \) . Show that \( f \) is an embedding onto the torus \( \mathbb{T} \) deprived from its greatest circle and from the set \( \mathbb{T} \cap \left( {{\mathbb{R}}_{ + }\times \{ 0\} \times \mathbb{R}}\right) \), where \[ \mathbb{T} \mathrel{\text{:=}} \left\{ {\left( {x, y, z}\right) \in {\mathbb{R}}^{3} : {\left( \sqrt{{x}^{2} + {y}^{2}} - r\right) }^{2} + {z}^{2} = {s}^{2}}\right\} . \] 4. Using the submersion theorem, show that \( \mathbb{T} \) is a \( {C}^{\infty } \) -submanifold of \( {\mathbb{R}}^{3} \) . 5. (a) (Beltrami’s tractricoid) Let \( f : \mathbb{R} \rightarrow {\mathbb{R}}^{2} \) be given by \( f\left( t\right) \mathrel{\text{:=}} (1/\cosh t, t - \) tanh \( t \) ). Determine the points of \( T \mathrel{\text{:=}} f\left( \mathbb{R}\right) \) that are smooth. (b) (Beltrami’s pseudosphere) Let \( g\left( {s, t}\right) \mathrel{\text{:=}} \left( {\cos s/\cosh t,\sin s/\cosh t, t - \tanh t}\right) \) . Determine the points of \( S \mathrel{\text{:=}} g\left( {\mathbb{R}}^{2}\right) \) that are smooth. They form a surface of (negative) constant Gaussian curvature. It can serve as a model for hyperbolic geometry. 6. Study the Roman surface of equation \( {x}^{2}{y}^{2} + {y}^{2}{z}^{2} + {z}^{2}{x}^{2} - {xyz} = 0 \) . Consider its parameterization \( \left( {\theta ,\varphi }\right) \mapsto \left( {\cos \theta \cos \varphi \sin \varphi ,\sin \theta \cos \varphi \sin \varphi ,\cos \theta \sin \theta {\cos }^{2}\varphi }\right) \) . 7. Study the cross-cap surface \( \{ \left( {1 + \cos v}\right) \cos u,\left( {1 + \cos v}\right) \sin u,\tanh \left( {u - \pi }\right) \sin v) \) : \( \left( {u, v}\right) \in \left\lbrack {0,{2\pi }}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \} \) and compare it with the self-intersecting disk, the image of \( \left\lbrack {0,{2\pi }}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \) by the parameterization \( \left( {u, v}\right) \mapsto \left( {v\cos {2u}, v\sin {2u}, v\cos u}\right) \) . 8. Study Whitney’s umbrella \( \left\{ {\left( {{uv}, u,{v}^{2}}\right) : \left( {u, v}\right) \in {\mathbb{R}}^{2}}\right\} \) . Check that it is determined by the equation \( {x}^{2} - {y}^{2}z = 0 \) . Such a surface is of interest in the theory of singularities. For this surface or the preceding one, make some drawings if you can or find some on the Internet. 9. Let \( O \mathrel{\text{:=}} \left( {0,1}\right) \cup \left( {1,\infty }\right) \subset \mathbb{R}, f : O \rightarrow {\mathbb{R}}^{2} \) being given by \( f\left( t\right) = \left( {t + {t}^{-1},{2t} + {t}^{-2}}\right) \) . Show that \( f \) is an embedding, but that its continuous extension to \( \left( {0, + \infty }\r

Proposition 2.92. Let \( X \) and \( Z \) be Banach spaces, \( Z \) being finite-dimensional, let \( W \) be an open subset of \( X \), and let \( g : W \rightarrow Z \) be Hadamard differentiable at \( a \in W \) , with \( \operatorname{Dg}\left( a\right) \left( X\right) = Z \) . Then there exist open neighborhoods \( U \) of a in \( W, V \) of \( g\left( a\right) \) in \( Z \) and a map \( h : V \rightarrow U \) that is differentiable at \( g\left( a\right) \) and such that \( h\left( {g\left( a\right) }\right) = a \) , \( g \circ h = {I}_{V} \) . In particular, \( g \) is open at a.

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Exercise 2.25. Let \( {x}_{1},\ldots ,{x}_{n} \) be atomic propositions and let \( A \) and \( B \) be two logic statements in CNF. The logic statement \( A \Rightarrow B \) is satisfied if any truth assignment that satisfies \( A \) also satisfies \( B \) . Prove that \( A \Rightarrow B \) is satisfied if and only if the logic statement \( \neg A \vee B \) is satisfied. Exercise 2.26. Consider a 0,1 set \( S \mathrel{\text{:=}} \left\{ {x \in \{ 0,1{\} }^{n} : {Ax} \leq b}\right\} \) where \( A \in {\mathbb{R}}^{m \times n} \) and \( b \in {\mathbb{R}}^{m} \) . Prove that \( S \) can be written in the form \( S = \{ x \in \) \( \left. {\{ 0,1{\} }^{n} : {Dx} \leq d}\right\} \) where \( D \) is a matrix all of whose entries are \( 0, + 1 \) or -1 (Matrices \( D \) and \( A \) may have a different number of rows). Exercise 2.27 (Excluding \( \left( {0,1}\right) \) -vectors). Find integer linear formulations for the following integer sets (Hint: Use the generalized set covering inequalities). - The set of all \( \left( {0,1}\right) \) -vectors in \( {\mathbb{R}}^{4} \) except \( \left( \begin{array}{l} 0 \\ 1 \\ 1 \\ 0 \end{array}\right) \) . - The set of all \( \left( {0,1}\right) \) -vectors in \( {\mathbb{R}}^{6} \) except \( \left( \begin{array}{l} 0 \\ 1 \\ 1 \\ 0 \\ 1 \\ 1 \end{array}\right) \left( \begin{array}{l} 0 \\ 1 \\ 0 \\ 1 \\ 1 \\ 0 \end{array}\right) \left( \begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}\right) \) . - The set of all \( \left( {0,1}\right) \) -vectors in \( {\mathbb{R}}^{6} \) except all the vectors having exactly two 1s in the first 3 components and one 1 in the last 3 components. - The set of all \( \left( {0,1}\right) \) -vectors in \( {\mathbb{R}}^{n} \) with an even number of \( 1\mathrm{\;s} \) . - The set of all \( \left( {0,1}\right) \) -vectors in \( {\mathbb{R}}^{n} \) with an odd number of 1 ’s. Exercise 2.28. Show that if \( P = \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) is such that \( P \cap {\mathbb{Z}}^{n} \) is the set of \( 0 - 1 \) vectors with an even number of 1’s, then \( {Ax} \leq b \) contains at least \( {2}^{n - 1} \) inequalities. Exercise 2.29. Given a Sudoku game and a solution \( \bar{x} \), formulate as an integer linear program the problem of certifying that \( \bar{x} \) is the unique solution. Exercise 2.30 (Crucipixel Game). Given a \( m \times n \) grid, the purpose of the game is to darken some of the cells so that in every row (resp. column) the darkened cells form distinct strings of the lengths and in the order prescribed by the numbers on the left of the row (resp. on top of the column). Two strings are distinct if they are separated by at least one white cell. For instance, in the figure below the tenth column must contain a string of length 6 followed by some white cells and then a sting of length 2 . The game consists in darkening the cells to satisfy the requirements.  - Formulate the game as an integer linear program. - Formulate the problem of certifying that a given solution is unique as an integer linear program. - Play the game in the figure. Exercise 2.31. Let \( P = \left\{ {{A}_{1}x \leq {b}_{1}}\right\} \) be a polytope and \( S = \left\{ {{A}_{2}x < {b}_{2}}\right\} \) . Formulate the problem of maximizing a linear function over \( P \smallsetminus S \) as a mixed 0,1 program. Exercise 2.32. Consider continuous variables \( {y}_{j} \) that can take any value between 0 and \( {u}_{j} \), for \( j = 1,\ldots, k \) . Write a set of mixed integer linear constraints to impose that at most \( \ell \) of the \( k \) variables \( {y}_{j} \) can take a nonzero value. [Hint: use \( k \) binary variables \( {x}_{j} \in \{ 0,1\} \) .] Either prove that your formulation is perfect, in the spirit of Proposition 2.6, or give an example showing that it is not. Exercise 2.33. Assume \( c \in {\mathbb{Z}}^{n}, A \in {\mathbb{Z}}^{m \times n}, b \in {\mathbb{Z}}^{m} \) . Give a polynomial transformation of the 0,1 linear program \( \max \;{cx} \) \[ {Ax} \leq b \] \[ x \in \{ 0,1{\} }^{n} \] into a quadratic program \[ \max \;{cx} - M{x}^{T}\left( {1 - x}\right) \] \[ {Ax} \leq b \] \[ 0 \leq x \leq 1, \] i.e., show how to choose the scalar \( M \) as a function of \( A, b \) and \( c \) so that an optimal solution of the quadratic program is always an optimal solution of the 0,1 linear program (if any).  The authors working on Chap. 2  Giacomo Zambelli at the US border. Immigration Officer: What is the purpose of your trip? Giacomo: Visiting a colleague; I am a mathematician. Immigration Officer: What do mathematicians do? Giacomo: Sit in a chair and think. ## Chapter 3 ## Linear Inequalities and Polyhedra The focus of this chapter is on the study of systems of linear inequalities \( {Ax} \leq b \) . We look at this subject from two different angles. The first, more algebraic, addresses the issue of solvability of \( {Ax} \leq b \) . The second studies the geometric properties of the set of solutions \( \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) of such systems. In particular, this chapter covers Fourier's elimination procedure, Farkas' lemma, linear programming, the theorem of Minkowski-Weyl, polarity, Carathéorory's theorem, projections and minimal representations of the set \( \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) . ## 3.1 Fourier Elimination The most basic question concerning a system of linear inequalities is whether or not it has a solution. Fourier [145] devised a simple method to address this problem. Fourier's method is similar to Gaussian elimination, in that it performs row operations to eliminate one variable at a time. Let \( A \in {\mathbb{R}}^{m \times n} \) and \( b \in {\mathbb{R}}^{m} \), and suppose we want to determine if the system \( {Ax} \leq b \) has a solution. We first reduce this question to one about a system with \( n - 1 \) variables. Namely, we determine necessary and sufficient conditions for which, given a vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \in {\mathbb{R}}^{n - 1} \), there exists \( {\bar{x}}_{n} \in \mathbb{R} \) such that \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n}}\right) \) satisfies \( {Ax} \leq b \) . Let \( I \mathrel{\text{:=}} \{ 1,\ldots, m\} \) and define \[ {I}^{ + } \mathrel{\text{:=}} \left\{ {i \in I : {a}_{in} > 0}\right\} ,\;{I}^{ - } \mathrel{\text{:=}} \left\{ {i \in I : {a}_{in} < 0}\right\} ,\;{I}^{0} \mathrel{\text{:=}} \left\{ {i \in I : {a}_{in} = 0}\right\} . \] (C) Springer International Publishing Switzerland 2014 M. Conforti et al., Integer Programming, Graduate Texts in Mathematics 271, DOI 10.1007/978-3-319-11008-0_3 Dividing the \( i \) th row by \( \left| {a}_{in}\right| \) for each \( i \in {I}^{ + } \cup {I}^{ - } \), we obtain the following system, which is equivalent to \( {Ax} \leq b \) : \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}^{\prime }{x}_{j}\; + {x}_{n}\; \leq {b}_{i}^{\prime },\;i \in {I}^{ + } \] \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}^{\prime }{x}_{j}\; - {x}_{n}\; \leq {b}_{i}^{\prime },\;i \in {I}^{ - } \] (3.1) \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}{x}_{j}\; \leq {b}_{i},\;i \in {I}^{0} \] where \( {a}_{ij}^{\prime } = {a}_{ij}/\left| {a}_{in}\right| \) and \( {b}_{i}^{\prime } = {b}_{i}/\left| {a}_{in}\right| \) for \( i \in {I}^{ + } \cup {I}^{ - } \) . For each pair \( i \in {I}^{ + } \) and \( k \in {I}^{ - } \), we sum the two inequalities indexed by \( i \) and \( k \), and we add the resulting inequality to the system (3.1). Furthermore, we remove the inequalities indexed by \( {I}^{ + } \) and \( {I}^{ - } \) . This way, we obtain the following system: \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}\left( {{a}_{ij}^{\prime } + {a}_{kj}^{\prime }}\right) {x}_{j} \leq {b}_{i}^{\prime } + {b}_{k}^{\prime },\;i \in {I}^{ + }, k \in {I}^{ - }, \] (3.2) \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}{x}_{j} \leq {b}_{i},\;i \in {I}^{0}. \] If \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1},{\bar{x}}_{n}}\right) \) satisfies \( {Ax} \leq b \), then \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfies (3.2). The next theorem states that the converse also holds. Theorem 3.1. A vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfies the system (3.2) if and only if there exists \( {\bar{x}}_{n} \) such that \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1},{\bar{x}}_{n}}\right) \) satisfies \( {Ax} \leq b \) . Proof. We already remarked the "if" statement. For the converse, assume there is a vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfying (3.2). Note that the first set of inequalities in (3.2) can be rewritten as \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{kj}^{\prime }{x}_{j} - {b}_{k}^{\prime } \leq {b}_{i}^{\prime } - \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}^{\prime }{x}_{j},\;i \in {I}^{ + }, k \in {I}^{ - }. \] (3.3) Let \( l : = \mathop{\max }\limits_{{k \in {I}^{ - }}}\{ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{kj}^{\prime }{\bar{x}}_{j} - {b}_{k}^{\prime }\} \) and \( u : = \mathop{\min }\limits_{{i \in {I}^{ + }}}\{ {b}_{i}^{\prime } - \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}^{\prime }{\bar{x}}_{j}\} , \) where we define \( l \mathrel{\text{:=}} - \infty \) if \( {I}^{ - } = \varnothing \) and \( u \mathrel{\text{:=}} + \infty \) if \( {I}^{ + } = \varnothing \) . Since \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfies (3.3), we have that \( l \leq u \) . Therefore, for any \( {\bar{x}}_{n} \) such that \( l \leq {\bar{x}}_{n} \leq u \), the

Exercise 2.25. Let \( {x}_{1},\ldots ,{x}_{n} \) be atomic propositions and let \( A \) and \( B \) be two logic statements in CNF. The logic statement \( A \Rightarrow B \) is satisfied if any truth assignment that satisfies \( A \) also satisfies \( B \) . Prove that \( A \Rightarrow B \) is satisfied if and only if the logic statement \( \neg A \vee B \) is satisfied.

To prove that \( A \Rightarrow B \) is satisfied if and only if the logic statement \( \neg A \vee B \) is satisfied, we need to show two things: 1. If \( A \Rightarrow B \) is satisfied, then \( \neg A \vee B \) is satisfied. 2. If \( \neg A \vee B \) is satisfied, then \( A \Rightarrow B \) is satisfied. **Part 1: If \( A \Rightarrow B \) is satisfied, then \( \neg A \vee B \) is satisfied.** Assume \( A \Rightarrow B \) is satisfied. This means that for any truth assignment where \( A \) is true, \( B \) must also be true. Now consider the truth table for \( \neg A \vee B \): - If \( A \) is true, then since \( A \Rightarrow B \) is satisfied, \( B \) must be true. Therefore, \( \neg A \vee B \) evaluates to true because \( B \) is true. - If \( A \) is false, then \( \neg A \) is true, making \( \neg A \vee B \) true regardless of the value of \( B \). Thus, in both cases, \( \neg A \vee B \) is satisfied whenever \( A \Rightarrow B \) is satisfied. **Part 2: If \( \neg A \vee B \) is satisfied, then \( A \Rightarrow B \) is satisfied.** Assume \( \neg A \vee B \) is satisfied. This means that for any truth assignment, either \( \neg A \) is true or \( B \) is true (or both). Now consider the truth table for \( A \Rightarrow B \): - If \( A \) is true, then for \( \neg A \vee B \) to be true, \( B \) must be true (since \( \neg A \) would be false). Therefore, if \( A \) is true and \( B \) is true, then \( A \Rightarrow B \) evaluates to true. - If \( A \) is false, then regardless of the value of \( B \), the implication \( A \Rightarrow B \) evaluates to true because the antecedent (A) is false. Thus, in both cases, whenever

Theorem 11.1.1. Let \( G \) be a linear algebraic group. For every \( g \in G \) the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is a linear isomorphism from \( \operatorname{Lie}\left( G\right) \) onto \( T{\left( G\right) }_{g} \) . Hence \( G \) is a smooth algebraic set and \( \dim \operatorname{Lie}\left( G\right) = \dim G \) . Proof. We first show that for fixed \( g \in G \), the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is injective from \( \operatorname{Lie}\left( G\right) \) to \( T{\left( G\right) }_{g} \) . Suppose \( {\left( {X}_{A}\right) }_{g} = 0 \) . Then for \( x \in G \) and \( f \in \mathcal{O}\left\lbrack {\mathbf{{GL}}\left( V\right) }\right\rbrack \) we have \[ \left( {{X}_{A}f}\right) \left( x\right) = \left( {{X}_{A}f}\right) \left( {x{g}^{-1}g}\right) = \left( {L\left( {g{x}^{-1}}\right) \left( {{X}_{A}f}\right) }\right) \left( g\right) \] \[ = \left( {{X}_{A}\left( {L\left( {g{x}^{-1}}\right) f}\right) }\right) \left( g\right) = 0. \] Here we have used the left invariance of the vector field \( {X}_{A} \) on the second line. This shows that \( {X}_{A}f \in {\mathcal{I}}_{G} \) for all \( f \in \mathcal{O}\left\lbrack {\mathbf{{GL}}\left( V\right) }\right\rbrack \) . In particular, since \( I \in G \), we must have \( \left( {{X}_{A}f}\right) \left( I\right) = 0 \) for all regular functions \( f \) on \( \mathbf{{GL}}\left( V\right) \) . Hence \( A = 0 \) by Lemma 1.4.7. The dual space \( \mathcal{O}{\left\lbrack G\right\rbrack }^{ * } \) is naturally identified with the subspace of \( \mathcal{O}\left\lbrack {\mathbf{{GL}}\left( V\right) }\right\rbrack \) consisting of the linear functionals that vanish on \( {\mathcal{J}}_{G} \) . In particular, each tangent vector to \( G \) at \( g \) is also a tangent vector to \( \mathbf{{GL}}\left( V\right) \) at \( g \) . To show that the map from \( \operatorname{Lie}\left( G\right) \) to \( T{\left( G\right) }_{g} \) is surjective, it suffices by left invariance to take \( g = I \) . If \( v \in T{\left( G\right) }_{I} \) then by Lemma 1.4.7 there is a unique \( A \in \operatorname{End}\left( V\right) \) such that \( v = {v}_{A} \) . We claim that \( A \in \operatorname{Lie}\left( G\right) \) . Take \( f \in {\mathcal{J}}_{G} \) and \( g \in G \) . Then \[ \left( {{X}_{A}f}\right) \left( g\right) = \left( {L\left( {g}^{-1}\right) {X}_{A}f}\right) \left( I\right) = {X}_{A}\left( {L\left( {g}^{-1}\right) f}\right) \left( I\right) = v\left( {L\left( {g}^{-1}\right) f}\right) . \] But \( L\left( {g}^{-1}\right) f \in {\mathcal{I}}_{G} \) and \( v \) vanishes on \( {\mathcal{J}}_{G} \) . Hence \( \left( {{X}_{A}f}\right) \left( g\right) = 0 \) . This shows that \( {X}_{A}{\mathcal{J}}_{G} \subset {\mathcal{J}}_{G} \), proving that \( A \in \operatorname{Lie}\left( G\right) \) . Every affine algebraic set has a unique decomposition into irreducible components (see Section A.1.5). For the case of a linear algebraic group, this decomposition can be described as follows: Theorem 11.1.2. Let \( G \) be a linear algebraic group. Then \( G \) contains a unique subgroup \( {G}^{ \circ } \) that is closed, irreducible, and of finite index in \( G \) . Furthermore, \( {G}^{ \circ } \) is a normal subgroup and its cosets in \( G \) are both the irreducible components and the connected components of \( G \) relative to the Zariski topology. Proof. We show the existence of a subgroup \( {G}^{ \circ } \) with the stated properties as follows: Let \( G = {X}_{1} \cup \cdots \cup {X}_{r} \) be an incontractible decomposition of \( G \) into irreducible components (cf. Lemma A.1.12). We label them so that \( 1 \in {X}_{i} \) for \( 1 \leq i \leq p \) and \( 1 \notin {X}_{i} \) for \( p < i \leq r \) . We first prove that \( p = 1 \) . Indeed, the set \( {X}_{1} \times \cdots \times {X}_{p} \) is irreducible, by Lemma A.1.14. Let \[ \mu : {X}_{1} \times \cdots \times {X}_{p} \rightarrow G,\;\mu \left( {{x}_{1},\ldots ,{x}_{p}}\right) = {x}_{1}\cdots {x}_{p}. \] Then \( \mu \) is a regular map. Set \( {X}_{0} = \mu \left( {{X}_{1} \times \cdots \times {X}_{p}}\right) \) . The Zariski closure \( \overline{{X}_{0}} \) is irreducible by Lemma A.1.15. Also, \( {X}_{i} \subset {X}_{0} \) for \( 1 \leq i \leq p \), since \( 1 \in {X}_{j} \) for \( 1 \leq j \leq p \) . Hence \( {X}_{i} = {X}_{0} = \overline{{X}_{0}} \) for \( i = 1,\ldots, p \) by the irreducibility of \( {X}_{i} \) . This is possible only if \( p = 1 \) . Thus \[ 1 \in {X}_{1}\;\text{ and }\;1 \notin {X}_{i}\;\text{ for }i = 2,\ldots, r. \] (11.1) Let \( g \in G \) . Since left multiplication by \( g \) is a homeomorphism, the decomposition \( G = {gG} = g{X}_{1} \cup \cdots \cup g{X}_{r} \) is also incontractible. Hence by the uniqueness of such decompositions (cf. Lemma A.1.12), there is a permutation \( \sigma \left( g\right) \in {\mathfrak{S}}_{r} \) such that \( g{X}_{i} = {X}_{\sigma \left( g\right) i} \) . Clearly the map \( \sigma : G \rightarrow {\mathfrak{S}}_{r} \) is a group homomorphism. If \( \sigma \left( g\right) 1 = i \), then \( g = g \cdot 1 \in {X}_{i} \) . Conversely, if \( g \in {X}_{i} \), then \( 1 \in {g}^{-1}{X}_{i} = {X}_{\sigma \left( {g}^{-1}\right) i} \) . Hence \( \sigma \left( {g}^{-1}\right) i = 1 \) by (11.1). This shows that \[ {X}_{i} = \{ g \in G \mid \sigma \left( g\right) 1 = i\} . \] (11.2) In particular, \( {X}_{1} \) is a subgroup of \( G \) . For any \( g \in G \), the set \( g{X}_{1}{g}^{-1} \) is irreducible and contains 1. Hence \( g{X}_{1}{g}^{-1} = {X}_{1} \), so \( {X}_{1} \) is a normal subgroup. It is now clear from (11.2) that each \( {X}_{i} \) is a coset of \( {X}_{1} \) . So setting \( {G}^{ \circ } = {X}_{1} \), we obtain a subgroup with the properties stated in the theorem. To prove uniqueness, assume that \( {G}_{1} \) is a subgroup of finite index in \( G \) that is also a closed, irreducible subset. Let \( {x}_{i}{G}_{1} \), for \( i = 1,\ldots, n \), be the cosets of \( {G}_{1} \) . Since left multiplication is a homeomorphism, each coset is closed and irreducible. But \( G \) is the disjoint union of these cosets, so the cosets must be the irreducible components of \( G \) . In particular, \( {G}_{1} \) is the unique component of \( G \) that contains 1 . The complement of \( {G}_{1} \) in \( G \) is a finite union of components, so it is also closed. Hence \( {G}_{1} \) is also open in \( G \), and hence connected, since it is irreducible. In Section 2.2.3 an algebraic group \( G \) is defined to be connected if the ring \( \mathcal{O}\left\lbrack G\right\rbrack \) has no zero divisors. Here are two other equivalent definitions. Corollary 11.1.3. Let \( G \) be a linear algebraic group. The following are equivalent: 1. \( G \) is a connected topological space in the Zariski topology. 2. \( G \) is irreducible as an affine algebraic set. 3. The ring \( \mathcal{O}\left\lbrack G\right\rbrack \) has no zero divisors. Proof. Apply Theorem 11.1.2 and Lemma A.1.10. ## 11.1.2 Subgroups and Homomorphisms Let \( G \subset \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) be a linear algebraic group. An algebraic subgroup of \( G \) is a Zariski-closed subset \( H \subset G \) that is also a subgroup. The definition of a linear algebraic group in Section 1.4.1 implies that an algebraic subgroup \( H \) of \( G \) in this sense is also a linear algebraic group as defined there. Furthermore, the inclusion map \( \iota : H \subset G \) is regular and \( \mathcal{O}\left\lbrack H\right\rbrack \cong \mathcal{O}\left\lbrack G\right\rbrack /{\mathcal{I}}_{H} \) . Here \[ {\mathcal{J}}_{H} = \left\{ {f \in \mathcal{O}\left\lbrack G\right\rbrack : {\left. f\right| }_{H} = 0}\right\} \] In Section 1.4.1 we used this definition of \( {\mathcal{J}}_{H} \) only when \( G = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) . Since the regular functions on \( G \) are the restrictions to \( G \) of the regular functions on \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) , the definition of the ideal \( {\mathcal{J}}_{H} \) is unambiguous as long as the ambient group \( G \) is understood. Lemma 11.1.4. Let \( K \) be a subgroup of \( G \) . Then the closure \( \bar{K} \) of \( K \) in the Zariski topology is a group, and hence \( \bar{K} \) is an algebraic subgroup of \( G \) . Furthermore, if \( K \) contains a nonempty open subset of \( \bar{K} \) then \( K \) is closed in the Zariski topology. Proof. Let \( x \in K \) . Then \( K = {xK} \subset x\bar{K} \) . Since left multiplication by \( x \) is a homeomorphism in the Zariski topology, we know that \( x\bar{K} \) is closed. Hence \( \bar{K} \subset x\bar{K} \), giving \( {x}^{-1}\bar{K} \subset \bar{K} \) . Thus \( K \cdot \bar{K} \subset \bar{K} \) . Repeating this argument for \( x \in \bar{K} \), we conclude that \( \bar{K} \cdot \bar{K} \subset \bar{K} \) . Since inversion is a homeomorphism, it is clear that \( \bar{K} \) is stable under \( x \mapsto {x}^{-1} \) . Thus \( \bar{K} \) is a subgroup. Let \( U \subset K \) be Zariski open in \( \bar{K} \) and nonempty. Take \( x \in U \) and set \( V = {x}^{-1}U \) . Then \( 1 \in V \subset K \) and \( V \) is Zariski open in \( \bar{K} \) . Suppose \( y \in \bar{K} \smallsetminus K \) . Then \( {yV} \subset \bar{K} \smallsetminus K \) , since \( V \subset K \) and \( K \) is a group. Furthermore, \( {yV} \) is an open neighborhood of \( y \) in \( \bar{K} \) . Hence \( \bar{K} \smallsetminus K \) is open in \( \bar{K} \) . Hence \( K \) is Zariski closed in \( G \) . Regular homomorphisms of algebraic groups always have the following desirable properties: Theorem 11.1.5. Let \( \varphi : G \rightarrow H \) be a regular homomorphism of linear algebraic groups. Then \( F = \operatorname{Ker}\left( \varphi \right) \) is a closed subgroup of \( G \) and \( \varphi \left( G\right) \) is a closed subgroup of \( H \) . Hence \( \varphi \left( G\right) \) is an algebraic group. Furthermore, \( \varphi \left( {G}^{ \circ }\right)

Theorem 11.1.1. Let \( G \) be a linear algebraic group. For every \( g \in G \) the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is a linear isomorphism from \( \operatorname{Lie}\left( G\right) \) onto \( T{\left( G\right) }_{g} \) . Hence \( G \) is a smooth algebraic set and \( \dim \operatorname{Lie}\left( G\right) = \dim G \) .

We first show that for fixed \( g \in G \), the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is injective from \( \operatorname{Lie}\left( G\right) \) to \( T{\left( G\right) }_{g} \) . Suppose \( {\left( {X}_{A}\right) }_{g} = 0 \) . Then for \( x \in G \) and \( f \in \mathcal{O}\left\lbrack {\mathbf{{GL}}\left( V\right) }\right\rbrack \) we have \[ \left( {{X}_{A}f}\right) \left( x\right) = \left( {{X}_{A}f}\right) \left( {x{g}^{-1}g}\right) = \left( {L\left( {g{x}^{-1}}\right) \left( {{X}_{A}f}\right) }\right) \left( g\right) \] \[ = \left( {{X}_{A}\left( {L\left( {g{x}^{-1}}\right) f}\right) }\right) \left( g\right) = 0. \] Here we have used the left invariance of the vector field \( {X}_{A} \) on the second line. This shows that \( {X}_{A}f \in {\mathcal{I}}_{G} \) for all \( f \in \mathcal{O}\left\lbrack {\mathbf{{GL}}\left( V\right) }\right\rbrack \) . In particular, since \( I \in G \), we must have \( \left( {{X}_{A}f}\right) \left( I\right) = 0 \) for all regular functions \( f \) on \( \mathbf{{GL}}\left( V\right) \) . Hence \( A = 0 \) by Lemma 1.4.7. The dual space \( \mathcal{O}{\left\lbrack G\right\rbrack }^{ * } \) is naturally identified with the subspace of \( \mathcal{O}\left\lbrack {\mathbf{{GL}}\left( V\right) }\right\rbrack \) consisting of the linear functionals that vanish on \( {\mathcal{J}}_{G} \) . In particular, each tangent vector to \( G \) at \( g \) is also a tangent vector to \( \mathbf{{GL}}\left( V\right) \) at \( g \) . To show that the map from \( \operatorname{Lie}\left( G\right) \) to \( T{\left( G\right) }_{g} \) is surjective, it suffices by left invariance to take \( g = I \) . If \( v \in T{\left( G\right) }_{I} \) then by Lemma 1.4.7 there is a unique \( A \in \operatorname{End}\left( V\right) \) such that \( v = {v}_{A} \) . We claim that \( A \in \operatorname{Lie}\left( G\right)

Theorem 5.51. Suppose \( f\left( {e}^{i\theta }\right) = \mathop{\sum }\limits_{{-\infty }}^{\infty }{a}_{n}{e}^{in\theta } \) lies in \( W \) . If \( f \) does not vanish on \( T \) , then \( 1/f \) is also in \( W \), that is, there exist \( \left\{ {b}_{n}\right\} \) with \( \mathop{\sum }\limits_{{-\infty }}^{\infty }\left| {b}_{n}\right| < \infty \) and \[ \frac{1}{f\left( {e}^{i\theta }\right) } = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{b}_{n}{e}^{in\theta }. \] The statement of the theorem can also be phrased as follows: If a function \( f \) in \( C\left( T\right) \) has an absolutely convergent Fourier series and is nonzero on \( T \), then \( 1/f \) has an absolutely convergent Fourier series. Proof (Theorem 5.51). The hypothesis on \( f \) says that \( {\varphi }_{\lambda }\left( f\right) = f\left( \lambda \right) \) does not vanish as \( \lambda \) ranges over \( T \) . Since we have shown that the functionals \( {\varphi }_{\lambda } \) exhaust \( {\mathcal{M}}_{W} \), we may apply Corollary 5.29 to conclude that \( f \) is invertible in \( W \), which is the desired conclusion. In Exercise 5.37 you are asked to show that the Gelfand transform on the Wiener algebra is not isometric, and as a consequence \( W \) cannot be made into a \( {C}^{ * } \) -algebra. ## 5.7 The Continuous Functional Calculus Let’s recap where we are from the last section: Given a normal element \( N \) in a unital \( {C}^{ * } \) -algebra \( \mathcal{A} \), the \( {C}^{ * } \) -algebra \( {C}^{ * }\left( N\right) \) generated by \( N,{N}^{ * } \), and \( I \), which is the closure of the polynomials in \( N \) and \( {N}^{ * } \), is a commutative \( {C}^{ * } \) -algebra. Spectral permanence says \[ {\sigma }_{\mathcal{A}}\left( A\right) = {\sigma }_{{C}^{ * }\left( N\right) }\left( A\right) \] for any \( A \in {C}^{ * }\left( N\right) \), so the subscript can be omitted without danger of misinterpretation. Theorem 5.46 says there is a unique isometric \( * \) -isomorphism \( \gamma \) from \( {C}^{ * }\left( N\right) \) onto \( C\left( {\sigma \left( N\right) }\right) \) which pairs \( N \) with the identity function \( z \) on \( \sigma \left( N\right) \) . Under this \( * \) - isomorphism, \( {N}^{ * } \) and \( \bar{z} \) are paired, as are \( {cI} \) and the constant function \( c \), and a polynomial \( p\left( {N,{N}^{ * }}\right) \) is paired with the corresponding polynomial \( p\left( {z,\bar{z}}\right) \) . Now if \( f \) is any continuous function on \( \sigma \left( N\right) \), we define \( f\left( N\right) \) to be the corresponding element \( {\gamma }^{-1}\left( f\right) \) of \( {C}^{ * }\left( N\right) \) ; note this is in agreement with our previous observations when \( f \) is a polynomial in \( z \) and \( \bar{z} \) . Clearly if \( f, g \in C\left( {\sigma \left( N\right) }\right) \), so that \( f + g \) and \( {fg} \) are also, then \[ \left( {f + g}\right) \left( N\right) = f\left( N\right) + g\left( N\right) \] and \[ \left( {fg}\right) \left( N\right) = f\left( N\right) g\left( N\right) \] since \( {\gamma }^{-1} \) is linear and multiplicative. This assignment \( f \mapsto f\left( N\right) \) is called the continuous functional calculus or the functional calculus for normal elements. It is illustrated schematically below. \[ {C}^{ * }\left( N\right) \underset{{\gamma }^{-1}}{\overset{\gamma }{ \leftrightarrow }}C\left( {\sigma \left( N\right) }\right) \] \[ I \leftrightarrow 1 \] \[ N \leftrightarrow z \] \[ {N}^{ * } \leftrightarrow \bar{z} \] \[ {N}^{j}{N}^{*k} \leftrightarrow {z}^{j}{\bar{z}}^{k} \] \[ p\left( {N,{N}^{ * }}\right) \leftrightarrow p\left( {z,\bar{z}}\right) \] \[ f\left( N\right) \leftrightarrow f \] Making this definition of \( f\left( N\right) \) immediately raises the issue of how it behaves with respect to spectra. The next result-the full version of the spectral mapping theorem-is to be compared with Theorem 5.14. As observed above, there is no need to distinguish between \( {\sigma }_{\mathcal{A}} \) and \( {\sigma }_{{C}^{ * }\left( N\right) } \) in this result. Theorem 5.52. Suppose \( N \) is a normal element in a unital \( {C}^{ * } \) -algebra \( \mathcal{A} \) and let \( f \in C\left( {\sigma \left( N\right) }\right) \) . We have \( \sigma \left( {f\left( N\right) }\right) = f\left( {\sigma \left( N\right) }\right) \) . Proof. Since \( f \mapsto f\left( N\right) \equiv {\gamma }^{-1}\left( f\right) \) is a \( * \) -isomorphism of \( C\left( {\sigma \left( N\right) }\right) \) onto \( {C}^{ * }\left( N\right) \) we have \[ \sigma \left( {f\left( N\right) }\right) = {\sigma }_{C\left( {\sigma \left( N\right) }\right) }\left( f\right) = \text{ range }f = f\left( {\sigma \left( N\right) }\right) . \] As an application, recall that we previously observed (as a consequence of the spectral radius formula) that for a self-adjoint element \( A \) in a unital \( {C}^{ * } \) -algebra, \( r\left( A\right) = \parallel A\parallel \) . We can now obtain a deeper result, namely that the equality of the norm and spectral radius holds for any normal element. If \( N \) is normal in a unital \( {C}^{ * } \) -algebra, then by Theorem 5.46 there is an isometric \( * \) -isomorphism of \( {C}^{ * }\left( N\right) \) with \( C\left( {\sigma \left( N\right) }\right) \) which pairs \( N \) with \( z \) . Thus \[ \parallel N\parallel = \parallel z{\parallel }_{\infty, C\left( {\sigma \left( N\right) }\right) } = \max \{ \left| z\right| : z \in \sigma \left( N\right) \} = r\left( N\right) . \] Thus we have shown the following result. Theorem 5.53. For any normal element \( N \) of a unital \( {C}^{ * } \) -algebra, \( \parallel N\parallel = r\left( N\right) \) . Definition 5.54. A self-adjoint element \( A \) of a unital \( {C}^{ * } \) -algebra \( \mathcal{A} \) is said to be positive if its (necessarily real) spectrum is contained in \( \lbrack 0,\infty ) \) ; we write \( A \geq 0 \) in this case. The positive elements of \( \mathcal{A} \) are denoted \( {\mathcal{A}}_{ + } \) . For example, in the \( {C}^{ * } \) -algebra \( C\left( X\right) \), where \( X \) is a compact Hausdorff space, the positive elements are precisely the nonnegative functions. The functional calculus for normal operators plays a role in the next result on roots of positive elements. Theorem 5.55. If \( A \in {\mathcal{A}}_{ + } \) and \( n \in \mathbb{N} \), there is a unique \( B \in {\mathcal{A}}_{ + } \) satisfying \( {B}^{n} = A \) . Proof. We establish existence first. Since by assumption, \( \sigma \left( A\right) \subseteq \lbrack 0,\infty ) \), the real-valued function \( f\left( t\right) = {t}^{1/n} \) is continuous on \( \sigma \left( A\right) \) . Thus \( f\left( A\right) \) is defined by the functional calculus; set \( B = f\left( A\right) \) . Is \( B \) self-adjoint? Yes, since \( f \) is real-valued, hence self-adjoint as an element of \( C\left( {\sigma \left( A\right) }\right) \) . Moreover, by Theorem 5.52, \[ \sigma \left( B\right) = \sigma \left( {f\left( A\right) }\right) = f\left( {\sigma \left( A\right) }\right) \subseteq \lbrack 0,\infty ) \] so that \( B \) is positive. Finally, \( {B}^{n} = {f}^{n}\left( A\right) = A \), since \( {f}^{n} \) is the identity function. For the uniqueness statement, we first note that uniqueness is clear if \( \mathcal{A} = C\left( X\right) \) for some compact Hausdorff space \( X \) ; this is simply the statement that a nonnegative number has a unique nonnegative \( n \) th root. Now suppose, in a general setting, that \( B \) and \( C \) are positive \( n \) th roots of the positive element \( A \), with \( B = f\left( A\right) \) for \( f\left( t\right) = {t}^{1/n} \) . Note that \( C \) commutes with \( A \) since \( {CA} = C{C}^{n} = {C}^{n}C = {AC} \), hence \( C \) commutes with any polynomial in \( A \) . Now by the Weierstrass approximation theorem, \( f\left( t\right) = {t}^{1/n} \) is a uniform limit of a sequence of polynomials \( {p}_{n} \), on say \( \left\lbrack {0,\parallel A\parallel }\right\rbrack \) . Since \( B = \) \( f\left( A\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{p}_{n}\left( A\right) \), where each \( {p}_{n}\left( A\right) \) commutes with \( C \), we can conclude that \( B \) commutes with \( C \) . Consider \( \mathcal{B} = {C}^{ * }\left( {B, C}\right) \), the smallest unital \( {C}^{ * } \) -algebra containing the self-adjoint elements \( B \) and \( C \) . It contains \( A = {B}^{n} \), and by the above argument is easily seen to be commutative, so that the Gelfand transform \( \Gamma \) is an isometric \( * \) -isomorphism of \( \mathcal{B} \) onto \( C\left( {\mathcal{M}}_{\mathcal{B}}\right) \) . Since \( A, B \), and \( C \) are positive elements of \( \mathcal{B},\widehat{A},\widehat{B} \), and \( \widehat{C} \) are positive elements of \( C\left( {\mathcal{M}}_{\mathcal{B}}\right) \), i.e., they are nonnegative functions. We have \[ {\left( \widehat{B}\right) }^{n} = \widehat{A} = \widehat{{C}^{n}} = {\left( \widehat{C}\right) }^{n}, \] so that by our observations on uniqueness in \( C\left( {\mathcal{M}}_{\mathcal{B}}\right) \) we must have \( \widehat{B} = \widehat{C} \), and hence \( B = C \) as desired. We can use the definition of positive element to give an ordering on the selfadjoint elements of a unital \( {C}^{ * } \) -algebra. Definition 5.56. If \( A \) and \( B \) are self-adjoint elements of a unital \( {C}^{ * } \) -algebra we say \( A \leq B \) if \( B - A \) is positive. Two important facts about positive elements in a unital \( {C}^{ * } \) -algebra \( \mathcal{A} \) are as follows: - If \( A \in \mathcal{A} \), then \( {A}^{ * }A \geq 0 \) . - If \( A \in \mathcal{A} \) and \( B \in \mathcal{A} \) are positive, then so is \( A + B \) . The converse of the first property, that any positive element of an arbitrary unital \( {C}^{ * } \) -algebra \( \mathcal{A} \) is of the form \( {A}^{ * }A \) for some \( A \in \mathcal{A} \) is a

Theorem 5.51. Suppose \( f\left( {e}^{i\theta }\right) = \mathop{\sum }\limits_{{-\infty }}^{\infty }{a}_{n}{e}^{in\theta } \) lies in \( W \) . If \( f \) does not vanish on \( T \) , then \( 1/f \) is also in \( W \), that is, there exist \( \left\{ {b}_{n}\right\} \) with \( \mathop{\sum }\limits_{{-\infty }}^{\infty }\left| {b}_{n}\right| < \infty \) and \[ \frac{1}{f\left( {e}^{i\theta }\right) } = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{b}_{n}{e}^{in\theta }. \]

Proof (Theorem 5.51). The hypothesis on \( f \) says that \( {\varphi }_{\lambda }\left( f\right) = f\left( \lambda \right) \) does not vanish as \( \lambda \) ranges over \( T \) . Since we have shown that the functionals \( {\varphi }_{\lambda } \) exhaust \( {\mathcal{M}}_{W} \), we may apply Corollary 5.29 to conclude that \( f \) is invertible in \( W \), which is the desired conclusion.

Theorem 2.40. If the function \( f \) has continuous first partial derivatives in a neighborhood of \( c \) that satisfy the \( \mathrm{{CR}} \) equations at \( c \), then \( f \) is (complex) differentiable at \( c \) . Proof. The theorem is an immediate consequence of (2.12), since in this case \( {f}_{\bar{z}}\left( c\right) = 0 \) and hence \( {f}^{\prime }\left( c\right) = {f}_{z}\left( c\right) \) . Corollary 2.41. If the function \( f \) has continuous first partial derivatives in an open neighborhood \( U \) of \( c \in \mathbb{C} \) and the CR equations hold at each point of \( U \), then \( f \) is holomorphic at \( c \) (in fact on \( U \) ). Remark 2.42. The converse to this corollary is also true. It will take us some time to prove it. Theorem 2.43. If \( f \) is holomorphic and real-valued on a domain \( D \), then \( f \) is constant. Proof. As usual we write \( f = u + {uv} \) ; in this case \( v = 0 \) . The CR equations say \( {u}_{x} = {v}_{y} = 0 \) and \( {u}_{y} = - {v}_{x} = 0 \) . Thus \( u \) is constant, since \( D \) is connected. Theorem 2.44. If \( f \) is holomorphic and \( {f}^{\prime } = 0 \) on a domain \( D \), then \( f \) is constant. Proof. As above \( f = u + {\iota v} \) and \( {f}^{\prime } = {u}_{x} + \iota {v}_{x} = 0 \) . The last equation together with the CR equations say \( 0 = {u}_{x} = {v}_{y} \) and; \( 0 = {v}_{x} = - {u}_{y} \) . Thus both \( u \) and \( v \) are constant, since \( D \) is connected. ## Exercises 2.1. (a) Let \( \left\{ {z}_{n}\right\} \) be a sequence of complex numbers and assume \[ \left| {{z}_{n} - {z}_{m}}\right| < \frac{1}{1 + \left| {n - m}\right| },\text{ for all }n\text{ and }m. \] Show that the sequence converges. Do you have enough information to evaluate \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{z}_{n} \) ? What else can you say about this sequence? (b) Let \( \left\{ {z}_{n}\right\} \) be a sequence with \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{z}_{n} = 0 \) and let \( \left\{ {w}_{n}\right\} \) be a bounded sequence. Show that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{w}_{n}{z}_{n} = 0 \] 2.2. (a) Let \( z \) and \( c \) denote two complex numbers. Show that \[ {\left| \bar{c}z - 1\right| }^{2} - {\left| z - c\right| }^{2} = \left( {1 - {\left| z\right| }^{2}}\right) \left( {1 - {\left| c\right| }^{2}}\right) . \] (b) Use (a) to conclude that if \( c \) is any fixed complex number with \( \left| c\right| < 1 \), then \[ \{ z \in \mathbb{C};\left| {z - c}\right| < \left| {\bar{c}z - 1}\right| \} = \{ z \in \mathbb{C};\left| z\right| < 1\} , \] \[ \{ z \in \mathbb{C};\left| {z - c}\right| = \left| {\bar{c}z - 1}\right| \} = \{ z \in \mathbb{C};\left| z\right| = 1\} \text{and} \] \[ \{ z \in \mathbb{C};\left| {z - c}\right| > \left| {\bar{c}z - 1}\right| \} = \{ z \in \mathbb{C};\left| z\right| > 1\} . \] 2.3. Let \( a, b \), and \( c \) be three distinct points on a straight line with \( b \) between \( a \) and \( c \) . Show that \[ \frac{a - b}{c - b} \in {\mathbb{R}}_{ < 0} \] 2.4. (a) Given two points \( {z}_{1},{z}_{2} \) such that \( \left| {z}_{1}\right| < 1 \) and \( \left| {z}_{2}\right| < 1 \), show that for every point \( z \neq 1 \) in the closed triangle with vertices \( {z}_{1},{z}_{2} \), and 1, \[ \frac{\left| 1 - z\right| }{1 - \left| z\right| } \leq K \] where \( K \) is a constant that depends only on \( {z}_{1} \) and \( {z}_{2} \) . (b) Determine the smallest value of \( K \) for \( {z}_{1} = \frac{1 + \iota }{2} \) and \( {z}_{2} = \frac{1 - \iota }{2} \) . 2.5. Verify the Cauchy-Riemann equations for the function \( f\left( z\right) = {z}^{3} \) by splitting \( f \) into its real and imaginary parts. 2.6. Suppose \( z = x + \imath y \) . Define \[ f\left( z\right) = \frac{x{y}^{2}\left( {x + \imath y}\right) }{{x}^{2} + {y}^{4}} \] for \( z \neq 0 \) and \( f\left( 0\right) = 0 \) . Show that \[ \lim \frac{f\left( z\right) - f\left( 0\right) }{z} = 0 \] as \( z \rightarrow 0 \) along any straight line. Show that as \( z \rightarrow 0 \) along the curve \( x = {y}^{2} \), the limit of the difference quotient is \( \frac{1}{2} \), thus showing that \( {f}^{\prime }\left( 0\right) \) does not exist. 2.7. Let \( x = r\cos \theta \) and \( y = r\sin \theta \) . Show that the Cauchy-Riemann equations in polar coordinates for \( F = U + {\iota V} \), where \( U = U\left( {r,\theta }\right) \) and \( V = V\left( {r,\theta }\right) \), are \[ r\frac{\partial U}{\partial r} = \frac{\partial V}{\partial \theta }\text{ and }r\frac{\partial V}{\partial r} = - \frac{\partial U}{\partial \theta }, \] or, in alternate notation, \[ r{U}_{r} = {V}_{\theta }\text{ and }r{V}_{r} = - {U}_{\theta }. \] 2.8. Let \( f \) be a complex-valued function defined on a region in the complex plane, and assume that both \( {f}_{x} \) and \( {f}_{y} \) exist in this region. Using the definitions of \( {f}_{z} \) and \( {f}_{\bar{z}} \), show that for \( {\mathbf{C}}^{1} \) -functions \( f \) , \[ f\text{is holomorphic if and only if}{f}_{\bar{z}} = 0 \] and that in this case \( {f}_{z} = {f}^{\prime } \) . 2.9. Let \( R \) and \( \Phi \) be two real-valued \( {\mathbf{C}}^{1} \) -functions of a complex variable \( z \) . Show that \( f = R{\mathrm{e}}^{\iota \Phi } \) is holomorphic if and only if \[ {R}_{\bar{z}} + \imath R{\Phi }_{\bar{z}} = 0. \] 2.10. Show that if \( f \) and \( g \) are \( {\mathbf{C}}^{1} \) -functions, then the (complex) chain rule is expressed as follows (here \( w = f\left( z\right) \) and \( g \) is viewed as a function of \( w \) ). \[ {\left( g \circ f\right) }_{z} = {g}_{w}{f}_{z} + {g}_{\bar{w}}{\bar{f}}_{z} \] and \[ {\left( g \circ f\right) }_{\bar{z}} = {g}_{w}{f}_{\bar{z}} + {g}_{\bar{w}}{\bar{f}}_{\bar{z}} \] 2.11. Let \( p \) be a complex-valued polynomial of two real variables: \[ p\left( z\right) = \sum {a}_{ij}{x}^{i}{y}^{j} \] Write \[ p\left( z\right) = \mathop{\sum }\limits_{{j \geq 0}}{P}_{j}\left( z\right) {\bar{z}}^{j} \] where each \( {P}_{j} \) is of the form \( {P}_{j}\left( z\right) = \sum {b}_{ij}{z}^{i} \) (a polynomial in \( z \) ). Prove that \( p \) is an entire function if and only if \[ 0 \equiv {P}_{1} \equiv {P}_{2} \equiv \ldots \] What can you conclude in this case for the matrix \( \left\lbrack {a}_{ij}\right\rbrack \) ? 2.12. Deduce the analogues of the CR equations for anti-holomorphic functions, in rectangular, polar, and complex coordinates. 2.13. Let \( f : \mathbb{C} \rightarrow \mathbb{C} \) be a holomorphic function, and set \( g\left( z\right) = \bar{f}\left( \bar{z}\right) \) and \( h\left( z\right) = f\left( \bar{z}\right) \), for \( z \) in \( \mathbb{C} \) . Show that \( g \) is holomorphic and \( h \) is anti-holomorphic on \( \mathbb{C} \) . Furthermore, \( h \) is holomorphic on \( \mathbb{C} \) if and only if \( f \) is a constant function. 2.14. Let \( D \) be an arbitrary (nonempty) open connected set in \( \mathbb{C} \) . Describe the class of complex-valued functions on \( D \) that are both holomorphic and anti-holomorphic. 2.15. Does there exist a holomorphic function \( f \) on \( \mathbb{C} \) whose real part is: (a) \( u\left( {x, y}\right) = {\mathrm{e}}^{x} \) ? Or (b) \( u\left( {x, y}\right) = {\mathrm{e}}^{x}\left( {x\cos y - y\sin y}\right) \) ? Justify your answer; that is, if yes, exhibit the holomorphic function(s) and if not, prove it. 2.16. Prove the fundamental theorem of algebra: If \( {a}_{0},\ldots ,{a}_{n - 1} \) are complex numbers \( \left( {n \geq 1}\right) \) and \( p\left( z\right) = {z}^{n} + {a}_{n - 1}{z}^{n - 1} + \cdots + {a}_{0} \), then there exists a number \( {z}_{0} \in \mathbb{C} \) such that \( p\left( {z}_{0}\right) = 0 \) . Hint: A standard method of attack: (a) Show that there are an \( M > 0 \) and an \( R > 0 \) such that for all \( \left| z\right| \geq R,\left| {p\left( z\right) }\right| \geq \) \( M \) holds. (b) Show next that there is a \( {z}_{0} \in \mathbb{C} \) such that \[ \left| {p\left( {z}_{0}\right) }\right| = \min \{ \left| {p\left( z\right) }\right| ;z \in \mathbb{C}\} . \] (c) By the change of variable \( p\left( {z + {z}_{0}}\right) = g\left( z\right) \), it suffices to show that \( g\left( 0\right) = 0 \) . (d) Write \( g\left( z\right) = \alpha + {z}^{m}\left( {\beta + {c}_{1}z + \cdots + {c}_{n - m}{z}^{n - m}}\right) \) with \( \beta \neq 0 \) . Choose \( \gamma \) such that \[ {\gamma }^{m} = - \frac{\alpha }{\beta } \] If \( \alpha \neq 0 \), obtain the contradiction \( \left| {g\left( {\gamma z}\right) }\right| < \left| \alpha \right| \) for some \( z \) . Note. We will later have several simpler proofs of this theorem using results from complex analysis, for instance, in Theorem 5.16 and Exercise 6.1. See also the April 2006 issue of The American Mathematical Monthly for yet other proofs of this fundamental result. 2.17. Conclude from the fundamental theorem of algebra that a nonconstant complex polynomial of degree \( n \) has \( n \) complex roots, counted with multiplicities. Use this result to show that a nonconstant real polynomial that cannot be factored as a product of two nonconstant real polynomials of lower degree (i.e., a real irreducible nonconstant polynomial) has degree one or two. 2.18. Using the fundamental theorem of algebra stated in Exercise 2.16, prove the Frobenius theorem: If \( F \) is a field containing the reals whose dimension as a real vector space is finite, then either \( F \) is the reals or \( F \) is (isomorphic to) \( \mathbb{C} \) . Hint: An outline of possible steps follows. (a) Assume \( {\dim }_{\mathbb{R}}F = n > 1 \) . Show that for \( \theta \) in \( F - \mathbb{R} \) there exists a nonzero real polynomial \( p \) with leading coefficient 1 and such that \( p\left( \theta \right) = 0 \) . (b) Show that there exist real numbers \( \beta \) and \( \gamma \) such that \[ {\theta }^{2} - {2\beta \theta } + \gamma = 0. \] (c) Show that there exists

Theorem 2.40. If the function \( f \) has continuous first partial derivatives in a neighborhood of \( c \) that satisfy the \( \mathrm{{CR}} \) equations at \( c \), then \( f \) is (complex) differentiable at \( c \).

The theorem is an immediate consequence of (2.12), since in this case \( {f}_{\bar{z}}\left( c\right) = 0 \) and hence \( {f}^{\prime }\left( c\right) = {f}_{z}\left( c\right) \).

Theorem 2.7. Let \( \mathrm{R} \) be a ring and \( \mathrm{I} \) an ideal of \( \mathrm{R} \) . Then the additive quotient group \( \mathrm{R}/\mathrm{I} \) is a ring with multiplication given by \[ \left( {a + I}\right) \left( {b + I}\right) = {ab} + I \] If \( \mathbf{R} \) is commutative or has an identity, then the same is true of \( \mathbf{R}/\mathbf{I} \) . SKETCH OF PROOF OF 2.7. Once we have shown that multiplication in \( R/I \) is well defined, the proof that \( R/I \) is a ring is routine. (For example, if \( R \) has identity \( {1}_{R} \), then \( {1}_{R} + I \) is the identity in \( R/I \) .) Suppose \( a + I = {a}^{\prime } + I \) and \( b + I = {b}^{\prime } + I \) . We must show that \( {ab} + I = {a}^{\prime }{b}^{\prime } + I \) . Since \( {a}^{\prime }\varepsilon {a}^{\prime } + I = a + I \) , \( {a}^{\prime } = a + i \) for some \( i \in I \) . Similarly \( {b}^{\prime } = b + j \) with \( j \in I \) . Consequently \( {a}^{\prime }{b}^{\prime } = \left( {a + i}\right) \left( {b + j}\right) = {ab} + {ib} + {aj} + {ij} \) . Since \( I \) is an ideal, \[ {a}^{\prime }{b}^{\prime } - {ab} = {ib} + {aj} + {ij\varepsilon I}. \] Therefore \( {a}^{\prime }{b}^{\prime } + I = {ab} + I \) by Corollary I.4.3, whence multiplication in \( R/I \) is well defined. As one might suspect from the analogy with groups, ideals and homomorphisms of rings are closely related. Theorem 2.8. Iff \( : \mathrm{R} \rightarrow \mathrm{S} \) is a homomorphism of rings, then the kernel of \( \mathrm{f} \) is an ideal in R. Conversely if \( \mathrm{I} \) is an ideal in \( \mathrm{R} \), then the map \( \pi : \mathrm{R} \rightarrow \mathrm{R}/\mathrm{I} \) given by \( \mathrm{r} \mapsto \mathrm{r} + \mathrm{I} \) is an epimorphism of rings with kernel I. The map \( \pi \) is called the canonical epimorphism (or projection). PROOF OF 2.8. Ker \( f \) is an additive subgroup of \( R \) . If \( x \in \operatorname{Ker}f \) and \( r \in R \), then \( f\left( {rx}\right) = f\left( r\right) f\left( x\right) = f\left( r\right) 0 = 0 \), whence \( {rx\varepsilon }\operatorname{Ker}f \) . Similarly, \( {xr\varepsilon }\operatorname{Ker}f \) . Therefore, Ker \( f \) is an ideal. By Theorem I.5.5 the map \( \pi \) is an epimorphism of groups with kernel I. Since \( \pi \left( {ab}\right) = {ab} + I = \left( {a + I}\right) \left( {b + I}\right) = \pi \left( a\right) \pi \left( b\right) \) for all \( a, b \in R,\pi \) is also an epimorphism of rings. - In view of the preceding results it is not surprising that the various isomorphism theorems for groups (Theorems I.5.6-I.5.12) carry over to rings with normal subgroups and groups replaced by ideals and rings respectively. In each case the desired isomorphism is known to exist for additive abelian groups. If the groups involved are, in fact, rings and the normal subgroups ideals, then one need only verify that the known isomorphism of groups is also a homomorphism and hence an isomorphism of rings. Caution: in the proofs of the isomorphism theorems for groups all groups and cosets are written multiplicatively, whereas the additive group of a ring and the cosets of an ideal are written additively. Theorem 2.9. If \( \mathrm{f} : \mathrm{R} \rightarrow \mathrm{S} \) is a homomorphism of rings and \( \mathrm{I} \) is an ideal of \( \mathrm{R} \) which is contained in the kernel of \( \mathrm{f} \), then there is a unique homomorphism of rings \( \widehat{\mathrm{f}} : \mathrm{R}/\mathrm{I} \rightarrow \mathrm{S} \) such that \( \widetilde{\mathrm{f}}\left( {\mathrm{a} + \mathrm{I}}\right) = \mathrm{f}\left( \mathrm{a}\right) \) for all \( \mathrm{a} \in \mathrm{R},\operatorname{Im}\widetilde{\mathrm{f}} = \operatorname{Im}\mathrm{f} \) and \( \operatorname{Ker}\widetilde{\mathrm{f}} = \left( {\operatorname{Ker}\mathrm{f}}\right) /\mathrm{I} \) . \( \widetilde{\mathrm{f}} \) is an isomorphism if and only if \( \mathrm{f} \) is an epimorphism and \( \mathrm{I} = \operatorname{Ker}\mathrm{f} \) . # PROOF. Exercise; see Theorem I.5.6. Corollary 2.10. (First Isomorphism Theorem) If \( \mathrm{f} : \mathrm{R} \rightarrow \mathrm{S} \) is a homomorphism of rings, then \( \mathrm{f} \) induces an isomorphism of rings \( \mathrm{R}/\operatorname{Ker}\mathrm{f} \cong \operatorname{Im}\mathrm{f} \) . PROOF. Exercise; see Corollary I.5.7. Corollary 2.11. If \( \mathrm{f} : \mathrm{R} \rightarrow \mathrm{S} \) is a homomorphism of rings, I is an ideal in \( \mathrm{R} \) and \( \mathrm{J} \) is an ideal in \( \mathrm{S} \) such that \( \mathrm{f}\left( \mathrm{I}\right) \subset \mathrm{J} \), then \( \mathrm{f} \) induces a homomorphism of rings \( \overline{\mathrm{f}} : \mathrm{R}/\mathrm{I} \rightarrow \mathrm{S}/\mathrm{J} \) , given by \( \mathrm{a} + \mathrm{I} \mapsto \mathrm{f}\left( \mathrm{a}\right) + \mathrm{J} \) . \( \overline{\mathrm{f}} \) is an isomorphism if and only if \( \operatorname{Im}\mathrm{f} + \mathrm{J} = \mathrm{S} \) and \( {\mathrm{f}}^{-1}\left( \mathrm{\;J}\right) \subset \mathrm{I} \) . In particular, if \( \mathrm{f} \) is an epimorphism such that \( \mathrm{f}\left( \mathrm{I}\right) = \mathrm{J} \) and \( \operatorname{Ker}\mathrm{f} \subset \mathrm{I} \), then \( \overline{\mathbf{f}} \) is an isomorphism. PROOF. Exercise; see Corollary I.5.8. Theorem 2.12. Let \( \mathrm{I} \) and \( \mathrm{J} \) be ideals in a ring \( \mathrm{R} \) . (i) (Second Isomorphism Theorem) There is an isomorphisms of rings \( \mathrm{I}/\left( {\mathrm{I} \cap \mathrm{J}}\right) \cong \) \( \left( {\mathrm{I} + \mathrm{J}}\right) /\mathrm{J} \) (ii) (Third Isomorphism Theorem) if \( \mathrm{I} \subset \mathrm{J} \), then \( \mathrm{J}/\mathrm{I} \) is an ideal in \( \mathrm{R}/\mathrm{I} \) and there is an isomorphism of rings \( \left( {\mathrm{R}/\mathrm{I}}\right) /\left( {\mathrm{J}/\mathrm{I}}\right) \cong \mathrm{R}/\mathrm{J} \) . PROOF. Exercise; see Corollaries I.5.9 and I.5.10. Theorem 2.13. If \( \mathrm{I} \) is an ideal in a ring \( \mathrm{R} \), then there is a one-to-one correspondence between the set of all ideals of \( \mathrm{R} \) which contain \( \mathrm{I} \) and the set of all ideals of \( \mathrm{R}/\mathrm{I} \), given by \( \mathrm{J} \mapsto \mathrm{J}/\mathrm{I} \) . Hence every ideal in \( \mathrm{R}/\mathrm{I} \) is of the form \( \mathrm{J}/\mathrm{I} \), where \( \mathrm{J} \) is an ideal of \( \mathrm{R} \) which contains I. PROOF. Exercise; see Theorem I.5.11, Corollary I.5.12 and Exercise 13. Next we shall characterize in several ways two kinds of ideals (prime and maximal), which are frequently of interest. Definition 2.14. An ideal \( \mathrm{P} \) in a ring \( \mathrm{R} \) is said to be prime if \( \mathrm{P} \neq \mathrm{R} \) and for any ideals A, B in R \[ \mathrm{{AB}} \subset \mathrm{P} \Rightarrow \mathrm{A} \subset \mathrm{P}\text{ or }\mathrm{B} \subset \mathrm{P}\text{. } \] The definition of prime ideal excludes the ideal \( R \) for both historical and technical reasons. Here is a very useful characterization of prime ideals; other characterizations are given in Exercise 14. Theorem 2.15. If \( \mathrm{P} \) is an ideal in a ring \( \mathrm{R} \) such that \( \mathrm{P} \neq \mathrm{R} \) and for all \( \mathrm{a},\mathrm{b}\varepsilon \mathrm{R} \) \[ \mathrm{{ab}}\varepsilon \mathrm{P} \Rightarrow \mathrm{a}\varepsilon \mathrm{P}\text{ or }\mathrm{b}\varepsilon \mathrm{P}, \] (1) then \( \mathrm{P} \) is prime. Conversely if \( \mathrm{P} \) is prime and \( \mathrm{R} \) is commutative, then \( \mathrm{P} \) satisfies condition (1). REMARK. Commutativity is necessary for the converse (Exercise 9 (b)). PROOF OF 2.15. If \( A \) and \( B \) are ideals such that \( {AB} \subset P \) and \( A ⊄ P \), then there exists an element \( a \in A - P \) . For every \( b \in B,{ab} \in {AB} \subset P \), whence \( a \in P \) or \( b \in P \) . Since \( a \nmid P \), we must have \( b \in P \) for all \( b \in B \) ; that is, \( B \subset P \) . Therefore, \( P \) is prime. Conversely, if \( P \) is any ideal and \( {ab} \in P \), then the principal ideal \( \left( {ab}\right) \) is contained in \( P \) by Definition 2.4. If \( R \) is commutative, then Theorem 2.5 implies that \( \left( a\right) \left( b\right) \subset \left( {ab}\right) \), whence \( \left( a\right) \left( b\right) \subset P \) . If \( P \) is prime, then either \( \left( a\right) \subset P \) or \( \left( b\right) \subset P \) , whence \( a \in P \) or \( b \in P \) . EXAMPLES. The zero ideal in any integral domain is prime since \( {ab} = 0 \) if and only if \( a = 0 \) or \( b = 0 \) . If \( p \) is a prime integer, then the principal ideal \( \left( p\right) \) in \( \mathbf{Z} \) is prime since \[ {ab\varepsilon }\left( p\right) \Rightarrow p \mid {ab} \Rightarrow p \mid a\text{ or }p \mid b \Rightarrow {a\varepsilon }\left( p\right) \text{ or }{b\varepsilon }\left( p\right) . \] Theorem 2.16. In a commutative ring \( \mathrm{R} \) with identity \( {1}_{\mathrm{R}} \neq 0 \) an ideal \( \mathrm{P} \) is prime if and only if the quotient ring \( \mathrm{R}/\mathrm{P} \) is an integral domain. PROOF. \( R/P \) is a commutative ring with identity \( {1}_{R} + P \) and zero element \( 0 + P = P \) by Theorem 2.7. If \( P \) is prime, then \( {1}_{R} + P \neq P \) since \( P \neq R \) . Furthermore, \( R/P \) has no zero divisors since \[ \left( {a + P}\right) \left( {b + P}\right) = P \Rightarrow {ab} + P = P \Rightarrow {ab\varepsilon P} \Rightarrow {a\varepsilon P}\text{ or } \] \[ b \in P \Rightarrow a + P = P\text{ or }b + P = P. \] Therefore, \( R/P \) is an integral domain. Conversely, if \( R/P \) is an integral domain, then \( {1}_{R} + P \neq 0 + P \), whence \( {1}_{R} : P \) . Therefore, \( P \neq R \) . Since \( R/P \) has no zero divisors, \[ {ab} \in P \Rightarrow {ab} + P = P \Rightarrow \left( {a + P}\right) \left( {b + P}\right) = P \Rightarrow a + P = P\text{ or } \] \[ b + P = P \Rightarrow {a\varepsilon P}\text{ or

Theorem 2.7. Let \( \mathrm{R} \) be a ring and \( \mathrm{I} \) an ideal of \( \mathrm{R} \). Then the additive quotient group \( \mathrm{R}/\mathrm{I} \) is a ring with multiplication given by \[ \left( {a + I}\right) \left( {b + I}\right) = {ab} + I \] If \( \mathbf{R} \) is commutative or has an identity, then the same is true of \( \mathbf{R}/\mathbf{I} \).

SKETCH OF PROOF OF 2.7. Once we have shown that multiplication in \( R/I \) is well defined, the proof that \( R/I \) is a ring is routine. (For example, if \( R \) has identity \( {1}_{R} \), then \( {1}_{R} + I \) is the identity in \( R/I \).) Suppose \( a + I = {a}^{\prime } + I \) and \( b + I = {b}^{\prime } + I \). We must show that \( {ab} + I = {a}^{\prime }{b}^{\prime } + I \). Since \( {a}^{\prime }\varepsilon {a}^{\prime } + I = a + I \), \( {a}^{\prime } = a + i \) for some \( i \in I \). Similarly \( {b}^{\prime } = b + j \) with \( j \in I \). Consequently \( {a}^{\prime }{b}^{\prime } = \left( {a + i}\right) \left( {b + j}\right) = {ab} + {ib} + {aj} + {ij} \). Since \( I \) is an ideal, \[ {a}^{\prime }{b}^{\prime } - {ab} = {ib} + {aj} + {ij\varepsilon I}. \] Therefore \( {a}^{\prime }{b}^{\prime } + I = {ab} + I \) by Corollary I.4.3, whence multiplication in \( R/I \) is well defined.

Exercise 2.16 If \( V \) is an irreducible finite-dimensional representation of a Lie group \( G \), show that \( {V}^{ * } \) is also irreducible. Exercise 2.17 This exercise considers a natural generalization of \( {V}_{n}\left( {\mathbb{C}}^{2}\right) \) . Let \( W \) be a representation of a Lie group \( G \) . Define \( {V}_{n}\left( W\right) \) to be the space of holomorphic polynomials on \( W \) that are homogeneous of degree \( n \) and let \( \left( {gP}\right) \left( \eta \right) = P\left( {{g}^{-1}\eta }\right) \) . Show that there is an equivalence of representations \( {S}^{n}\left( {W}^{ * }\right) \cong {V}_{n}\left( W\right) \) induced by viewing \( {T}_{1}\cdots {T}_{n},{T}_{i} \in {W}^{ * } \), as a function on \( W \) . Exercise 2.18 (a) Show that the map \( \pi : t \rightarrow \left( \begin{array}{ll} 1 & t \\ 0 & 1 \end{array}\right) \) produces a representation of \( \mathbb{R} \) on \( {\mathbb{C}}^{2} \) . (b) Show that this representation is not unitary. (c) Find all invariant submodules. (b) Show that the representation is reducible and yet not completely reducible. Exercise 2.19 Use Schur's Lemma (Theorem 2.12) to quickly calculate the centers of the groups having standard representations listed in §2.1.2.1. Exercise 2.20 Let \( \left( {\cdot , \cdot }\right) \) be an inner product on \( {\mathbb{C}}^{n} \) . Show that \( U\left( n\right) \cong \{ g \in \) \( {GL}\left( {n,\mathbb{C}}\right) \mid \left( {{gv}, g{v}^{\prime }}\right) = \left( {v,{v}^{\prime }}\right) \) for \( \left. {v,{v}^{\prime } \in {\mathbb{C}}^{n}}\right\} \) . Exercise 2.21 (a) Use Equation 2.13 to show that all irreducible finite-dimensional representations of an Abelian Lie group are 1-dimensional (c.f. Exercise 3.18). (b) Classify all irreducible representations of \( {S}^{1} \) and show that \( \widehat{{S}^{1}} \cong \mathbb{Z} \) . (c) Find the irreducible summands of the representation of \( {S}^{1} \) on \( {\mathbb{C}}^{2} \) generated by the isomorphism \( {S}^{1} \cong {SO}\left( 2\right) \) . (d) Show that a smooth homomorphism \( \varphi : \mathbb{R} \rightarrow \mathbb{C} \) satisfies the differential equation \( {\varphi }^{\prime } = \left\lbrack {{\varphi }^{\prime }\left( 0\right) }\right\rbrack \varphi \) . Use this to show that the set of irreducible representations of \( \mathbb{R} \) is indexed by \( \overline{\mathbb{C}} \) and that the unitary ones are indexed by \( i\mathbb{R} \) . (e) Use part (d) to show that the set of irreducible representations of \( {\mathbb{R}}^{ + } \) under its multiplicative structure is indexed by \( \mathbb{C} \) and that the unitary ones are indexed by \( i\mathbb{R} \) . (f) Classify all irreducible representations of \( \mathbb{C} \cong {\mathbb{R}}^{2} \) under its additive structure and of \( \mathbb{C} \smallsetminus \{ 0\} \) under its multiplicative structure. Exercise 2.22 Let \( V \) be a finite-dimensional representation of a compact Lie group \( G \) . Show the set of \( G \) -invariant inner products on \( G \) is isomorphic to \( {\operatorname{Hom}}_{G}\left( {{V}^{ * },{V}^{ * }}\right) \) . Exercise 2.23 (a) Let \( {\pi }_{i} : {V}_{i} \rightarrow U\left( n\right) \) be two (unitary) equivalent irreducible representations of a compact Lie group \( G \) . Use Corollary 2.20 to show that there exists a unitary transformation intertwining \( {\pi }_{1} \) and \( {\pi }_{2} \) . (b) Repeat part (a) without the hypothesis of irreducibility. Exercise 2.24 Let \( V \) be a finite-dimensional representation of a compact Lie group \( G \) and let \( W \subseteq V \) be a subrepresentation. Show that \( {W}_{\left\lbrack \pi \right\rbrack } \subseteq {V}_{\left\lbrack \pi \right\rbrack } \) for \( \left\lbrack \pi \right\rbrack \in \widehat{G} \) . Exercise 2.25 Suppose \( V \) is a finite-dimensional representation of a compact Lie group \( G \) . Show that the set of \( G \) -intertwining automorphisms of \( V \) is isomorphic to \( \mathop{\prod }\limits_{{\left\lbrack \pi \right\rbrack \in \widehat{G}}}{GL}\left( {{m}_{\pi },\mathbb{C}}\right) \) where \( {m}_{\pi } \) is the multiplicity of the isotypic component \( {V}_{\left\lbrack \pi \right\rbrack } \) . ## 2.3 Examples of Irreducibility ## 2.3.1 \( {SU}\left( 2\right) \) and \( {V}_{\mathrm{n}}\left( {\mathbb{C}}^{2}\right) \) In this section we show that the representation \( {V}_{n}\left( {\mathbb{C}}^{2}\right) \) from \( §{2.1.2.2} \) of \( {SU}\left( 2\right) \) is irreducible. In fact, we will later see (Theorem 3.32) these are, up to equivalence, the only irreducible representations of \( {SU}\left( 2\right) \) . The trick employed here points towards the powerful techniques that will be developed in \( §4 \) where derivatives, i.e., the tangent space of \( G \), are studied systematically (c.f. Lemma 6.6). Let \( H \subseteq {V}_{n}\left( {\mathbb{C}}^{2}\right) \) be a nonzero invariant subspace. From Equation 2.3, (2.25) \[ \operatorname{diag}\left( {{e}^{i\theta },{e}^{-{i\theta }}}\right) \cdot \left( {{z}_{1}^{k}{z}_{2}^{n - k}}\right) = {e}^{i\left( {n - {2k}}\right) \theta }{z}_{1}^{k}{z}_{2}^{n - k}. \] As the joint eigenvalues \( {e}^{i\left( {n - {2k}}\right) \theta } \) are distinct and since \( H \) is preserved by \( \left\{ {\operatorname{diag}\left( {{e}^{i\theta },{e}^{-{i\theta }}}\right) }\right\}, H \) is spanned by some of the joint eigenvectors \( {z}_{1}^{k}{z}_{2}^{n - k} \) . In particular, there is a \( {k}_{0} \), so \( {z}_{1}^{{k}_{0}}{z}_{2}^{n - {k}_{0}} \in H \) . \[ \text{Let}{K}_{t} = \left( \begin{matrix} \cos t & - \sin t \\ \sin t & \cos t \end{matrix}\right) \in {SU}\left( 2\right) \text{and let}{\eta }_{t} = \left( \begin{matrix} \cos t & i\sin t \\ i\sin t & \cos t \end{matrix}\right) \in {SU}\left( 2\right) \text{.} \] Since \( H \) is \( {SU}\left( 2\right) \) invariant, \( \frac{1}{2}\left( {{K}_{t} \pm i{\eta }_{t}}\right) {z}_{1}^{{k}_{0}}{z}_{2}^{n - {k}_{0}} \in H \) . Thus, when the limits exist, \( {\left. \frac{d}{dt}\left\lbrack \frac{1}{2}\left( {K}_{t} \pm i{\eta }_{t}\right) {z}_{1}^{{k}_{0}}{z}_{2}^{n - {k}_{0}}\right\rbrack \right| }_{t = 0} \in H \) . Using Equation 2.3, a simple calculation (Exercise 2.26) shows that (2.26) \[ {\left. \frac{1}{2}\frac{d}{dt}\left\lbrack \left( {K}_{t} \pm i{\eta }_{t}\right) {z}_{1}^{{k}_{0}}{z}_{2}^{n - {k}_{0}}\right\rbrack \right| }_{t = 0} = \left\{ \begin{matrix} {k}_{0}{z}_{1}^{{k}_{0} - 1}{z}_{2}^{n - {k}_{0} + 1} & \text{ for } + \\ \left( {{k}_{0} - n}\right) {z}_{1}^{{k}_{0} + 1}{z}_{2}^{n - {k}_{0} - 1} & \text{ for } - . \end{matrix}\right. \] Induction therefore implies that \( {V}_{n}\left( \mathbb{C}\right) \subseteq H \), and so \( {V}_{n}\left( \mathbb{C}\right) \) is irreducible. ## 2.3.2 \( \mathrm{{SO}}\left( n\right) \) and Harmonic Polynomials In this section we show that the representation of \( {SO}\left( n\right) \) on the harmonic polynomials \( {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \subseteq {V}_{m}\left( {\mathbb{R}}^{n}\right) \) is irreducible (see \( §{2.1.2.3} \) for notation). Let \( {D}_{m}\left( {\mathbb{R}}^{n}\right) \) be the space of complex constant coefficient differential operators on \( {\mathbb{R}}^{n} \) of degree \( m \) . Recall that the algebra isomorphism from \( {\bigoplus }_{m}{V}_{m}\left( {\mathbb{R}}^{n}\right) \) to \( {\bigoplus }_{m}{D}_{m}\left( {\mathbb{R}}^{n}\right) \) is generated by mapping \( {x}_{i} \rightarrow {\partial }_{{x}_{i}} \) . In general, if \( q \in {\bigoplus }_{m}{V}_{m}\left( {\mathbb{R}}^{n}\right) \), write \( {\partial }_{q} \) for the corresponding element of \( {\bigoplus }_{m}{D}_{m}\left( {\mathbb{R}}^{n}\right) \) . Define \( \langle \cdot , \cdot \rangle \) a Hermitian form on \( {V}_{m}\left( {\mathbb{R}}^{n}\right) \) by \( \langle p, q\rangle = {\partial }_{\bar{q}}\left( p\right) \in \mathbb{C} \) for \( p, q \in \) \( {V}_{m}\left( {\mathbb{R}}^{n}\right) \) . Since \( \left\{ {{x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\ldots {x}_{n}^{{k}_{n}} \mid {k}_{i} \in \mathbb{N}\text{and}{k}_{1} + {k}_{2} + \cdots + {k}_{n} = m}\right\} \) turns out to be an orthogonal basis for \( {V}_{m}\left( {\mathbb{R}}^{n}\right) \), it is easy to see that \( \langle \cdot , \cdot \rangle \) is an inner product. In fact, \( \langle \cdot , \cdot \rangle \) is actually \( O\left( n\right) \) -invariant (Exercise 2.27), although we will not need this fact. Lemma 2.27. With respect to the inner product \( \langle \cdot , \cdot \rangle \) on \( {V}_{m}\left( {\mathbb{R}}^{n}\right) ,{\mathcal{H}}_{m}{\left( {\mathbb{R}}^{n}\right) }^{ \bot } = \) \( {\left| x\right| }^{2}{V}_{m - 2}\left( {\mathbb{R}}^{n}\right) \) where \( {\left| x\right| }^{2} = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}^{2} \in {V}_{2}\left( {\mathbb{R}}^{n}\right) \) . As \( O\left( n\right) \) -modules, \[ {V}_{m}\left( {\mathbb{R}}^{n}\right) \cong {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \oplus {\mathcal{H}}_{m - 2}\left( {\mathbb{R}}^{n}\right) \oplus {\mathcal{H}}_{m - 4}\left( {\mathbb{R}}^{n}\right) \oplus \cdots . \] Proof. Let \( p \in {V}_{m}\left( {\mathbb{R}}^{n}\right) \) and \( q \in {V}_{m - 2}\left( {\mathbb{R}}^{n}\right) \) . Then \( \left\langle {p,{\left| x\right| }^{2}q}\right\rangle = {\partial }_{{\left| x\right| }^{2}}{}_{\bar{q}}p = {\partial }_{\bar{q}}{\Delta p} = \) \( \langle {\Delta p}, q\rangle \) . Thus \( {\left\lbrack {\left| x\right| }^{2}{V}_{m - 2}\left( {\mathbb{R}}^{n}\right) \right\rbrack }^{ \bot } = {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \) so that (2.28) \[ {V}_{m}\left( {\mathbb{R}}^{n}\right) = {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \oplus {\left| x\right| }^{2}{V}_{m - 2}\left( {\mathbb{R}}^{n}\right) . \] Induction therefore shows that \[ {V}_{m}\left( {\mathbb{R}}^{n}\right) = {\mathcal{H}}_{m}\left( {\mathbb{R}}^{n}\right) \oplus {\left| x\right| }^{2}{\mathcal{H}}_{m - 2}\left( {\mathbb{R}}^{n}\right) \oplus {\left| x\right| }^{4}{\ma

Exercise 2.16 If \( V \) is an irreducible finite-dimensional representation of a Lie group \( G \), show that \( {V}^{ * } \) is also irreducible.

To show that \( {V}^{ * } \) is irreducible, we need to demonstrate that any invariant subspace of \( {V}^{ * } \) is either the zero subspace or the whole space \( {V}^{ * } \). Let \( W \) be a nonzero invariant subspace of \( {V}^{ * } \). This means that for any \( w \in W \) and any \( g \in G \), the action of \( g \) on \( w \) remains in \( W \). In other words, if \( \rho: G \to GL(V) \) is the representation of \( G \) on \( V \), then the dual representation \( \rho^{*}: G \to GL(V^{*}) \) satisfies \( \rho^{*}(g)(w) \in W \) for all \( g \in G \) and \( w \in W \). Consider the natural pairing between \( V \) and \( V^{*} \), given by \( \langle v, f \rangle = f(v) \) for \( v \in V \) and \( f \in V^{*} \). For any nonzero invariant subspace \( W \subseteq V^{*} \), define the annihilator of \( W \) in \( V \) as: \[ W^{\perp} = \{ v \in V \, | \, \langle v, f \rangle = 0 \, \text{for all} \, f \in W \}. \] Since \( W \) is invariant under the action of \( G \), for any \( g \in G \) and any \( v \in W^{\perp} \), we have: \[ \langle gv, f \rangle = f(gv) = (g^{-1}f)(v) = 0 \] for all \( f \in W \). This implies that \( gv \in W^{\perp} \), so \( W^{\perp} \) is an invariant subspace of \( V \). Given that \( V \) is irreducible, either \( W^{\perp} = \{0\} \) or \( W^{\perp} = V \). If \( W^{\perp} = \{0\} \), then for any nonzero vector \( v_0 \in V \) there exists some linear functional \( f_0 \in W \) such that \( f_0(v_0) \neq 0\). Since every linear functional in \( V^{*} \) can be written as a linear combination of such functionals, this implies that every linear functional in \( V^{*} \) belongs to some nonzero element in the span of these functionals, hence every element in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionals belongs to some nonzero element in the span of these functionals. Therefore, every linear functional in the span of these functionalities belongs to some nonzero element in

Proposition 6.45. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a Lattès map that fits into a commutative diagram (6.22). Then \[ {\operatorname{CritVal}}_{\pi } = {\operatorname{PostCrit}}_{\phi } \] In particular, a Lattès map is postcritically finite. Proof. The key to the proof of this proposition is the fact that the map \( \psi : E \rightarrow E \) is unramified, i.e., it has no critical points, see Remark 6.20. (In the language of modern algebraic geometry, the map \( \psi \) is étale.) More precisely, the map \( \psi \) is the composition of an endomorphism of \( E \) and a translation (Remark 6.19), both of which are unramified. For any \( n \geq 1 \) we compute \[ {\operatorname{CritVal}}_{\pi } = {\operatorname{CritVal}}_{\pi {\psi }^{n}} \] because \( \psi \) is unramified, \[ = {\operatorname{CritVal}}_{{\phi }^{n}\pi } \] from the commutativity of (6.22), \( = {\operatorname{CritVal}}_{{\phi }^{n}} \cup {\phi }^{n}\left( {\operatorname{CritVal}}_{\pi }\right) \; \) from the definition of critical value, \[ \supseteq {\operatorname{CritVal}}_{{\phi }^{n}} \] This holds for all \( n \geq 1 \), which gives the inclusion \[ {\operatorname{CritVal}}_{\pi } \supseteq \mathop{\bigcup }\limits_{{n = 0}}^{\infty }{\phi }^{n}\left( {\operatorname{CritVal}}_{\phi }\right) = {\operatorname{PostCrit}}_{\phi }. \] In order to prove the opposite inclusion, suppose that there exists a point \( {P}_{0} \in E \) satisfying \[ {P}_{0} \in {\operatorname{CritPt}}_{\pi }\;\text{ and }\;\pi \left( {P}_{0}\right) \notin {\operatorname{PostCrit}}_{\phi }. \] (6.23) Consider any point \( Q \in {\psi }^{-1}\left( {P}_{0}\right) \) . Then \( Q \) is a critical point of \( {\pi \psi } \), since \( \psi \) is unramified and \( \pi \) is ramified at \( \psi \left( Q\right) \) by assumption. But \( {\pi \psi } = {\phi \pi } \), so we see that \( Q \) is a critical point for \( {\phi \pi } \) . On the other hand, \[ \phi \left( {\pi \left( Q\right) }\right) = \pi \left( {P}_{0}\right) \notin {\operatorname{CritVal}}_{\phi } = \phi \left( {\operatorname{CritPt}}_{\phi }\right) , \] so \( \pi \left( Q\right) \) is not a critical point for \( \phi \) . It follows that \( Q \) is a critical point of \( \pi \) . Further, we claim that no iterate of \( \phi \) is ramified at \( \pi \left( Q\right) \) . To see this, we use the given fact that \( \pi \left( {P}_{0}\right) \) is not in the postcritical set of \( \phi \) to compute \[ \pi \left( {P}_{0}\right) \notin {\mathsf{{PostCrit}}}_{\phi } \Rightarrow \pi \left( {P}_{0}\right) \notin {\phi }^{n}\left( {\mathsf{{CritPt}}}_{\phi }\right) \;\text{ for all }n \geq 1, \] \[ \Rightarrow \pi \left( {\psi \left( Q\right) }\right) \notin {\phi }^{n}\left( {\mathsf{{CritPt}}}_{\phi }\right) \;\text{ for all }n \geq 1, \] \[ \Rightarrow \phi \left( {\pi \left( Q\right) }\right) \notin {\phi }^{n}\left( {\operatorname{CritPt}}_{\phi }\right) \;\text{ for all }n \geq 1, \] \[ \Rightarrow \pi \left( Q\right) \notin {\phi }^{n}\left( {\mathsf{{CritPt}}}_{\phi }\right) \;\text{ for all }n \geq 0, \] \[ \Rightarrow \pi \left( Q\right) \notin {\operatorname{PostCrit}}_{\phi }. \] To recapitulate, we have now proven that every \( Q \in {\psi }^{-1}\left( {P}_{0}\right) \) satisfies \[ Q \in {\operatorname{CritPt}}_{\pi }\;\text{ and }\;\pi \left( Q\right) \notin {\operatorname{PostCrit}}_{\psi }. \] In other words, every point \( Q \in {\psi }^{-1}\left( {P}_{0}\right) \) satisfies the same two conditions (6.23) that are satisfied by \( {P}_{0} \) . Hence by induction we find that if there is any point \( {P}_{0} \) satisfying (6.23), then the full backward orbit of \( \psi \) is contained in the set of critical points of \( \pi \), i.e., \[ {\operatorname{CritPt}}_{\pi } \supset \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{\psi }^{-n}\left( {P}_{0}\right) \] But \( \psi \) is unramified and has degree at least 2 (note that \( \deg \psi = \deg \phi \) ), so \[ \text{#}{\operatorname{CritPt}}_{\pi } \geq \# \left( {{\psi }^{-n}\left( {P}_{0}\right) }\right) = {\left( \deg \psi \right) }^{n}\xrightarrow[{n \rightarrow \infty }]{}\infty \text{.} \] This is a contradiction, since \( \pi \) has only finitely many critical points, so we conclude that there are no points \( {P}_{0} \) satisfying (6.23). Hence \[ {P}_{0} \in {\operatorname{CritPt}}_{\pi } \Rightarrow \pi \left( {P}_{0}\right) \in {\operatorname{PostCrit}}_{\phi }, \] which gives the other inclusion \( {\operatorname{CritVal}}_{\pi } \subseteq {\operatorname{PostCrit}}_{\phi } \) . As an application of Proposition 6.45, we show that Lattès maps associated to distinct elliptic curves are not conjugate to one another. Theorem 6.46. Let \( K \) be an algebraically closed field of characteristic not equal to 2 and let \( \phi \) and \( {\phi }^{\prime } \) be Lattès maps defined over \( K \) that are associated, respectively, to elliptic curves \( E \) and \( {E}^{\prime } \) . Assume further that the projection maps \( \pi \) and \( {\pi }^{\prime } \) associated to \( \phi \) and \( {\phi }^{\prime } \) both have degree 2. If \( \phi \) and \( {\phi }^{\prime } \) are \( {\operatorname{PGL}}_{2}\left( K\right) \) -conjugate to one another, then \( E \) and \( {E}^{\prime } \) are isomorphic. Proof. Let \( f \in {\operatorname{PGL}}_{2}\left( K\right) \) be a linear fractional transformation conjugating \( {\phi }^{\prime } \) to \( \phi \) . Then we have a commutative diagram  We let \( {\pi }^{\prime \prime } = f \circ {\pi }^{\prime } \) . Note that since \( f \) is an isomorphism, the map \( {\pi }^{\prime \prime } \) still has degree 2. This yields the simplified commutative diagram  showing that \( \phi \) is a Lattès map associated to both elliptic curves \( E \) and \( {E}^{\prime } \) . Applying Proposition 6.45, first to \( E \) and then to \( {E}^{\prime } \), we find that \[ {\operatorname{CritVal}}_{\pi } = {\operatorname{PostCrit}}_{\phi } = {\operatorname{CritVal}}_{{\pi }^{\prime \prime }}. \] (6.24) In other words, the degree-2 maps \( \pi : E \rightarrow {\mathbb{P}}^{1} \) and \( {\pi }^{\prime \prime } : {E}^{\prime } \rightarrow {\mathbb{P}}^{1} \) have the exact same set of critical values. Then Lemma 6.38 tells us that \( E \) and \( {E}^{\prime } \) are isomorphic. ## 6.5 Flexible Lattès Maps A Lattès map is a rational map that is obtained by projecting an elliptic curve endomorphism down to \( {\mathbb{P}}^{1} \) . For any integer \( m \geq 2 \), every elliptic curve has a multiplication-by- \( m \) map and a projection \( E \rightarrow E/\{ \pm 1\} \cong {\mathbb{P}}^{1} \), so every elliptic curve has a corresponding Lattès map. As \( E \) varies, this collection of Lattès maps varies continuously, which prompts the following definition. Definition. A flexible Lattès map is a Lattès map \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) that fits into a Lattès commutative diagram (6.22) in which the map \( \psi : E \rightarrow E \) has the form \[ \psi \left( P\right) = \left\lbrack m\right\rbrack \left( P\right) + T\;\text{ for some }m \in \mathbb{Z}\text{ and some }T \in E \] and such that the projection map \( \pi : E \rightarrow {\mathbb{P}}^{1} \) satisfies \[ \deg \left( \pi \right) = 2\;\text{ and }\;\pi \left( P\right) = \pi \left( {-P}\right) \text{ for all }P \in E. \] Remark 6.47. The condition that \( \pi \) be even, i.e., that it satisfy \( \pi \left( {-P}\right) = \pi \left( P\right) \), is included for convenience. In general, if \( \pi : E \rightarrow {\mathbb{P}}^{1} \) is any map of degree 2, then there exists a point \( {P}_{0} \in E \) such that \( \pi \left( {-\left( {P + {P}_{0}}\right) }\right) = \pi \left( {P + {P}_{0}}\right) \) for all \( P \in E \) . Thus \( \pi \) becomes an even function if we use \( {P}_{0} \) as the identity element for the group law on \( E \) . See Exercise 6.16. Remark 6.48. We show in this section that the Lattès maps of a given degree have identical multiplier spectra. This is one reason that these Lattès maps are called "flexible," since they vary in continuous families whose periodic points have identical sets of multipliers. We saw in Section 4.5 that symmetric polynomials in the multipliers give rational functions on the moduli space \( {M}_{d} \) of rational maps modulo \( {\mathrm{{PGL}}}_{2} \) - conjugation. Flexible families of rational maps thus cannot be distinguished from one another in \( {M}_{d} \) solely through the values of their multipliers. Example 6.49. We saw in Example 6.41 that the Lattès function associated to the duplication map \( \psi \left( P\right) = \left\lbrack 2\right\rbrack \left( P\right) \) on the elliptic curve \( E : {y}^{2} = {x}^{3} + {ax} + b \) is given by the formula \[ {\phi }_{a, b}\left( x\right) = x\left( {2P}\right) = \frac{{x}^{4} - {2a}{x}^{2} - {8bx} + {a}^{2}}{4{x}^{3} + {4ax} + {4b}}. \] It is clear that if \( a \) and \( b \) vary continuously, subject to \( 4{a}^{3} + {27}{b}^{2} \neq 0 \), then the Lattès maps \( {\phi }_{a, b} \) vary continuously in the space of rational maps of degree 4 . More precisely, the set of maps \( {\phi }_{a, b} \) is a two-dimensional algebraic family of points in the space \( {\operatorname{Rat}}_{4} \), given explicitly by \[ {\mathbb{A}}^{2} \rightarrow {\operatorname{Rat}}_{4} \subset {\mathbb{P}}^{9},\;\left( {a, b}\right) \mapsto \left\lbrack {1,0, - {2a}, - {8b},{a}^{2},0,4,0,{4a},{4b}}\right\rbrack . \] If we conjugate by \( {f}_{u}\left( x\right) = {ux} \), the Lattès map \( {\phi }_{a, b} \) transforms into \[ {\phi }_{a, b}^{{f}_{u}}\left( x\right) = {u}^{-1}{\phi }_{a, b}\left( {ux}\right) = {\phi }_{{u}^{-2}a,{u}^{-3}b}\left( x\right) . \] Thus assuming (say) that \( {ab} \neq 0

Proposition 6.45. Let \( \phi : {\mathbb{P}}^{1} \rightarrow {\mathbb{P}}^{1} \) be a Lattès map that fits into a commutative diagram (6.22). Then \[ {\operatorname{CritVal}}_{\pi } = {\operatorname{PostCrit}}_{\phi } \] In particular, a Lattès map is postcritically finite.

The key to the proof of this proposition is the fact that the map \( \psi : E \rightarrow E \) is unramified, i.e., it has no critical points, see Remark 6.20. (In the language of modern algebraic geometry, the map \( \psi \) is étale.) More precisely, the map \( \psi \) is the composition of an endomorphism of \( E \) and a translation (Remark 6.19), both of which are unramified. For any \( n \geq 1 \) we compute \[ {\operatorname{CritVal}}_{\pi } = {\operatorname{CritVal}}_{\pi {\psi }^{n}} \] because \( \psi \) is unramified, \[ = {\operatorname{CritVal}}_{{\phi }^{n}\pi } \] from the commutativity of (6.22), \( = {\operatorname{CritVal}}_{{\phi }^{n}} \cup {\phi }^{n}\left( {\operatorname{CritVal}}_{\pi }\right) \; \) from the definition of critical value, \[ \supseteq {\operatorname{CritVal}}_{{\phi }^{n}} \] This holds for all \( n \geq 1 \), which gives the inclusion \[ {\operatorname{CritVal}}_{\pi } \supseteq \mathop{\bigcup }\limits_{{n = 0}}^{\infty }{\phi }^{n}\left( {\operatorname{CritVal}}_{\phi }\right) = {\operatorname{PostCrit}}_{\phi }. \] In order to prove the opposite inclusion, suppose that there exists a point \( {P}_{0} \in E \) satisfying \[ {P}_{0} \in {\operatorname{CritPt}}_{\pi }\;\text{ and }\;\pi \left( {P}_{0}\right) \notin {\operatorname{PostCrit}}_{\phi }. \] (6.23) Consider any point \( Q \in {\psi }^{-1}\left( {P}_{0}\right) \) . Then \( Q \) is a critical point of \( {\pi \psi } \), since \( \psi \) is unramified and \( \pi \) is ramified at \( \psi \left( Q\right) \) by assumption. But \( {\pi \psi } = {\phi \pi } \), so we see that \( Q \) is a critical point for \( {\phi \pi } \) . On the other hand, \[ \phi \left( {\pi \left( Q\right) }\right) = \pi \left( {P}_{0}\right) \notin {\operatorname{CritVal}}_{\phi } = \phi \left( {\operatorname{CritPt}}_{\phi }\right) , \] so \( \pi \left( Q\right) \) is not a critical point for \( \phi \). It follows that \( Q \) is a critical point of \( \pi \). Further, we claim that no iterate of \( \phi \) is ramified at \( \pi \left( Q\right) \). To see this, we use the given fact that \( \pi \left( {P}_{0}\right) \) is not in the postcritical set of \( \phi \) to compute \[ \pi \left( {P}_{0}\right) \notin {\mathsf{{PostCrit}}}_{\phi } \Rightarrow \pi \left( {P}_{0}\right) \notin {\phi }^{n}\left( {\mathsf{{CritPt}}}_{\phi }\right) \;\text{ for all }n \geq 1, \] \[ \Rightarrow \pi \left( {\psi \left( Q\right) }\right) \notin {\phi }^{n}\left( {\mathsf{{CritPt}}}_{\phi }\right) \;\text{ for all }n \geq 1, \] \[

Theorem 7.3.12 If \( G \) is an infinite connected solvable group of finite Morley rank with finite center, then \( G \) interprets an algebraically closed field. Proof We will prove this by induction on the rank of \( G \) . We first argue that we may, without loss of generality, assume that \( G \) is centerless. Suppose that \( Z\left( G\right) \) is finite. We claim that \( G/Z\left( G\right) \) is centerless. Let \( a \in G \) such that \( a/Z\left( G\right) \in Z\left( {G/Z\left( G\right) }\right) \) . For all \( g \in G,{a}^{-1}{g}^{-1}{ag} \in Z\left( G\right) \) . Thus, \( {a}^{-1}{a}^{G} \subseteq Z\left( G\right) \), and, hence, \( {a}^{G} \) is finite. Thus, \( \left\lbrack {G : C\left( a\right) }\right\rbrack \) is finite. Because \( G \) is connected, \( C\left( a\right) = G \) and \( a \in Z\left( G\right) \) . Thus, \( G/Z\left( G\right) \) is solvable and centerless. By Exercise 7.6.4, \( G/Z\left( G\right) \) is connected. Because \( G/Z\left( G\right) \) is interpretable in \( G \), if \( G/Z\left( G\right) \) interprets an algebraically closed field, so does \( G \) . Let \( A \trianglelefteq G \) be a minimal infinite definable normal subgroup. By Lemma 7.3.11 and Corollary 7.3.5, \( A \) is solvable and \( {A}^{\prime } \) is a proper connected normal definable subgroup of \( A \) . Moreover, any automorphism of \( A \) fixes \( {A}^{\prime } \) setwise, thus \( {A}^{\prime } \trianglelefteq G \) . By choice of \( A,{A}^{\prime } = \{ 1\} \) and \( A \) is Abelian. Let \( C\left( A\right) = \{ g \in G : {ga} = {ag} \) for all \( a \in A\} \) . Because \( G \) is centerless, \( C\left( A\right) \neq G \) . Because \( A \) is normal, so is \( C\left( A\right) \) . Let \( {G}_{1} = G/C\left( A\right) \) . If \( g, h \in G \) and \( g = {hc} \) where \( c \in C\left( A\right) \), then for \( a \in A \) \[ {ha}{h}^{-1} = {gca}{c}^{-1}{g}^{-1} = {ga}{g}^{-1}. \] Thus, \( {G}_{1} \) acts on \( A \) by conjugation. If \( {ga}{g}^{-1} = {ha}{h}^{-1} \) for all \( a \in A \), then \( {g}^{-1}{ha}{h}^{-1}{ga} = a \) and \( {g}^{-1}h \in C\left( A\right) \) . Thus, \( g/C\left( A\right) = h/C\left( A\right) \) and \( {G}_{1} \) acts faithfully on \( A \) . Moreover, because \( A \) is the smallest infinite normal definable subgroup, no infinite definable subgroup of \( A \) is \( {G}_{1} \) -invariant. The group \( {G}_{1} \) is solvable and, because \( A \leq C\left( A\right) ,\operatorname{RM}\left( {G}_{1}\right) < \operatorname{RM}\left( G\right) \) . If \( Z\left( {G}_{1}\right) \) is finite, then \( {G}_{1} \) (and hence \( G \) ) interprets an algebraically closed field by induction. Thus, we may assume that \( Z\left( {G}_{1}\right) \) is infinite. Let \( H \) be a minimal definable infinite subgroup of \( Z\left( {G}_{1}\right) \) . Then, \( H \) is Abelian, \( H \trianglelefteq {G}_{1} \) , and \( H \) acts faithfully on \( A \) by conjugation. If there are no definable infinite \( H \) -invariant proper subgroups of \( A \) , then by Theorem 7.3.9 there is an interpretable algebraically closed field. Otherwise, let \( B < A \) be a minimal infinite \( H \) -invariant definable subgroup. Let \( {H}_{0} = \left\{ {h \in H : {b}^{h} = b}\right. \) for all \( \left. {b \in B}\right\} \) . Suppose that \( H = {H}_{0} \) (i.e., \( H \) acts trivially on \( B \) ). Because \( B \) is a minimal \( H \) -invariant subgroup, \( B \) is indecomposable by 7.3.3 and, by Lemma 7.6.12, \( {B}^{g} \) is indecomposable for all \( g \in {G}_{1} \) . Because \( H \leq Z\left( {G}_{1}\right) \), if \( h \in H \) and \( b \in B \), then \[ {\left( {b}^{g}\right) }^{h} = {\left( {b}^{h}\right) }^{g} = {b}^{g} \] Thus, \( H \) acts trivially on \( {B}^{g} \) as well. By Zil’ber’s Indecomposability Theorem, the group generated by \( \left\langle {{B}^{g} : g \in {G}_{1}}\right\rangle \) is a definable \( {G}_{1} \) -invariant subgroup of \( A \) . But \( A \) is a minimal \( {G}_{1} \) -invariant subgroup, thus \( A \) is generated by \( \left\langle {{B}^{g} : g \in {G}_{1}}\right\rangle \) . Because \( H \) acts trivially on each \( {B}^{g}, H \) acts trivially on \( A \), a contradiction. Thus, \( {H}_{0} \) is a proper subgroup of \( H \) and, because \( H \) is minimal, \( {H}_{0} \) is finite. But then \( H/{H}_{0} \) acts faithfully on \( B \) and there are no infinite definable \( H/{H}_{0} \) -invariant subgroups of \( B \) . Thus, by Theorem 7.3.9, we can interpret an algebraically closed field. We have already seen a concrete example of Theorem 7.3.12 in Section 1.3. Let \( K \) be an algebraically closed field and \( G \) be the group of matrices \[ G = \left\{ {\left( \begin{array}{ll} a & b \\ 0 & 1 \end{array}\right) : a, b \in K, a \neq 0}\right\} \] then \( G \) is a connected, solvable, centerless group of finite Morley rank (see Exercise 7.6.19). The proof of Theorem 7.3.12 is an abstraction of the concrete interpretation of the field in Section 1.3. We will give one more extension of this result. Definition 7.3.13 A group \( G \) is nilpotent if there is a chain of normal subgroups \( G = {G}_{0} \trianglerighteq \ldots \trianglerighteq {G}_{n} = \{ 1\} \) such that \( {G}_{i}/{G}_{i + 1} \leq Z\left( {G/{G}_{i + 1}}\right) \) for all \( i < n \) . For a group \( G \), we define the lower central series as \( {\Gamma }_{0}\left( G\right) = G \) , \( {\Gamma }_{n + 1}\left( G\right) = \left\lbrack {{\Gamma }_{n}\left( G\right) : G}\right\rbrack \) the group generated by commutators \( \{ \left\lbrack {a, b}\right\rbrack : a \in \) \( \left. \left. {{\Gamma }_{n}\left( G\right), b \in G}\right\rbrack \right\} \) . Then \( G \trianglerighteq {\Gamma }_{1}\left( G\right) \trianglerighteq {\Gamma }_{2}\left( G\right) \ldots \) . We define the upper central series by \( {Z}_{0}\left( G\right) = \{ 1\} \) and \( {Z}_{n}\left( G\right) = \left\{ {g \in G : g/{Z}_{n - 1} \in Z\left( {G/{Z}_{n - 1}}\right) }\right\} \) . We will use the following facts about nilpotent groups. See, for example, [89] 7.54. Lemma 7.3.14 A group \( G \) is nilpotent if and only if there is an \( n \) such that \( {\Gamma }_{n}\left( G\right) = \{ 1\} \) if and only if there is an \( n \) such that \( {Z}_{n}\left( G\right) = G \) . Theorem 7.3.15 If \( G \) is an infinite connected, solvable, nonnilpotent group of finite Morley rank, then \( G \) interprets an algebraically closed field. Proof Let \( {Z}_{0}\left( G\right) \trianglelefteq {Z}_{1}\left( G\right) \ldots \) be the upper central series of \( G \) . Because \( G \) has finite Morley rank, there is an \( n \) such that \( \operatorname{RM}\left( {{Z}_{n}\left( G\right) }\right) \) is maximal. Then, \( {Z}_{n + 1}\left( G\right) /{Z}_{n}\left( G\right) \) is finite and, because \( G \) is nonnilpotent, \( {Z}_{n}\left( G\right) \neq G \) . Consider \( G/{Z}_{n}\left( G\right) \) . By Lemma 7.3.11 and Exercise 7.6.4 \( G/{Z}_{n}\left( G\right) \) is a connected solvable group of finite Morley rank. Because \( Z\left( {G/{Z}_{n + 1}\left( G\right) }\right) = \) \( {Z}_{n + 1}\left( G\right) /{Z}_{n}\left( G\right) \) is finite, \( G/{Z}_{n}\left( G\right) \) has finite center. By Theorem 7.3.12, \( G/{Z}_{n}\left( G\right) \), and hence \( G \), interprets an algebraically closed field. ## 7.4 Definable Groups in Algebraically Closed Fields In this section, we will investigate groups interpretable in algebraically closed fields. Our goal is to show that any such group is definably isomorphic to an algebraic group. If \( G \) is interpretable in an algebraically closed field \( K \), then, by elimination of imaginaries, there is a definable \( X \subseteq {K}^{n} \) and a definable \( f : X \times X \rightarrow X \) such that \( \left( {G, \cdot }\right) \) is definably isomorphic to \( \left( {X, f}\right) \) . Thus, to study interpretable groups it suffices to study groups where the underlying set and multiplication are definable sets. In algebraically closed fields, the definable subsets are exactly the constructible subsets, so our goal is to show that any constructible group is definably isomorphic to an algebraic group. ## Varieties We have already encountered two types of algebraic groups: linear algebraic groups and elliptic curves. We define the category of algebraic groups to include both types of examples. We begin by defining an abstract algebraic variety. The idea is that we build abstract varieties from Zariski closed subsets of \( {K}^{n} \) in the same way that we build manifolds from open balls in \( {\mathbb{R}}^{n} \) or \( {\mathbb{C}}^{n} \) . Definition 7.4.1 A variety \( {}^{1} \) is a topological space \( V \) such that \( V \) has a finite open cover \( V = {V}_{1} \cup \ldots \cup {V}_{n} \) where for \( i = 1,\ldots, n \) there is \( {U}_{i} \subseteq {K}^{{n}_{i}} \) a Zariski closed set and a homeomorphism \( {f}_{i} : {V}_{i} \rightarrow {U}_{i} \) such that: i) \( {U}_{i, j} = {f}_{i}\left( {{V}_{i} \cap {V}_{j}}\right) \) is an open subset of \( {U}_{i} \), and ii) \( {f}_{i, j} = {f}_{i} \circ {f}_{j}^{-1} : {U}_{j, i} \rightarrow {U}_{i, j} \) is a rational map. We call \( {f}_{1},\ldots ,{f}_{n} \) charts for \( V \) . Varieties arise in many natural ways. Let \( K \) be an algebraically closed field. Lemma 7.4.2 i) If \( V \subseteq {K}^{n} \) is Zariski closed, then \( V \) is a variety. ii) If \( V \subseteq {K}^{n} \) is Zariski closed and \( O \subseteq {K}^{n} \) is Zariski open, then \( V \cap O \) is a variety. iii) \( {\mathbb{P}}^{1}\left( K\right) \) is a variety. iv) If \( V \subseteq {\mathbb{P}}^{n}\left( K\right) \) is Zariski closed and \( O \subseteq {\mathbb{P}}^{n}\left( K\right) \) is Zariski open, then \( V \cap O \) is a variety. ## Proof i) Clear. ii) Let \( O = \mathop{\bigcup }\limits_{{i = 1}}^{m}{O}_{i} \) where \( {O}_{i} = \left\{ {x \in {K}^{n} : {g}_{i}\left( x\right) \neq 0}\right\} \) for some \( {g}_{i} \in \) \( K\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right

Theorem 7.3.12 If \( G \) is an infinite connected solvable group of finite Morley rank with finite center, then \( G \) interprets an algebraically closed field.

We will prove this by induction on the rank of \( G \). We first argue that we may, without loss of generality, assume that \( G \) is centerless. Suppose that \( Z\left( G\right) \) is finite. We claim that \( G/Z\left( G\right) \) is centerless. Let \( a \in G \) such that \( a/Z\left( G\right) \in Z\left( {G/Z\left( G\right) }\right) \). For all \( g \in G,{a}^{-1}{g}^{-1}{ag} \in Z\left( G\right) \). Thus, \( {a}^{-1}{a}^{G} \subseteq Z\left( G\right) \), and, hence, \( {a}^{G} \) is finite. Thus, \( \left\lbrack {G : C\left( a\right) }\right\rbrack \) is finite. Because \( G \) is connected, \( C\left( a\right) = G \) and \( a \in Z\left( G\right) \). Thus, \( G/Z\left( G\right) \) is solvable and centerless. By Exercise 7.6.4, \( G/Z\left( G\right) \) is connected. Because \( G/Z\left( G\right) \) is interpretable in \( G \), if \( G/Z\left( G\right) \) interprets an algebraically closed field, so does \( G \). Let \( A \trianglelefteq G \) be a minimal infinite definable normal subgroup. By Lemma 7.3.11 and Corollary 7.3.5, \( A \) is solvable and \( {A}^{\prime } \) is a proper connected normal definable subgroup of \( A \). Moreover, any automorphism of \( A \) fixes \( {A}^{\prime } \) setwise, thus \( {A}^{\prime } \trianglelefteq G \). By choice of \( A,{A}^{\prime } = \{ 1\} \) and \( A \) is Abelian. Let \( C\left( A\right) = \{ g \in G : {ga} = {ag} \) for all \( a \in A\} \). Because \( G \) is centerless, \( C\left( A\right) \neq G \). Because \( A \) is normal, so is \( C\left( A\right) \). Let \( {G}_{1} = G/C\left( A\right) \). If \( g, h \in G \) and \( g = {hc} \) where \( c \in C\left( A\right) \), then for \( a \in A \) \[ {ha}{h}^{-1} = {gca}{c}^{-1}{g}^{-1} = {ga}{g}^{-1}. \] Thus, \( {G}_{1} \) acts on \( A \) by conjugation. If \( {ga}{g}^{-1} = {ha}{h}^{-1} \) for all \( a \in A \), then \( {g}^{-1}{ha}{h}^{-1}{ga} = a \) and \( {g}^{-1}h \in C\left( A\right) \). Thus, \( g/C\left( A\right) = h/C\left( A\right) \) and \( {G}_{1} \) acts faithfully on

Lemma 3. If Dedekind's functional equation (5) \[ \eta \left( {A\tau }\right) = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \tau \right) , \] is satisfied for some \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in \( \Gamma \) with \( c > 0 \) and \( \varepsilon \left( A\right) \) given by (2), then it is also satisfied for \( A{T}^{m} \) and for \( {AS} \) . That is,(5) implies (6) \[ \eta \left( {A{T}^{m}\tau }\right) = \varepsilon \left( {A{T}^{m}}\right) \{ - i\left( {{c\tau } + d + {mc}}\right) {\} }^{1/2}\eta \left( \tau \right) , \] and (7) \[ \eta \left( {AS\tau }\right) = \varepsilon \left( {AS}\right) \{ - i\left( {{d\tau } - c}\right) {\} }^{1/2}\eta \left( \tau \right) \;\text{ if }d > 0, \] whereas (8) \[ \eta \left( {AS\tau }\right) = \varepsilon \left( {AS}\right) \{ - i\left( {-{d\tau } + c}\right) {\} }^{1/2}\eta \left( \tau \right) \;\text{ if }d < 0. \] 192 Proof. Replace \( \tau \) by \( {T}^{m}\tau \) in (5) to obtain \[ \eta \left( {A{T}^{m}\tau }\right) = \varepsilon \left( A\right) {\left\{ -i\left( c{T}^{m}\tau + d\right) \right\} }^{1/2}\eta \left( {{T}^{m}\tau }\right) \] \[ = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + {mc} + d}\right) {\} }^{1/2}{e}^{{\pi im}/{12}}\eta \left( \tau \right) . \] Using Lemma 2 we obtain (6). Now replace \( \tau \) by \( {S\tau } \) in (5) to get \[ \eta \left( {AS\tau }\right) = \varepsilon \left( A\right) \{ - i\left( {{cS\tau } + d}\right) {\} }^{1/2}\eta \left( {S\tau }\right) . \] Using Theorem 3.1 we can write this as (9) \[ \eta \left( {AS\tau }\right) = \varepsilon \left( A\right) \{ - i\left( {{cS\tau } + d}\right) {\} }^{1/2}\{ - {i\tau }{\} }^{1/2}\eta \left( \tau \right) . \] If \( d > 0 \), we write \[ {cS\tau } + d = - \frac{c}{\tau } + d = \frac{{d\tau } - c}{\tau } \] hence, \[ - i\left( {{cS\tau } + d}\right) = \frac{-i\left( {{d\tau } - c}\right) }{-{i\tau }}{e}^{-{\pi i}/2}, \] and therefore, \( \{ - i\left( {{cS\tau } + d}\right) {\} }^{1/2}\{ - {i\tau }{\} }^{1/2} = {e}^{-{\pi i}/4}\{ - i\left( {{d\tau } - c}\right) {\} }^{1/2} \) . Using this in (9) together with Lemma 3, we obtain (7). If \( d < 0 \), we write \[ {cS\tau } + d = - \frac{c}{\tau } + d = \frac{-{d\tau } + c}{-\tau } \] so that in this case we have \[ - i\left( {{cS\tau } + d}\right) = \frac{-i\left( {-{d\tau } + c}\right) }{-{i\tau }}{e}^{{\pi i}/2}, \] and therefore, \( \{ - i\left( {{cS\tau } + d}\right) {\} }^{1/2}\{ - {i\tau }{\} }^{1/2} = {e}^{{\pi i}/4}\{ - i\left( {-{d\tau } + c}\right) {\} }^{1/2} \) . Using this in (9) together with Lemma 3, we obtain (8). Remark on the root of unity \( \varepsilon \left( A\right) \) Dedekind's functional equation (1), with an unspecified 24th root of unity \( \varepsilon \left( A\right) \), follows immediately by extracting 24th roots in the functional equation for \( \Delta \left( \tau \right) \) . Much of the effort in this theory is directed at showing that the root of unity \( \varepsilon \left( A\right) \) has the form given in (2). It is of interest to note that a simple argument due to Dedekind gives the following theorem: Theorem. If (1) holds whenever \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \in \Gamma \) and \( c \neq 0 \), then \[ \varepsilon \left( A\right) = \exp \left\{ {{\pi i}\left( {\frac{a + d}{12c} - f\left( {d, c}\right) }\right) }\right\} \] for some rational number \( f\left( {d, c}\right) \) depending only on \( d \) and \( c \) . Proof. Let \[ {A\tau } = \frac{{a\tau } + b}{{c\tau } + d}\;\text{ and }\;{A}^{\prime }\tau = \frac{{a}^{\prime }\tau + {b}^{\prime }}{{c\tau } + d} \] be two transformations in \( \Gamma \) having the same denominator \( {c\tau } + d \) . Then \[ {ad} - {bc} = 1\;\text{ and }\;{a}^{\prime }d - {b}^{\prime }c = 1, \] so both pairs \( a, b \) and \( {a}^{\prime },{b}^{\prime } \) are solutions of the linear Diophantine equation \[ {xd} - {yc} = 1\text{.} \] Consequently, there is an integer \( n \) such that \[ {a}^{\prime } = a + {nc},\;{b}^{\prime } = b + {nd}. \] Hence, \[ {A}^{\prime }\tau = \frac{\left( {a + {nc}}\right) \tau + \left( {b + {nd}}\right) }{{c\tau } + d} = \frac{{a\tau } + b}{{c\tau } + d} + n = {A\tau } + n. \] Therefore, we have \[ \eta \left( {{A}^{\prime }\tau }\right) = \eta \left( {{A\tau } + n}\right) = {e}^{{\pi in}/{12}}\eta \left( {A\tau }\right) = {e}^{{\pi in}/{12}}\varepsilon \left( A\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \tau \right) , \] because of (1). On the other hand, (1) also gives us \[ \eta \left( {{A}^{\prime }\tau }\right) = \varepsilon \left( {A}^{\prime }\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \tau \right) . \] Comparing the two expressions for \( \eta \left( {{A}^{\prime }\tau }\right) \), we find \( \varepsilon \left( {A}^{\prime }\right) = {e}^{{\pi in}/{12}}\varepsilon \left( A\right) \) . But \( n = \left( {{a}^{\prime } - a}\right) /c \), so \[ \varepsilon \left( {A}^{\prime }\right) = \exp \left( \frac{{\pi i}\left( {{a}^{\prime } - a}\right) }{12c}\right) \varepsilon \left( A\right) , \] or \[ \exp \left( {-\frac{{\pi i}{a}^{\prime }}{12c}}\right) \varepsilon \left( {A}^{\prime }\right) = \exp \left( {-\frac{\pi ia}{12c}}\right) \varepsilon \left( A\right) . \] This shows that the product \( \exp \left( {-\frac{\pi ia}{12c}}\right) \varepsilon \left( A\right) \) depends only on \( c \) and \( d \) . Therefore, the same is true for the product \[ \exp \left( {-\frac{{\pi i}\left( {a + d}\right) }{12c}}\right) \varepsilon \left( A\right) \] This complex number has absolute value 1 and can be written as \[ \exp \left( {-\frac{{\pi i}\left( {a + d}\right) }{12c}}\right) \varepsilon \left( A\right) = \exp \left( {-{\pi if}\left( {d, c}\right) }\right) \] for some real number \( f\left( {d, c}\right) \) depending only on \( c \) and \( d \) . Hence, \[ \varepsilon \left( A\right) = \exp \left\{ {{\pi i}\left( {\frac{a + d}{12c} - f\left( {d, c}\right) }\right) }\right\} . \] Because \( {\varepsilon }^{24} = 1 \), it follows that \( {12cf}\left( {d, c}\right) \) is an integer, so \( f\left( {d, c}\right) \) is a rational number. ## Bibliography 1. Apostol, Tom M. Sets of values taken by Dirichlet's L-series. Proc. Sympos. Pure Math., Vol. VIII, 133-137. Amer. Math. Soc., Providence, R.I., 1965. MR 31 # 1229. 2. Apostol, Tom M. Calculus, Vol. II, 2nd Edition. John Wiley and Sons, Inc. 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Princeton Univ. Press, Princeton, New Jersey, 1962. MR 24 #A2664. 12. Gupta, Hansraj. An identity. Res. Bull. Panjab Univ. (N.S.) 15 (1964), 347-349 (1965). MR 32 #4070. 13. Hardy, G. H. and Ramanujan, S. Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. (2) 17 (1918), 75-115. 14. Haselgrove, C. B. A disproof of a conjecture of Pólya. Mathematika 5 (1958), 141-145. MR 21 #3391. 15. Hecke, E. Über die Bestimmung Dirichletscher Reihen durch ihre Funktional-gleichung. Math. Ann. 112 (1936), 664-699. 16. Hecke, E. Über Modulfunktionen und die Dirichlet Reihen mit Eulerscher Produkt-entwicklung. I. Math. Ann. 114 (1937), 1-28; II. 316-351. 17. Iseki, Shô. The transformation formula for the Dedekind modular function and related functional equations. Duke Math. J. 24 (1957), 653-662. MR 19, 943a. 18. Knopp, Marvin I. Modular Functions in Analytic Number Theory. Markham Mathematics Series, Markham Publishing Co., Chicago, 1970. MR 42 #198. 19. Lehmer, D. H. Ramanujan’s function \( \tau \left( n\right) \) . Duke Math. J. 10 (1943), 483-492. MR 5, 35b. 20. Lehmer, D. H. Properties of the coefficients of the modular invariant \( J\left( \tau \right) \) . Amer. J. Math. 64 (1942), 488-502. MR 3, 272c. 21. Lehmer, D. H. On the Hardy-Ramanujan series for the partition function. J. London Math. Soc. 12 (1937), 171-176. 22. Lehmer, D. H. On the remainders and convergence of the series for the partition function. Trans. Amer. Math. Soc. 46 (1939), 362-373. MR 1, 69c. 23. Lehner, Joseph. Divisibility properties of the Fourier coefficients of the modular invariant \( j\left( \tau \right) \) . Amer. J. Math. 71 (1949),136-148. MR 10,357a. 24. Lehner, Joseph. Further congruence properties of the Fourier coefficients of the modular invariant \( j\left( \tau \right) \) . Amer. J. Math. 71 (1949),373-386. MR 10,357b. 25. Lehner, Joseph, and Newman, Morris. Sums involving Farey fractions. Acta Arith. 15 (1968/69), 181-187. MR 39 #134. 26. Lehner, Joseph. Lectures on Modular Forms. National Bureau of Standards, Applied Mathematics Series, 61, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1969. MR 41 #8666. 27. LeVeque, William Judson. Reviews in Number Theory, 6 volumes. American Math. Soc., Providence, Rhode Island, 1974. 28. Mordell, Louis J. On Mr. Ramanujan’s empirical expansions of modular functions. Proc. Cambridge Phil. Soc. 19 (1917), 117-124. 29. Neville, Eric H. The structure of Farey series. P

Lemma 3. If Dedekind's functional equation \[ \eta \left( {A\tau }\right) = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + d}\right) {\} }^{1/2}\eta \left( \tau \right) , \] is satisfied for some \( A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in \( \Gamma \) with \( c > 0 \) and \( \varepsilon \left( A\right) \) given by (2), then it is also satisfied for \( A{T}^{m} \) and for \( {AS} \) . That is,(5) implies \[ \eta \left( {A{T}^{m}\tau }\right) = \varepsilon \left( {A{T}^{m}}\right) \{ - i\left( {{c\tau } + d + {mc}}\right) {\} }^{1/2}\eta \left( \tau \right) , \] and \[ \eta \left( {AS\tau }\right) = \varepsilon \left( {AS}\right) \{ - i\left( {{d\tau } - c}\right) {\} }^{1/2}\eta \left( \tau \right) \;\text{ if }d > 0, \] whereas \[ \eta \left( {AS\tau }\right) = \varepsilon \left( {AS}\right) \{ - i\left( {-{d\tau } + c}\right) {\} }^{1/2}\eta \left( \tau \right) \;\text{ if }d < 0. \]

Proof. Replace \( \tau \) by \( {T}^{m}\tau \) in (5) to obtain \[ \eta \left( {A{T}^{m}\tau }\right) = \varepsilon \left( A\right) {\left\{ -i\left( c{T}^{m}\tau + d\right) \right\} }^{1/2}\eta \left( {{T}^{m}\tau }\right) \] \[ = \varepsilon \left( A\right) \{ - i\left( {{c\tau } + {mc} + d}\right) {\} }^{1/2}{e}^{{\pi im}/{12}}\eta \left( \tau \right) . \] Using Lemma 2 we obtain (6). Now replace \( \(\tau\) by \(S\(\tau\) in (5 to get \[ \(\eta (AS\(\tau\) = \(\varepsilon (A\) \{ -i({cS\(\tau\) + d\} ^{1/2}\(\eta (S\(\tau\) . Using Theorem 3.1 we can write this as (9 \[ \(\eta (AS\(\tau\) = \(\varepsilon (A\) \{ -i({cS\(\τ\) + d\} ^{1/2}\{ -i\(\τ\} ^{1/2}\(\η (\τ\). If d > 0, we write \[ cS\(\τ\) + d = -\frac{c}{\τ} + d = \(\frac{{d\(\τ} - c}{\τ}\) hence, \[ -i({cS\(\τ\) + d\} = \(\frac{-i({d\(\τ} - c}{-i\(\τ\}\{e}^{-\pi i}/2\}, and therefore, \{ -i({cS\(\τ\) + d\} ^{1/2}\{ -i\(\τ\} ^{1/2} = \{e}^{-\pi i}/4\{ -i({d\(\τ} - c\} ^{1/2}. Using this in (9 together with Lemma 3, we obtain (7. If d < 0, we write \[ cS\(\τ\) + d = -\frac{c}{\τ} + d = \(\frac{-{d\(\τ} + c}{-\τ}\) so that in this case we have \[ -i({cS\(\τ\) + d\} = \(\frac{-i(-{d\(\τ} + c}{-i\(\τ\}\{e}^{\pi i}/2\}, and therefore, \{ -i({cS\(\τ\) + d\} ^{1/2}\{ -i(\(\τ\} ^{1/2} = \{e}^{\pi i}/4\{ -i(-{d(\(\

Corollary 4.1.1. The discrete Dirichlet problem \[ {\Delta }_{h}{u}^{h} = 0\;\text{ in }{\Omega }_{h} \] \[ {u}^{h} = {g}^{h}\;\text{ on }{\Gamma }^{h}, \] for given \( {g}^{h} \) has at most one solution. Proof. This follows in the usual manner by applying the maximum principle to the difference of two solutions. It is remarkable that in the discrete case this uniqueness result already implies an existence result: Corollary 4.1.2. The discrete Dirichlet problem \[ {\Delta }_{h}{u}^{h} = 0\;\text{ in }{\Omega }_{h} \] \[ {u}^{h} = {g}^{h}\;\text{ on }{\Gamma }^{h}, \] admits a unique solution for each \( {g}^{h} : {\Gamma }_{h} \rightarrow \mathbb{R} \) . Proof. As already observed, the discrete problem constitutes a finite system of linear equations with the same number of equations and unknowns. Since by Corollary 4.1.1, for homogeneous boundary data \( {g}^{h} = 0 \), the homogeneous solution \( {u}^{h} = 0 \) is the unique solution, the fundamental theorem of linear algebra implies the existence of a solution for an arbitrary right-hand side, i.e., for arbitrary \( {g}^{h} \) . The solution of the discrete Poisson equation \[ {\Delta }_{h}{u}^{h} = {f}^{h}\;\text{ in }{\Omega }^{h} \] (4.1.17) with given \( {f}^{h} \) is similarly simple; here, without loss of generality, we consider only the homogeneous boundary condition \[ {u}^{h} = 0\;\text{ on }{\Gamma }^{h}, \] (4.1.18) because an inhomogeneous condition can be treated by adding a solution of the corresponding discrete Laplace equation. In order to represent the solution, we shall now construct a Green function \( {G}^{h}\left( {x, y}\right) \) . For that purpose, we consider a particular \( {f}^{h} \) in (4.1.17), namely, \[ {f}^{h}\left( x\right) = \left\{ \begin{array}{ll} 0 & \text{ for }x \neq y \\ \frac{1}{{h}^{2}} & \text{ for }x = y \end{array}\right. \] for given \( y \in {\Omega }_{h} \) . Then \( {G}^{h}\left( {x, y}\right) \) is defined as the solution of (4.1.17) and (4.1.18) for that \( {f}^{h} \) . The solution for an arbitrary \( {f}^{h} \) is then obtained as \[ {u}^{h}\left( x\right) = {h}^{2}\mathop{\sum }\limits_{{y \in {\Omega }_{h}}}{G}^{h}\left( {x, y}\right) {f}^{h}\left( y\right) . \] (4.1.19) In order to show that solutions of the discrete Laplace equation \( {\Delta }_{h}{u}^{h} = 0 \) in \( {\Omega }_{h} \) for \( h \rightarrow 0 \) converge to a solution of the Laplace equation \( {\Delta u} = 0 \) in \( \Omega \) , we need estimates for the \( {u}^{h} \) that do not depend on \( h \) . It turns out that as in the continuous case, such estimates can be obtained with the help of the maximum principle. Namely, for the symmetric difference quotient \[ {u}_{i}\left( x\right) \mathrel{\text{:=}} \frac{1}{2h}\left( {u\left( {{x}^{1},\ldots ,{x}^{i - 1},{x}^{i} + h,{x}^{i + 1},\ldots ,{x}^{d}}\right) }\right. \] \[ \left. {-u\left( {{x}^{1},\ldots ,{x}^{i - 1},{x}^{i} - h,{x}^{i + 1},\ldots ,{x}^{d}}\right) }\right) \] \[ = \frac{1}{2}\left( {{u}_{i}\left( x\right) + {u}_{\bar{\imath }}\left( x\right) }\right) \] (4.1.20) we may prove in complete analogy with Corollary 2.2.7 the following result: Lemma 4.1.1. Suppose that in \( {\Omega }_{h} \) , \[ {\Delta }_{h}{u}^{h}\left( x\right) = {f}^{h}\left( x\right) \] (4.1.21) Let \( {x}_{0} \in {\Omega }_{h} \), and suppose that \( {x}_{0} \) and all its neighbors have distance greater than or equal to \( R \) from \( {\Gamma }_{h} \) . Then \[ \left| {{u}_{\widetilde{\imath }}^{h}\left( {x}_{0}\right) }\right| \leq \frac{d}{R}\mathop{\max }\limits_{{\Omega }_{h}}\left| {u}^{h}\right| + \frac{R}{2}\mathop{\max }\limits_{{\Omega }_{h}}\left| {f}^{h}\right| . \] (4.1.22) Proof. Without loss of generality \( i = 1,{x}_{0} = 0 \) . We put \[ \mu \mathrel{\text{:=}} \mathop{\max }\limits_{{\Omega }_{h}}\left| {u}^{h}\right|, M \mathrel{\text{:=}} \mathop{\max }\limits_{{\Omega }_{h}}\left| {f}^{h}\right| . \] We consider once more the auxiliary function \[ {v}^{h}\left( x\right) \mathrel{\text{:=}} \frac{\mu }{{R}^{2}}{\left| x\right| }^{2} + {x}^{1}\left( {R - {x}^{1}}\right) \left( {\frac{\mathrm{d}\mu }{{R}^{2}} + \frac{M}{2}}\right) . \] Because of \[ {\Delta }_{h}{\left| x\right| }^{2} = \mathop{\sum }\limits_{{i = 1}}^{d}\frac{1}{{h}^{2}}\left( {{\left( {x}^{i} + h\right) }^{2} + {\left( {x}^{i} - h\right) }^{2} - 2{\left( {x}^{i}\right) }^{2}}\right) = {2d}, \] we have again \[ {\Delta }_{h}{v}^{h}\left( x\right) = - M \] as well as \[ {v}^{h}\left( {0,{x}^{2},\ldots ,{x}^{d}}\right) \geq 0\;\text{ for all }{x}^{2},\ldots ,{x}^{d}, \] \[ {v}^{h}\left( x\right) \geq \mu \;\text{ for }\left| x\right| \geq R,\;0 \leq {x}^{1} \leq R. \] Furthermore, for \( {\bar{u}}^{h}\left( x\right) \mathrel{\text{:=}} \frac{1}{2}\left( {{u}^{h}\left( {{x}^{1},\ldots ,{x}^{d}}\right) - {u}^{h}\left( {-{x}^{1},{x}^{2},\ldots ,{x}^{d}}\right) }\right) \) , \( \left| {{\Delta }_{h}{\bar{u}}^{h}\left( x\right) }\right| \leq M \) for those \( x \in {\Omega }_{h} \), for which this expression is defined, \[ {\bar{u}}^{h}\left( {0,{x}^{2},\ldots ,{x}^{d}}\right) = 0\;\text{ for all }{x}^{2},\ldots ,{x}^{d}, \] \[ \left| {{\bar{u}}^{h}\left( x\right) }\right| \leq \mu \;\text{ for }\left| x\right| \geq R,\;{x}^{1} \geq 0. \] On the discretization \( {B}_{h}^{ + } \) of the half-ball \( {B}^{ + } \mathrel{\text{:=}} \left\{ {\left| x\right| \leq R,{x}^{1} > 0}\right\} \), we thus have \[ {\Delta }_{h}\left( {{v}^{h} \pm {\bar{u}}^{h}}\right) \leq 0 \] as well as \[ {v}^{h} \pm {\bar{u}}^{h} \geq 0\;\text{ on the discrete boundary of }{B}_{h}^{ + } \] (in order to be precise, here one should take as the discrete boundary all vertices in the exterior of \( {B}^{ + } \) that have at least one neighbor in \( {B}^{ + } \) ). The maximum principle (Theorem 4.1.1) yields \[ \left| {\bar{u}}^{h}\right| \leq {v}^{h}\;\text{ in }{B}_{h}^{ + }, \] and hence \[ \left| {{u}_{\widetilde{i}}^{h}\left( 0\right) }\right| = \frac{1}{h}\left| {{\bar{u}}^{h}\left( {h,0,\ldots ,0}\right) }\right| \leq \frac{1}{h}{v}^{h}\left( {h,0,\ldots ,0}\right) \] \[ \leq \frac{\mathrm{d}\mu }{R} + \frac{R}{2}M + \frac{\mu }{{R}^{2}}\left( {1 - d}\right) h. \] For solutions of the discrete Laplace equation \[ {\Delta }_{h}{u}^{h} = 0\;\text{ in }{\Omega }_{h} \] (4.1.23) we then inductively get estimates for higher-order difference quotients, because if \( {u}^{h} \) is a solution, so are all difference quotients \( {u}_{i}^{h},{u}_{\bar{i}}^{h},{u}_{\bar{i}}^{h}{u}_{i\bar{i}}^{h},{u}_{\bar{i}\bar{i}}^{\bar{h}} \), etc. For example, from (4.1.22) we obtain for a solution of (4.1.23) that if \( {x}_{0} \) is far enough from the boundary \( {\Gamma }_{h} \), then \[ \left| {{u}_{\widetilde{u}}^{h}\left( {x}_{0}\right) }\right| \leq \frac{d}{R}\mathop{\max }\limits_{{\Omega }_{h}}\left| {u}_{\widetilde{i}}^{h}\right| \leq \frac{{d}^{2}}{{R}^{2}}\mathop{\max }\limits_{{\bar{\Omega }}_{h}}\left| {u}^{h}\right| = \frac{{d}^{2}}{{R}^{2}}\mathop{\max }\limits_{{\Gamma }_{h}}\left| {u}^{h}\right| . \] (4.1.24) Thus, by induction, we can bound difference quotients of any order, and we obtain the following theorem: Theorem 4.1.2. Ifall solutions \( {u}^{h} \) of \[ {\Delta }_{h}{u}^{h} = 0\;\text{ in }{\Omega }_{h} \] are bounded independently of \( h \) (i.e., \( \mathop{\max }\limits_{{\Gamma }_{h}}\left| {u}^{h}\right| \leq \mu \) ), then in any subdomain \( \widetilde{\Omega } \subset \subset \Omega \), some subsequence of \( {u}^{h} \) converges to a harmonic function as \( h \rightarrow 0 \) . Convergence here first means convergence with respect to the supremum norm, i.e., \[ \mathop{\lim }\limits_{{n \rightarrow 0}}\mathop{\max }\limits_{{x \in {\Omega }_{n}}}\left| {{u}_{n}\left( x\right) - u\left( x\right) }\right| = 0 \] with harmonic \( u \) . By the preceding considerations, however, the difference quotients of \( {u}_{n} \) converge to the corresponding derivatives of \( u \) as well. We wish to briefly discuss some aspects of difference equations that are important in numerical analysis. There, for theoretical reasons, one assumes that one already knows the existence of a smooth solution of the differential equation under consideration, and one wants to approximate that solution by solutions of difference equations. For that purpose, let \( L \) be an elliptic differential operator and consider discrete operators \( {L}_{h} \) that are applied to the restriction of a function \( u \) to the lattice \( {\Omega }_{h} \) . Definition 4.1.1. The difference scheme \( {L}_{h} \) is called consistent with \( L \) if \[ \mathop{\lim }\limits_{{h \rightarrow 0}}\left( {{Lu} - {L}_{h}u}\right) = 0 \] for all \( u \in {C}^{2}\left( \bar{\Omega }\right) \) . The scheme \( {L}_{h} \) is called convergent to \( L \) if the solutions \( u,{u}^{h} \) of \[ {Lu} = f\;\text{ in }\Omega, u = \varphi \text{ on }\partial \Omega , \] \( {L}_{h}{u}^{h} = {f}^{h}\; \) in \( {\Omega }_{h} \), where \( {f}^{h} \) is the restriction of \( f \) to \( {\Omega }_{h} \) , \( {u}^{h} = {\varphi }^{h}\; \) on \( {\Gamma }_{h} \), where \( {\varphi }^{h} \) is the restriction to \( {\Omega }_{h} \) of a continuous extension of \( \varphi \) , satisfy \[ \mathop{\lim }\limits_{{h \rightarrow 0}}\mathop{\max }\limits_{{x \in {\Omega }_{h}}}\left| {{u}^{h}\left( x\right) - u\left( x\right) }\right| = 0. \] In order to see the relation between convergence and consistency we consider the "global error" \[ \sigma \left( x\right) \mathrel{\text{:=}} {u}^{h}\left( x\right) - u\left( x\right) \] and the "local error" \[ s\left( x\right) \mathrel{\text{:=}} {L}_{h}u\left( x\right) - {Lu}\left( x\right) \] and compute, for \( x \in {\Omega }_{h} \) , \[ {L}_{h}\sigma \left( x\right) = {L}_{h}{u}^{h}\left( x\right) - {L}_{h}u\left( x\right) = {f}^{h}\left( x\right) - {Lu}\left( x\right) - s\left( x\right) \] \[ = - s\left( x\right) \text{, since}{f}^{h}\left( x\right) = f\left( x\right) =

Corollary 4.1.1. The discrete Dirichlet problem \[ {\Delta }_{h}{u}^{h} = 0\;\text{ in }{\Omega }_{h} \] \[ {u}^{h} = {g}^{h}\;\text{ on }{\Gamma }^{h}, \] for given \( {g}^{h} \) has at most one solution.

This follows in the usual manner by applying the maximum principle to the difference of two solutions.

Theorem 11.6.9. Let \( g \) and \( h \) be hyperbolic with axes and translation lengths \( {A}_{g},{A}_{h},{T}_{g} \) and \( {T}_{h} \) respectively. Suppose that \( \langle g, h\rangle \) is discrete and nonelementary and that no images of \( {A}_{g} \) and \( {A}_{h} \) cross. Then \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \cosh \rho \left( {{A}_{g},{A}_{h}}\right) \geq \cosh \left( {\frac{1}{2}{T}_{g}}\right) \cosh \left( {\frac{1}{2}{T}_{h}}\right) - \frac{1}{2}. \] If \( \langle g, h\rangle \) has no elliptic elements, we can replace \( - \frac{1}{2}{by} + 1 \) (and the lower bound by 2). If \( g \) is a simple hyperbolic element in \( \langle g, h\rangle \) this result can be applied with \( h \) being any conjugate, say \( {fg}{f}^{-1} \), of \( g \) . Thus (by elementary manipulation) we obtain the next inequality. Corollary 11.6.10. If \( g \) and \( h \) are hyperbolic elements generating a discrete non-elementary group and if \( g \) is a simple hyperbolic element in this group, then for all \( f \) in \( \langle g, h\rangle \), either \( f\left( {A}_{g}\right) = {A}_{g} \) or \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \frac{1}{2}\rho \left( {{A}_{g}, f{A}_{g}}\right) \geq \frac{1}{2}. \] This bound is best possible. The next example shows that the lower bound of \( \frac{1}{2} \) is best possible. Example 11.6.11. Construct the polygon \( D \) as in Figure 11.6.3 where \( f \) (elliptic of order two) and \( g \) (hyperbolic) pair the sides of \( D \) . By Poincaré’s Theorem, \( D \) is a fundamental polygon for \( \langle f, g\rangle \) and as \( g \) pairs the sides of \( D, g \) must be a simple hyperbolic element. Finally, \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \frac{1}{2}\rho \left( {{A}_{g}, f{A}_{g}}\right) = \sinh \frac{1}{2}\rho \left( {L,{L}^{\prime }}\right) \sinh \rho \left( {0,{A}_{g}}\right) \] \[ = \cos \left( {\pi /3}\right) \text{.} \]  Figure 11.6.3  Figure 11.6.4 Proof of Theorem 11.6.9. Consider Figure 11.6.4. As \( g \) (or \( {g}^{-1} \) ) is \( {\sigma }_{3}{\sigma }_{1} \) and \( h\left( {\text{or}{h}^{-1}}\right) \) is \( {\sigma }_{2}{\sigma }_{3} \) we see that \( {\sigma }_{2}{\sigma }_{1} \) is in \( G \) . If \( G \) has no elliptic elements, then \( {L}_{1} \) and \( {L}_{2} \) cannot intersect (this case is not illustrated) and from Theorem 7.19.2 we obtain \( \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \cosh \rho \left( {{A}_{g},{A}_{h}}\right) = \cosh \left( {\frac{1}{2}{T}_{g}}\right) \cosh \left( {\frac{1}{2}{T}_{h}}\right) + \cosh \rho \left( {{L}_{1},{L}_{2}}\right) . \) This yields the second inequality. If \( {L}_{1} \) and \( {L}_{2} \) intersect, say at an angle \( \theta \), then \( \theta = {2\pi p}/q \) for some coprime integers \( p \) and \( q \) . If \( \theta > {2\pi }/q \) we can rotate \( {A}_{h} \) about the point of intersection to an image of itself which is closer to (but, by assumption, not intersecting) \( {A}_{g} \) . Thus if, in the argument above, we replace \( h \) by a conjugate \( {fh}{f}^{-1} \) of \( h \) with the property that its axis \( f\left( {A}_{h}\right) \) is as close as possible to (but distinct from) \( {A}_{g} \), we find that \[ \rho \left( {{A}_{g},{A}_{h}}\right) \geq \rho \left( {{A}_{g}, f{A}_{h}}\right) \] and the corresponding \( \theta \) satisfies \( \theta \leq {2\pi }/q \leq {2\pi }/3 \) as obviously \( \theta < \pi \) . Thus from Theorem 7.18.1 we obtain the first inequality, namely \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \cosh \rho \left( {{A}_{g},{A}_{h}}\right) \geq \cosh \left( {\frac{1}{2}{T}_{g}}\right) \cosh \left( {\frac{1}{2}{T}_{h}}\right) + \cos \left( {{2\pi }/3}\right) . \] Theorems 11.6.8 and 11.6.9 yield the following bound on \( P\left( {g, h}\right) \) . Theorem 11.6.12. Let \( g \) and \( h \) be hyperbolic elements which generate a discrete non-elementary group. Then \( P\left( {g, h}\right) \geq \cos \left( {{3\pi }/7}\right) \) . Proof. If the axes of \( g \) and \( h \) cross at \( w \), say, then obviously (using the notation of Theorems 11.6.8 and 11.6.9) \[ \begin{array}{l} P\left( {g, h}\right) = \sinh \frac{1}{2}\rho \left( {w,{gw}}\right) \sinh \frac{1}{2}\rho \left( {w,{hw}}\right) \end{array} \] \[ = \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \] \[ \geq \cos \left( {{3\pi }/7}\right) \text{.} \] The same inequality holds if any images of \( {A}_{g} \) and \( {A}_{h} \) cross. If not, then Theorem 11.6.9 is applicable and we obtain \( \sinh \frac{1}{2}\rho \left( {z,{gz}}\right) \sinh \frac{1}{2}\rho \left( {z,{hz}}\right) = \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \cosh \rho \left( {z,{A}_{g}}\right) \cosh \rho \left( {z,{A}_{h}}\right) \) \[ \geq \frac{1}{2}\mathrm{{sinh}}\left( {\frac{1}{2}{T}_{g}}\right) \mathrm{{sinh}}\left( {\frac{1}{2}{T}_{h}}\right) \mathrm{{cosh}}\left\lbrack {\rho \left( {z,{A}_{g}}\right) + \rho \left( {z,{A}_{h}}\right) }\right\rbrack \] \[ \geq \frac{1}{2}\mathrm{{sinh}}\left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \cosh \rho \left( {{A}_{g},{A}_{h}}\right) \] \[ \geq \frac{1}{2}\left\lbrack {\cosh \left( {\frac{1}{2}{T}_{g}}\right) \cosh \left( {\frac{1}{2}{T}_{h}}\right) - \frac{1}{2}}\right\rbrack \] \[ \geq \frac{1}{4} \] \[ > \cos \left( {{3\pi }/7}\right) \text{.} \] Finally, we consider \( M\left( {g, h}\right) \) for one elliptic and one hyperbolic element. Theorem 11.6.13. Let \( g \) be hyperbolic and let \( h \) be elliptic of order \( q\left( {q \geq 2}\right) \) . If \( \langle g, h\rangle \) is discrete and non-elementary, then \( M\left( {g, h}\right) \geq 1/\sqrt{8} \) . Proof. If \( g \) is a non-simple hyperbolic element of \( \langle g, h\rangle \) then (from Theorem 11.6.8) \[ M\left( {g, h}\right) \geq \sinh \left( {\frac{1}{2}{T}_{g}}\right) \] \[ \geq {\left\lbrack \cos \left( 3\pi /7\right) \right\rbrack }^{1/2} \] \[ > 1/\sqrt{8}\text{.} \] We may now assume that \( g \) is a simple hyperbolic element. In this case, the fixed point \( v \) of the elliptic \( h \) cannot lie on \( {A}_{g} \) and a rotation of \( {A}_{g} \) of an angle \( {2\pi }/q \) about \( v \) must map \( {A}_{g} \) onto a disjoint image which we may assume is \( h\left( {A}_{g}\right) \) : see Figure 11.6.5. From Section 7.17 we have \[ \cosh \rho \left( {v,{A}_{g}}\right) \sin \left( {\pi /q}\right) = \cosh \frac{1}{2}\rho \left( {{A}_{g}, h{A}_{g}}\right) \] \[ \geq \sinh \frac{1}{2}\rho \left( {{A}_{g}, h{A}_{g}}\right) \] and, from Corollary 11.6.10 (applied to \( \left\langle {g,{hg}{h}^{-1}}\right\rangle \) ), \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \frac{1}{2}\rho \left( {{A}_{g}, h{A}_{g}}\right) \geq \frac{1}{2}. \] Thus \[ \cosh \rho \left( {v,{A}_{g}}\right) \sin \left( {\pi /q}\right) \sinh \left( {\frac{1}{2}{T}_{g}}\right) \geq \frac{1}{2}. \] This expresses a geometric constraint between the parameters \( {T}_{g},{2\pi }/q \) and the separation of \( g \) and \( h \) as measured by \( \rho \left( {v,{A}_{g}}\right) \) . Writing \[ m = \max \left\{ {\sinh \frac{1}{2}\rho \left( {z,{gz}}\right) ,\sinh \frac{1}{2}\rho \left( {z,{hz}}\right) }\right\} \]  Figure 11.6.5 ## we have \[ \frac{1}{2} \leq \sin \left( {\pi /q}\right) \sinh \left( {\frac{1}{2}{T}_{g}}\right) \cosh \left\lbrack {\rho \left( {v, z}\right) + \rho \left( {z,{A}_{g}}\right) }\right\rbrack \] \[ = \sin \left( {\pi /q}\right) \sinh \left( {\frac{1}{2}{T}_{g}}\right) \left\lbrack {\cosh \rho \left( {v, z}\right) \cosh \rho \left( {z,{A}_{g}}\right) + \sinh \rho \left( {v, z}\right) \sinh \rho \left( {z,{A}_{g}}\right) }\right\rbrack \] \[ \leq m\sin \left( {\pi /q}\right) {\left\lbrack 1 + {\sinh }^{2}\rho \left( v, z\right) \right\rbrack }^{1/2} + {m}^{2} \] \[ \leq m{\left\lbrack {\sin }^{2}\left( \pi /q\right) + {m}^{2}\right\rbrack }^{1/2} + {m}^{2} \] \[ \leq m{\left( 1 + {m}^{2}\right) }^{1/2} + {m}^{2} \] which certainly implies that \( m \geq 1/\sqrt{8} \) . Collecting together all the results in this section we obtain a universal lower bound on \( M\left( {g, h}\right) \) . Theorem 11.6.14. If \( g \) and \( h \) generate a non-elementary discrete group, then \( M\left( {g, h}\right) \geq 0 \cdot {1318}\ldots \) and this lower bound is attained by two elliptic generators of the \( \left( {0 : 2,3,7}\right) \) -Triangle group. We end this section by completing an earlier proof. Proof of Theorem 11.2.4(2). We consider an accidental cycle of four vertices, say \[ {v}_{1},\;f\left( {v}_{1}\right) = {v}_{2},\;g\left( {v}_{1}\right) = {v}_{3},\;h\left( {v}_{1}\right) = {v}_{4} \] on the boundary of a Dirichlet polygon: thus the \( {v}_{j} \) lie on a circle with, say, centre \( w \) and radius \( r \) . If \( \langle f, g\rangle \) is non-elementary, then, as we have just seen, \[ M\left( {f, g}\right) \geq {0.1318}\ldots \] and so for some \( j\left( { = 2\text{or 3}}\right) \) , \[ {0.1318}\ldots \leq \sinh \frac{1}{2}\rho \left( {{v}_{1},{v}_{j}}\right) \] \[ \leq \sinh \frac{1}{2}\left\lbrack {\rho \left( {{v}_{1}, w}\right) + \rho \left( {w,{v}_{j}}\right) }\right\rbrack \] \[ \leq \sinh r\text{.} \] The same is true of \( \langle g, h\rangle \) or \( \langle h, f\rangle \) is non-elementary: thus it is sufficient to consider the case when all three groups \( \langle g, h\rangle ,\langle h, f\rangle \) and \( \langle f, g\rangle \) are elementary. We assume that these three two-generator groups are elementary. As \( {v}_{1},{v}_{2}

Theorem 11.6.9. Let \( g \) and \( h \) be hyperbolic with axes and translation lengths \( {A}_{g},{A}_{h},{T}_{g} \) and \( {T}_{h} \) respectively. Suppose that \( \langle g, h\rangle \) is discrete and nonelementary and that no images of \( {A}_{g} \) and \( {A}_{h} \) cross. Then \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \cosh \rho \left( {{A}_{g},{A}_{h}}\right) \geq \cosh \left( {\frac{1}{2}{T}_{g}}\right) \cosh \left( {\frac{1}{2}{T}_{h}}\right) - \frac{1}{2}. \] If \( \langle g, h\rangle \) has no elliptic elements, we can replace \( - \frac{1}{2} \) by \( + 1 \) (and the lower bound by 2).

Consider Figure 11.6.4. As \( g \) (or \( {g}^{-1} \)) is \( {\sigma }_{3}{\sigma }_{1} \) and \( h \) (or \( {h}^{-1} \)) is \( {\sigma }_{2}{\sigma }_{3} \), we see that \( {\sigma }_{2}{\sigma }_{1} \) is in \( G \). If \( G \) has no elliptic elements, then \( {L}_{1} \) and \( {L}_{2} \) cannot intersect (this case is not illustrated) and from Theorem 7.19.2 we obtain \[ \sinh \left( {\frac{1}{2}{T}_{g}}\right) \sinh \left( {\frac{1}{2}{T}_{h}}\right) \cosh \rho \left( {{A}_{g},{A}_{h}}\right) = \cosh \left( {\frac{1}{2}{T}_{g}}\right) \cosh \left( {\frac{1}{2}{T}_{h}}\right) + \cosh \rho \left( {{L}_{1},{L}_{2}}\right). \] This yields the second inequality. If \( {L}_{1} \) and \( {L}_{2} \) intersect, say at an angle \( \theta \), then \( \theta = {2\pi p}/q \) for some coprime integers \( p \) and \( q \). If \( \theta > {2\pi }/q \) we can rotate \( {A}_{h} \) about the point of intersection to an image of itself which is closer to (but, by assumption, not intersecting) \( {A}_{g} \). Thus if, in the argument above, we replace \( h \) by a conjugate \( {fh}{f}^{-1} \) of \( h \) with the property that its axis \( f\left( {A}_{h}\right) \) is as close as possible to (but distinct from) \( {A}_{g} \), we find that \[ \rho \left( {{A}_{g},{A}_{h}}\right) \geq \rho \left( {{A}_{g}, f{A}_{h}}\right) \] and the corresponding \( θ\) satisfies \( θ ≤ {2π}/q ≤ {2π}/3\) as obviously θ < π. Thus from Theorem 7.18.1 we obtain the first inequality, namely \[ \sinh

Theorem 1 Given an edge \( {ab} \), denote by \( N\left( {s, a, b, t}\right) \) the number of spanning trees of \( G \) in which the (unique) path from \( s \) to \( t \) contains a and \( b \), in this order. Define \( N\left( {s, b, a, t}\right) \) analogously and write \( N \) for the total number of spanning trees. Finally, let \( {w}_{ab} = \{ N\left( {s, a, b, t}\right) - N\left( {s, b, a, t}\right) \} /N \) . Distribute currents in the edges of \( G \) by sending a current of size \( {w}_{ab} \) from a to \( b \) for every edge \( {ab} \) . Then there is a total current size 1 from \( s \) to \( t \) satisfying the Kirchhoff laws. Proof. To simplify the situation, multiply all currents by \( N \) . Also, for every spanning tree \( T \) and edge \( {ab} \in E\left( G\right) \), let \( {w}^{\left( T\right) } \) be the current of size 1 along the unique \( s - t \) path in \( T \) : \[ {w}_{ab}^{\left( T\right) } = \left\{ \begin{array}{ll} 1 & \text{ if }T\text{ has a path }s\cdots {ab}\cdots t, \\ - 1 & \text{ if }T\text{ has a path }s\cdots {ba}\cdots t, \\ 0 & \text{ otherwise. } \end{array}\right. \] Then \[ N\left( {s, a, b, t}\right) - N\left( {s, b, a, t}\right) = \mathop{\sum }\limits_{T}{w}_{ab}^{\left( T\right) }, \] where the summation is over all spanning trees \( T \) . Therefore, our task is to show that if we send a current of size \( \mathop{\sum }\limits_{T}{w}_{ab}^{\left( T\right) } \) from \( a \) to \( b \) for every edge \( {ab} \), then we obtain a total current of size \( N \) from \( s \) to \( t \) satisfying the Kirchhoff laws. Now, each \( {w}^{\left( T\right) } \) is a current of size 1 from \( s \) to \( t \) satisfying Kirchhoff’s current law, and so their sum is a current of size \( N \) from \( s \) to \( t \) satisfying Kirchhoff’s current law. All we have to show then is that the potential law is also satisfied. As all edges have the same resistance, the potential law claims that the total current in a cycle with some orientation is zero. To show this, we proceed as earlier, but first we reformulate slightly the definition of \( N\left( {s, a, b, t}\right) \) . Call a spanning forest \( F \) of \( G \) a thicket if it has exactly two components, say \( {F}_{s} \) and \( {F}_{t} \), such that \( s \) is in \( {F}_{s} \) and \( t \) is in \( {F}_{t} \) . Then \( N\left( {s, a, b, t}\right) \) is the number of thickets \( F = {F}_{s} \cup {F}_{t} \) for which \( a \in {F}_{s} \) and \( b \in {F}_{t} \), and \( N\left( {s, b, a, t}\right) \) is defined analogously. What is then the contribution of a thicket \( F = {F}_{s} \cup {F}_{t} \) to the total current in a cycle? It is the number of cycle edges from \( {F}_{s} \) to \( {F}_{t} \) minus the number of cycle edges from \( {F}_{t} \) to \( {F}_{s} \) ; so it is zero. Let us write out the second part of the proof more formally, to make it even more evident that we use the basic and powerful combinatorial principle of double counting, or reversing the order of summation. For a thicket \( F = {F}_{s} \cup {F}_{t} \) and an edge \( {ab} \in E\left( G\right) \), set \[ {w}_{ab}^{\left( F\right) } = \left\{ \begin{array}{ll} {w}_{ab}^{\left( F + ab\right) } & \text{ if }F + {ab}\text{ is a spanning tree,} \\ 0 & \text{ otherwise. } \end{array}\right. \] Then \[ \mathop{\sum }\limits_{T}{w}_{ab}^{\left( T\right) } = \mathop{\sum }\limits_{F}{w}_{ab}^{\left( F\right) } \] where the second summation is over all thickets \( F \) . Finally, the total current around a cycle \( {x}_{1}{x}_{2}\cdots {x}_{k} \) of \( G \), with \( {x}_{k + 1} = {x}_{1} \), is \[ \mathop{\sum }\limits_{{i = 1}}^{k}\mathop{\sum }\limits_{F}{w}_{{x}_{i}{x}_{i + 1}}^{\left( F\right) } = \mathop{\sum }\limits_{F}\mathop{\sum }\limits_{{i = 1}}^{k}{w}_{{x}_{i}{x}_{i + 1}}^{\left( F\right) } = 0 \] since \( \mathop{\sum }\limits_{{i = 1}}^{k}{w}_{{x}_{i}{x}_{i + 1}}^{\left( F\right) } = 0 \) for every thicket \( F \) . More importantly, the proof of Theorem 1 can be rewritten to give a solution in the case when the edges have arbitrary conductances. For a spanning tree \( T \) define the weight \( w\left( T\right) \) of \( T \) as the product of the conductances of its edges. Let \( {N}^{ * } \) be the sum of the weights of all the spanning trees, let \( {N}^{ * }\left( {s, a, b, t}\right) \) be the sum of the weights of all the spanning trees in which \( b \) follows \( a \) on the (unique) \( s - t \) path in the tree, and let \( {N}^{ * }\left( {s, b, a, t}\right) = {N}^{ * }\left( {t, a, b, s}\right) \) . Theorem 2 There is a distribution of currents satisfying Ohm's law and Kirchhoff’s laws in which a current of size 1 enters at \( s \) and leaves at \( t \) . The value of the current in an edge \( {ab} \) is given by \( \left\{ {{N}^{ * }\left( {s, a, b, t}\right) - {N}^{ * }\left( {s, b, a, t}\right) }\right\} /{N}^{ * } \) . Let us note an immediate consequence of this result. Corollary 3 If the conductances of the edges are rational and a current of size 1 goes through the network then the current in each edge has rational value. The star-triangle transformation tells us that no matter what the rest of the network is, every 'star' may be replaced by a suitable 'triangle', and vice versa. On an even simpler level, if two networks, \( N \) and \( M \), share only two vertices, say \( a \) and \( b \), and nothing else, and the total resistance of \( M \) from \( a \) to \( b \) is \( r \), then in \( N \cup M \) we may replace \( M \) by an edge \( {ab} \) of resistance \( r \) . In fact, similar transformations can be carried out for networks with any number of vertices of attachment, not only two or three, as above. To be precise, if a part \( M \) of a network is attached to the rest of the network only at a set \( U \) of vertices, then we may replace \( M \) by edges of certain resistances joining the vertices of \( U \) (and introducing no other vertices) without changing the distribution of currents outside \( M \) . We leave this as an exercise (Exercise \( {13}^{ + } \) ). In estimating the resistance of a network, it is frequently convenient to make use of the fact that if the resistance of a wire is increased then the total resistance does not decrease. In particular, if some wires are cut then the total resistance does not decrease; similarly, if some vertices are shorted, i.e., are identified, then the total resistance does not increase. This is obvious if we appeal to physical intuition; however, the problem is that the Kirchhoff laws, together with Ohm's law, determine all currents, potential differences, and so on: having accepted these three laws, we have no right to appeal to any physical intuition. In this chapter we leave this assertion as an exercise (Exercise \( {14}^{ + } \) ), but we shall prove it, several times over, in Chapter IX, when we give a less superficial treatment of electrical networks. ## II. 2 Squaring the Square This is a diversion within a diversion; we feel bound to draw attention to a famous problem arising from recreational mathematics that is related to the theory of electrical networks. Is there a perfect squared square? In other words, is it possible to subdivide a closed square into finitely many (but at least two) square regions of distinct sizes that intersect only at their boundaries?  FIGURE II.6. The perfect squaring of the \( {33} \times {32} \) rectangle, due to Moroń. The answer to this question is far from obvious: on the one hand, there seems to be no reason why there should not be a perfect squared square; on the other hand, it is not easy to find even a perfect squared rectangle, a rectangle divided into finitely many (but at least two) squares of distinct sizes. As it happens, there are perfect squared rectangles: in 1925 Moroń found the perfect squaring of the \( {33} \times {32} \) rectangle shown in Fig. II.6. This squared rectangle has order 9: there are 9 squares in the subdivision; in the figure the number associated with a square is the length of its side. We shall use Moroń’s squared rectangle to illustrate an argument. Let us cut this rectangle out of a sheet of nichrome (or any other material with low conductivity) and let us put rods made of silver (or some other material of high conductivity) at the top and bottom. What happens if we ensure that the silver rod at the top is at 32 volts while the rod at the bottom is kept at 0 ? Trivially, a uniform current will flow from top to bottom. In fact, the potential at a point of the rectangle will depend only on the height of the point: the potential at height \( x \) will be \( x \) volts. Furthermore, there will be no current across the rectangle, only from top to bottom. Thus the current will not change at all if (i) we place silver rods on the horizontal sides of the squares and (ii) cut narrow slits along the vertical sides, as shown in the first picture of Fig. II.7. Now, since silver is a very good conductor, the points of each silver rod have been shortened, so they can be identified. Thus as an electric conductor the whole rectangle behaves like the plane network shown in the second picture of Fig. II.7, in which the conductance of an edge is equal to the conductance of the corresponding square from top to bottom. Clearly, the conductance of a rectangle from top to bottom is proportional to the length of a horizontal side and the resistance is proportional to a vertical side. Consequently, all squares have the same resistance, say unit resistance, so all edges in Fig. II. 7 have unit resistance. What is the potential drop in an edge? It is the side length of the corresponding square. What is the resistance of the whole system? The ratio of the vertical side of the original big rectangle to the horizontal side, that is, \( {32}/{33} \) . ![447b7a74-9b7c-4caa-a5da-f1ea5f61bc7a_62_0.jpg](images/447b7

Theorem 1 Given an edge \( {ab} \), denote by \( N\left( {s, a, b, t}\right) \) the number of spanning trees of \( G \) in which the (unique) path from \( s \) to \( t \) contains a and \( b \), in this order. Define \( N\left( {s, b, a, t}\right) \) analogously and write \( N \) for the total number of spanning trees. Finally, let \( {w}_{ab} = \{ N\left( {s, a, b, t}\right) - N\left( {s, b, a, t}\right) \} /N \) . Distribute currents in the edges of \( G \) by sending a current of size \( {w}_{ab} \) from a to \( b \) for every edge \( {ab} \). Then there is a total current size 1 from \( s \) to \( t \) satisfying the Kirchhoff laws.

To simplify the situation, multiply all currents by \( N \). Also, for every spanning tree \( T \) and edge \( {ab} \in E\left( G\right) \), let \( {w}^{\left( T\right) } \) be the current of size 1 along the unique \( s - t \) path in \( T \): \[ {w}_{ab}^{\left( T\right) } = \left\{ \begin{array}{ll} 1 & \text{ if }T\text{ has a path }s\cdots {ab}\cdots t, \\ - 1 & \text{ if }T\text{ has a path }s\cdots {ba}\cdots t, \\ 0 & \text{ otherwise. } \end{array}\right. \] Then \[ N\left( {s, a, b, t}\right) - N\left( {s, b, a, t}\right) = \mathop{\sum }\limits_{T}{w}_{ab}^{\left( T\right) }, \] where the summation is over all spanning trees \( T \). Therefore, our task is to show that if we send a current of size \( \mathop{\sum }\limits_{T}{w}_{ab}^{\left( T\right) } \) from \( a \) to \( b \) for every edge \( {ab} \), then we obtain a total current of size \( N \) from \( s \) to \( t \) satisfying the Kirchhoff laws. Now, each \( {w}^{\left( T\right) } \) is a current of size 1 from \( s \) to \( t \) satisfying Kirchhoff’s current law, and so their sum is a current of size \( N \) from \( s \) to \( t \) satisfying Kirchhoff’s current law. All we have to show then is that the potential law is also satisfied. As all edges have the same resistance, the potential law claims that the total current in a cycle with some orientation is zero. To show this, we proceed as earlier, but first we reformulate slightly the definition of \( N\left( {s, a, b, t}\right) \). Call a spanning forest \( F \) of \( G \) a thicket if it has exactly two components, say \( {F}_{s} \) and \( {F}_{t} \), such that \( s \) is in \( {F}_{s} \) and \( t \) is in \( {F}_{t} \). Then \( N\left( {s, a, b, t}\right) \) is the number of thickets \( F = {F}_{s} \cup {F}_{t} \) for which \( a \in {F}_{s} \) and \( b \in {F}_{t} \), and

Proposition 1.2 Assume \( \varphi \in {\mathcal{D}}^{m} \), for some \( m \in \mathbb{N} \) . For every integer \( n \geq 1 \), the convolution \( \varphi * {\chi }_{n} \) belongs to \( \mathcal{D} \) and \[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}\varphi * {\chi }_{n} = \varphi \;\text{ in }{\mathcal{D}}^{m} \] Proof. Since the functions \( \varphi \) and \( {\chi }_{n} \) have compact support, so does \( \varphi * {\chi }_{n} \) . More precisely, \[ \operatorname{Supp}\left( {\varphi * {\chi }_{n}}\right) \subset \operatorname{Supp}\varphi + \operatorname{Supp}{\chi }_{n} \subset \operatorname{Supp}\varphi + \bar{B}\left( {0,1/n}\right) \subset \operatorname{Supp}\varphi + \bar{B}\left( {0,1}\right) . \] At the same time, a classical theorem about differentiation under the integral sign easily implies, on the one hand, that \( \varphi * {\chi }_{n} \) is of class \( {C}^{\infty } \) and so \( \varphi * {\chi }_{n} \in \mathcal{D} \), and, on the other, that \( {D}^{p}\left( {\varphi * {\chi }_{n}}\right) = \left( {{D}^{p}\varphi }\right) * {\chi }_{n} \) for \( \left| p\right| \leq m \) . Now, since the support of \( {\chi }_{n} \) is contained in \( \bar{B}\left( {0,1/n}\right) \) and \( \int {\chi }_{n}\left( y\right) {dy} = 1 \) , we get \[ \left( {{D}^{p}\varphi }\right) * {\chi }_{n}\left( x\right) - \left( {{D}^{p}\varphi }\right) \left( x\right) = {\int }_{\left| y\right| \leq 1/n}\left( {{D}^{p}\varphi \left( {x - y}\right) - {D}^{p}\varphi \left( x\right) }\right) {\chi }_{n}\left( y\right) {dy} \] and \[ \mathop{\sup }\limits_{{x \in {\mathbb{R}}^{d}}}\left| {\left( {{D}^{p}\varphi }\right) * {\chi }_{n}\left( x\right) - \left( {{D}^{p}\varphi }\right) \left( x\right) }\right| \leq \mathop{\sup }\limits_{\substack{{x, z \in {\mathbb{R}}^{d}} \\ {\left| {z - x}\right| \leq 1/n} }}\left| {{D}^{p}\varphi \left( z\right) - {D}^{p}\varphi \left( x\right) }\right| . \] Since \( {D}^{p}\varphi \) is uniformly continuous (being continuous and having compact support), we deduce that the sequence \( {\left( {D}^{p}\left( \varphi * {\chi }_{n}\right) \right) }_{n \in \mathbb{N}} \) converges uniformly to \( {D}^{p}\varphi \) . Corollary 1.3 For every \( n \in \mathbb{N} \), the space \( \mathcal{D}\left( \Omega \right) \) is dense in \( {\mathcal{D}}^{m}\left( \Omega \right) \) . In particular, \( \mathcal{D}\left( \Omega \right) \) is dense in \( {C}_{c}\left( \Omega \right) \) . Proof. If \( \varphi \in {\mathcal{D}}^{m}\left( \Omega \right) \), we can consider \( \varphi \) as an element of \( {\mathcal{D}}^{m} \) (by extending it with the value 0 on \( {\mathbb{R}}^{d} \smallsetminus \Omega \) ). Now \[ \operatorname{Supp}\left( {\varphi * {\chi }_{n}}\right) \subset \operatorname{Supp}\varphi + \bar{B}\left( {0,1/n}\right) \] therefore \( \operatorname{Supp}\left( {\varphi * {\chi }_{n}}\right) \subset \Omega \) for \( n \) large enough \( - \) say \( n > 1/d\left( {\operatorname{Supp}\varphi ,{\mathbb{R}}^{d} \smallsetminus \Omega }\right) \) . Then, by the preceding proposition, \( \varphi * {\chi }_{n} \) belongs to \( \mathcal{D}\left( \Omega \right) \) for \( n \) large enough, and \( \mathop{\lim }\limits_{{n \rightarrow + \infty }}\varphi * {\chi }_{n} = \varphi \) in \( {\mathcal{D}}^{m}\left( \Omega \right) \) . Remark. The approximating sequence just constructed preserves positivity. Therefore, if \( \varphi \) is a positive element of \( {\mathcal{D}}^{m}\left( \Omega \right) \), there exists a sequence \( {\left( {\varphi }_{n}\right) }_{n \in \mathbb{N}} \) of positive elements of \( \mathcal{D}\left( \Omega \right) \) that converges to \( \varphi \) in \( {\mathcal{D}}^{m}\left( \Omega \right) \) (namely, \( {\varphi }_{n} = \varphi * {\chi }_{n} \) ). ## 1D \( {C}^{\infty } \) Partitions of Unity We now sharpen Proposition 1.8 on page 53 in the case of \( {\mathbb{R}}^{d} \) . Proposition 1.4 If \( K \) is a compact subset of \( {\mathbb{R}}^{d} \) and \( {O}_{1},\ldots ,{O}_{n} \) are open sets in \( {\mathbb{R}}^{d} \) such that \( K \subset \mathop{\bigcup }\limits_{{j = 1}}^{n}{O}_{j} \), there exist functions \( {\varphi }_{1},\ldots ,{\varphi }_{n} \) in \( \mathcal{D} \) such that \[ 0 \leq {\varphi }_{j} \leq 1\;\text{ and }\;\operatorname{Supp}{\varphi }_{j} \subset {O}_{j}\;\text{ for }j \in \{ 1,\ldots, n\} \] and such that \( \mathop{\sum }\limits_{{j = 1}}^{n}{\varphi }_{j}\left( x\right) = 1 \) for every \( x \in K \) . Proof. Set \( d = d\left( {K,{\mathbb{R}}^{d} \smallsetminus O}\right) \), with \( O = \mathop{\bigcup }\limits_{{j = 1}}^{n}{O}_{j} \) (the metric being the canonical euclidean metric in \( \left. {\mathbb{R}}^{d}\right) \) . Set \( {K}^{\prime } = \{ x : d\left( {x, K}\right) \leq d/2\} \) . The set \( {K}^{\prime } \) is compact and, since \( d > 0 \) , \[ {\mathring{K}}^{\prime } \supset \{ x : d\left( {x, K}\right) < d/2\} \supset K. \] Thus \( K \subset {\mathring{K}}^{\prime } \subset {K}^{\prime } \subset O \) . By Proposition 1.8 on page 53, there exist functions \( {h}_{1},\ldots ,{h}_{n} \) in \( {C}_{c} \) such that \[ 0 \leq {h}_{j} \leq 1\;\text{ and }\;\operatorname{Supp}{h}_{j} \subset {O}_{j}\;\text{ for }j \in \{ 1,\ldots, n\} , \] and such that \( \mathop{\sum }\limits_{{j = 1}}^{n}{h}_{j}\left( x\right) = 1 \) for every \( x \in {K}^{\prime } \) . Define \( \delta = d\left( {K,{\mathbb{R}}^{d} \smallsetminus {\mathring{K}}^{\prime }}\right) \) , \( {\eta }_{j} = d\left( {\operatorname{Supp}{h}_{j},{\mathbb{R}}^{d} \smallsetminus {O}_{j}}\right) \) for \( 1 \leq j \leq n \), and \[ \varepsilon = \frac{1}{2}\min \left( {\delta ,{\eta }_{1},\ldots ,{\eta }_{n}}\right) \] Let \( \chi \) be the function defined on page 261 and let \( u \) be defined by \[ u\left( x\right) = {\varepsilon }^{-d}\chi \left( {x/\varepsilon }\right) \] Then \( u \in \mathcal{D}, u \geq 0,\int u\left( x\right) {dx} = 1 \), and \( \operatorname{Supp}u = \bar{B}\left( {0,\varepsilon }\right) \) . For \( 1 \leq j \leq n \), set \( {\varphi }_{j} = {h}_{j} * u \) . Then \( {\varphi }_{j} \) is of class \( {C}^{\infty } \) (this follows immediately from the theorem on differentiation under the integral sign) and \[ \operatorname{Supp}{\varphi }_{j} \subset \operatorname{Supp}{h}_{j} + \bar{B}\left( {0,\varepsilon }\right) \subset {O}_{j}. \] In particular, \( {\varphi }_{j} \in \mathcal{D} \) . Moreover, \( 0 \leq {\varphi }_{j} \leq 1 \) . Finally, if \( x \in K \) and \( y \in \bar{B}\left( {0,\varepsilon }\right) \), we have \( x - y \in {K}^{\prime } \) and so \[ \mathop{\sum }\limits_{{j = 1}}^{n}{h}_{j}\left( {x - y}\right) u\left( y\right) = u\left( y\right) \] Integrating we obtain \[ \mathop{\sum }\limits_{{j = 1}}^{n}{\varphi }_{j}\left( x\right) = \int u\left( y\right) {dy} = 1\;\text{ for all }x \in K. \] We deduce the following denseness result: Proposition 1.5 The space \( \mathcal{D}\left( \Omega \right) \) is dense in \( \mathcal{E}\left( \Omega \right) \) and in \( {\mathcal{E}}^{m}\left( \Omega \right) \), for every \( m \in \mathbb{N} \) . Proof. Let \( {\left( {K}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of compact subsets of \( \Omega \) exhausting \( \Omega \) . By the previous proposition, there exists, for every integer \( n \in \mathbb{N} \), an element \( {\varphi }_{n} \in \mathcal{D}\left( \Omega \right) \) such that \[ 0 \leq {\varphi }_{n} \leq 1,\;{\varphi }_{n} = 1\text{ on }{K}_{n},\;\operatorname{Supp}{\varphi }_{n} \subset {K}_{n + 1}. \] If \( f \in \mathcal{E}\left( \Omega \right) \), we have \( f{\varphi }_{n} \in \mathcal{D}\left( \Omega \right) \) for every \( n \in \mathbb{N} \) . If \( K \) is a compact subset of \( \Omega \), there exists \( N \in \mathbb{N} \) such that \( K \subset {\mathring{K}}_{N} \) (see Proposition 1.6 on page 52); thus, for every \( n \geq N \) and every \( p \in {\mathbb{N}}^{d} \), we have \( {D}^{p}\left( {f{\varphi }_{n}}\right) = {D}^{p}f \) on \( K \) . By the definition of convergence in \( \mathcal{E}\left( \Omega \right) \), we deduce that \( \mathop{\lim }\limits_{{n \rightarrow + \infty }}\left( {f{\varphi }_{n}}\right) = f \) in \( \mathcal{E}\left( \Omega \right) \) . Using the same reasoning, one shows that \( {\mathcal{D}}^{m}\left( \Omega \right) \) is dense in \( {\mathcal{E}}^{m}\left( \Omega \right) \) . Moreover, as we saw in Corollary \( {1.3},\mathcal{D}\left( \Omega \right) \) is dense in \( {\mathcal{D}}^{m}\left( \Omega \right) \) . Thus every element of \( {\mathcal{D}}^{m}\left( \Omega \right) \) is the limit of a sequence of elements of \( \mathcal{D}\left( \Omega \right) \) in the sense of convergence in \( {\mathcal{E}}^{m}\left( \Omega \right) \) (since the canonical injection from \( {\mathcal{D}}^{m}\left( \Omega \right) \) into \( {\mathcal{E}}^{m}\left( \Omega \right) \) is continuous: see page 260). This implies, finally, that \( \mathcal{D}\left( \Omega \right) \) is dense in \( {\mathcal{E}}^{m}\left( \Omega \right) \) (because \( {\mathcal{E}}^{m}\left( \Omega \right) \) is a metric space: see Exercise 7 below). Remark. This proof also shows that every positive element of \( {\mathcal{E}}^{em}\left( \Omega \right) \) (or \( \mathcal{E}\left( \Omega \right) ) \) is the limit in \( {\mathcal{E}}^{m}\left( \Omega \right) \) (or in \( \mathcal{E}\left( \Omega \right) \), respectively) of a sequence of positive elements of \( \mathcal{D}\left( \Omega \right) \) . ## Exercises Throughout the exercises, \( \Omega \) stands for an open subset of \( {\mathbb{R}}^{d} \) . Many of the exercises use the result of Exercise 1. 1. Taylor’s formula with integral remainder. Let \( f \) be an element of \( {\mathcal{E}}^{n}\left( \Omega \right) \) (where \( n \geq 1 \) ) and let \( x \in \Omega \) . Take \( h \in {\mathbb{R}}^{d} \) such that \( \left\lbrack {x, x + h}\right\rbrack \subset \Omega \) . Show that \[ f\left( {x + h}\right) = f\left( x\right) + \mathop{\sum }\limits_{{k = 1}}^{{n - 1}}\left( {\frac{1}{k!}\math

Proposition 1.2 Assume \( \varphi \in {\mathcal{D}}^{m} \), for some \( m \in \mathbb{N} \). For every integer \( n \geq 1 \), the convolution \( \varphi * {\chi }_{n} \) belongs to \( \mathcal{D} \) and \[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}\varphi * {\chi }_{n} = \varphi \;\text{ in }{\mathcal{D}}^{m} \]

Proof. Since the functions \( \varphi \) and \( {\chi }_{n} \) have compact support, so does \( \varphi * {\chi }_{n} \). More precisely, \[ \operatorname{Supp}\left( {\varphi * {\chi }_{n}}\right) \subset \operatorname{Supp}\varphi + \operatorname{Supp}{\chi }_{n} \subset \operatorname{Supp}\varphi + \bar{B}\left( {0,1/n}\right) \subset \operatorname{Supp}\varphi + \bar{B}\left( {0,1}\right). \] At the same time, a classical theorem about differentiation under the integral sign easily implies, on the one hand, that \( \varphi * {\chi }_{n} \) is of class \( {C}^{\infty } \) and so \( \varphi * {\chi }_{n} \in \mathcal{D} \), and, on the other, that \( {D}^{p}\left( {\varphi * {\chi }_{n}}\right) = \left( {{D}^{p}\varphi }\right) * {\chi }_{n} \) for \( \left| p\right| \leq m \) . Now, since the support of \( {\chi }_{n} \) is contained in \( \bar{B}\left( {0,1/n}\right) \) and \( \int {\chi }_{n}\left( y\right) {dy} = 1 \) , we get \[ \left( {{D}^{p}\varphi }\right) * {\chi }_{n}\left( x\right) - \left( {{D}^{p}\varphi }\right) \left( x\right) = {\int }_{\left| y\right| \leq 1/n}\left( {{D}^{p}\varphi \left( {x - y}\right) - {D}^{p}\varphi \left( x\right) }\right) {\chi }_{n}\left( y\right) {dy} \] and \[ \mathop{\sup }\limits_{{x \in {\mathbb{R}}^{d}}}\left| {\left( {{D}^{p}\varphi }\right) * {\chi }_{n}\left( x\right) - \left( {{D}^{p}\varphi }\right) \left( x\right) }\right| \leq

Corollary 10.67. If no free occurrence of \( \alpha \) in \( \varphi \) is within the scope of a quantifier on a variable occurring in \( \sigma \), then \( {\mathrm{{FSubf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi } \) . Proof \[ \vdash \forall \alpha \sqsupset \varphi \rightarrow {\operatorname{Subf}}_{\sigma }^{\alpha } \sqsupset \varphi \;\text{ universal specification } \] \[ { \vdash }^{ \Vdash }{\mathrm{{Subf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi } \] suitable tautology Corollary 10.68. \( \; \vdash \varphi \rightarrow \exists {\alpha \varphi } \) . Corollary 10.69. \( \mathrm{H}\forall {\alpha \varphi } \rightarrow \exists {\alpha \varphi } \) . Proposition 10.70. \( \; \vdash \exists \alpha \forall {\beta \varphi } \rightarrow \forall \beta \exists {\alpha \varphi } \) . Proof \[ \vdash \varphi \rightarrow \exists {\alpha \varphi } \] \[ \mathcal{H}\beta \xrightarrow[]{\varphi }\xrightarrow[]{\varphi \rightarrow \forall \beta }\exists {\alpha \varphi } \] 10.23(2) \[ \vdash \neg \forall \beta \exists {\alpha \varphi } \rightarrow \neg \forall {\beta \varphi } \] tautology \[ \vdash \neg \forall \beta \exists {\alpha \varphi } \rightarrow \forall \alpha \neg \forall {\beta \varphi } \] \( {10.23}\left( 2\right) ,{10.64} \) \[ \vdash \exists \alpha \forall {\beta \varphi } \rightarrow \forall \beta \exists {\alpha \varphi } \] tautology The formula \( \forall \beta \exists {\alpha \varphi } \rightarrow \exists \alpha \forall {\beta \varphi } \) is not, in general, valid. The following important syntactical theorem is a version for formulas of the principle of substitution of equals for equals: Theorem 10.71 (Substitutivity of equivalence). Let \( \varphi ,\psi ,\chi \) be formulas and \( \alpha \in {}^{m}\operatorname{Rng}v \) . Suppose that if \( \beta \) occurs free in \( \varphi \) or in \( \psi \) but bound in \( \chi \) then \( \beta \in \left\{ {{\alpha }_{i} : i < m}\right\} \) . Let \( \theta \) be obtained from \( \chi \) by replacing zero or more occurrences of \( \varphi \) in \( \chi \) by \( \psi \) . Then \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {\chi \leftrightarrow \theta }\right) . \] Proof. We proceed by induction on \( \chi \) . We may assume that \( \theta \neq \chi \) . If \( \chi \) is atomic, then \( \chi = \varphi \) and \( \psi = \theta \) ; this case is trivial. Suppose \( \chi \) is \( \neg {\chi }^{\prime } \) . Then \( \theta \) is of the form \( \neg {\theta }^{\prime } \), and the induction hypothesis easily gives the desired result. The induction steps involving \( \mathbf{v} \) and \( \land \) are similar. Now suppose \( \chi = \forall \beta {\chi }^{\prime } \) . Then by the induction hypothesis, \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {{\chi }^{\prime } \leftrightarrow {\theta }^{\prime }}\right) . \] Note that \( \beta \) does not occur free in \( \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \) . Hence, using 10.61, we easily obtain \[ \vdash \forall {\alpha }_{0}\cdots \forall {\alpha }_{m - 1}\left( {\varphi \leftrightarrow \psi }\right) \rightarrow \left( {\forall \beta {\chi }^{\prime } \leftrightarrow \forall \beta {\theta }^{\prime }}\right) , \] as desired. Again note that implicit in 10.71 is the assertion that the expression \( \theta \) formed from \( \chi \) is again a formula; this is easily established. Now we introduce a notation which will be frequently used in the remainder of this book. Definition 10.72. Let \( \varphi \) be a formula, \( m \in \omega ,\sigma \in {}^{m} \) Trm. Choose \( k \) maximum such that \( {v}_{k} \) occurs in \( \varphi \) or in \( {\sigma }_{j} \) for some \( j < m, k = 0 \) if no variable occurs in \( \varphi \) or in any \( {\sigma }_{j} \) . Let \( {\alpha }_{0},\ldots ,{\alpha }_{n - 1} \) be a list of all variables which occur bound in \( \varphi \) but also occur in some \( {\sigma }_{j} \), with \( {\alpha }_{0} < \cdots < {\alpha }_{n - 1} \) in the natural order \( {v}_{0},{v}_{1},\ldots \) of the variables. Let \( \psi \) be the formula \[ {\operatorname{Subb}}_{v\left( {k + 1}\right) }^{⓪}{\operatorname{Subb}}_{v\left( {k + 2}\right) }^{⓫}\cdots {\operatorname{Subb}}_{v\left( {k + n}\right) }^{\alpha \left( {n - 1}\right) }\varphi , \] and let \( \varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \) be the formula obtained from \( \psi \) by simultaneously replacing all free occurrences of \( {v}_{0},\ldots ,{v}_{m - 1} \) by \( {\sigma }_{0},\ldots ,{\sigma }_{m - 1} \) respectively. The purpose of first forming \( \psi \) is to eliminate any possible clash of bound variables. The following two corollaries give the essential properties of this notation. Corollary 10.73. \( \; \vdash \forall {v}_{0}\cdots \forall {v}_{m - 1}\varphi \rightarrow \varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \) . Corollary 10.74. \( \;\mathrm{h}\varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \rightarrow \exists {v}_{0}\cdots \exists {v}_{m - 1}\varphi \) . Both of these corollaries are immediate consequences of earlier results, upon noticing that simultaneous substitution can be obtained by iterated ordinary substitution; in the notation of 10.72 , \[ \varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) = {\operatorname{Subf}}_{▞}^{v\left( {k + n + 1}\right) }{\operatorname{Subf}}_{▞}^{v\left( {k + n + 2}\right) }\cdots {\operatorname{Subf}}_{\sigma \left( {m - 1}\right) }^{v\left( {k + n + m}\right) } \] \[ {\operatorname{Subf}}_{v\left( {k + n + 1}\right) }^{v0}{\operatorname{Subf}}_{v\left( {k + n + 2}\right) }^{v1}\cdots {\operatorname{Subf}}_{v\left( {k + n + m}\right) }^{v\left( {m - 1}\right) }\psi . \] This fact is also useful in checking formally that this substitution notion is effective: Proposition 10.75. If \( x \) is the Gödel number of a formula \( \varphi, m \in \omega ,{y}_{0},\ldots \) , \( {y}_{m - 1} \) are Gödel numbers of terms \( {\sigma }_{0},\ldots ,{\sigma }_{m - 1} \) respectively, let \( s\left( {x,{y}_{0},\ldots }\right. \) , \( \left. {y}_{m - 1}\right) = {\mathcal{J}}^{ + }\varphi \left( {{\sigma }_{0},\ldots ,{\sigma }_{m - 1}}\right) \) ; if \( x \) and \( {y}_{0},\ldots ,{y}_{m - 1} \) do not satisfy these conditions, let \( s\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) = 0 \) . Then \( s \) is recursive. Proof. First we need a function picking out the integer \( k \) described in 10.72. For any \( x,{y}_{0},\ldots ,{y}_{m - 1} \in \omega \), let \[ g\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) = {\mu k}\left\lbrack \left( {\exists i \leq \operatorname{lx}\left\{ {\left( {{\left( x\right) }_{i} = \operatorname{gv}k + 1}\right) \text{ or }}\right. }\right. \right. \] \[ \mathop{\bigvee }\limits_{{j < m}}\exists i \leq \mathrm{l}{y}_{j}\left\lbrack {{\left( {y}_{j}\right) }_{i} = {gvk} + 1}\right\rbrack \} \text{ and }\forall i \leq \mathrm{l}x\lbrack {\left( x\right) }_{i} \div \] \[ \left. {1 \in \operatorname{Rng}\left( {g \circ v}\right) \Rightarrow {v}^{-1}{g}^{-1}\left( {{\left( x\right) }_{i} - 1}\right) \leq k}\right\rbrack \text{and} \] \[ \mathop{\bigwedge }\limits_{{j < m}}\forall i \leq 1{y}_{j}\left\lbrack {{\left( {y}_{j}\right) }_{i} - 1 \in \operatorname{Rng}\left( {g \circ v}\right) \Rightarrow {v}^{-1}{g}^{-1}\left( {{\left( {y}_{j}\right) }_{i} - 1}\right) \leq k}\right\rbrack ) \] \[ \text{or}(\forall i \leq \operatorname{lx}\left\lbrack {{\left( x\right) }_{i} \div 1 \notin \operatorname{Rng}\left( {g \circ v}\right) }\right. \text{and}\mathop{\bigwedge }\limits_{{j < m}}\forall i \leq \] \[ \left. \left. {\lg \left\lbrack {{\left( {y}_{j}\right) }_{i} \div 1 \notin \operatorname{Rng}\left( {g \circ v}\right) }\right\rbrack }\right) \right\rbrack \] Now let \( f \) be the function of 10.52, let \( {f}^{\prime } \) be the function of 10.57, and let \( S \) be the relation of 10.43. We now define a function \( {f}^{\prime \prime } \) which codes the set of variables which occur bound in \( \varphi \) but also occur in some \( {\sigma }_{j} \) : \[ {f}^{\prime \prime }\left( {x,{z}_{0},\ldots ,{z}_{m - 1}}\right) = {\mu y}\left( {{\left( y\right) }_{1y} = 1}\right. \text{and}\forall i \leq \lg (\exists j \leq \] \[ {lx}\left\lbrack {\left( {{\left( y\right) }_{i}, j, x}\right) \in S}\right\rbrack \; \land \;\mathop{\bigvee }\limits_{{j\; < \;m}}\exists u \leq {z}_{j}\;\exists w \leq {z}_{j}\lbrack {z}_{j} = \mathrm{{Cat}}\;(u,\mathrm{{Cat}} \] \[ \left( {{2}^{\left( y\right) i + 1}, w}\right) )\rbrack )\text{ and }\forall j \leq \mathrm{l}x\;\forall k \leq x\{ \left( {k, j, x}\right) \in S\text{ and } \] \[ \mathop{\bigvee }\limits_{{n < m}}\exists u \leq {z}_{n}\;\exists w \leq {z}_{n}\left\lbrack {{z}_{n} = \mathrm{{Cat}}\left( {u,\mathrm{{Cat}}\left( {{2}^{k + 1}, w}\right) }\right) }\right\rbrack \Rightarrow \exists i < \] \[ \left. {\lg \left\lbrack {{\left( y\right) }_{i} = k}\right\rbrack }\right\} \text{ and }\forall i, j < \lg \left\lbrack {i < j \Rightarrow {v}^{-1}{\mathcal{G}}^{-1}{\left( y\right) }_{i} < {v}^{-1}{\mathcal{G}}^{-1}{\left( y\right) }_{j}}\right\rbrack \text{. } \] Next we define a function \( {f}^{iv} \) yielding the formula \( \psi \) of 10.72, via an auxiliary function \( {f}^{m} \) : \[ {f}^{m}\left( {x,{y}_{0},\ldots ,{y}_{m - 1},0}\right) = x, \] \[ {f}^{\prime \prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}, i + 1}\right) = {f}^{\prime }\left( {\left( {f}^{\prime \prime }\left( x,{y}_{0},\ldots ,{y}_{m - 1}\right) \right) }_{i}\right. \text{,} \] \[ \operatorname{gv}\left( {h\left( {x,{y}_{0},\ldots ,{y}_{m - 1}}\right) + i + 1}\right) , \] \[\left. {{f}^{\prime \prime \prime }\left( {x,{y}_{0},\ldots ,{y}_{m - 1}, i}\right) }\right) \] and \[ {f}^{iv}\lef

Corollary 10.67. If no free occurrence of \( \alpha \) in \( \varphi \) is within the scope of a quantifier on a variable occurring in \( \sigma \), then \( {\mathrm{{FSubf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi } \) .

\[ \vdash \forall \alpha \sqsupset \varphi \rightarrow {\operatorname{Subf}}_{\sigma }^{\alpha } \sqsupset \varphi \;\text{ universal specification } \] \[ { \vdash }^{ \Vdash }{\mathrm{{Subf}}}_{\sigma }^{\alpha }\varphi \rightarrow \exists {\alpha \varphi } \] suitable tautology

Lemma 4.4.4. Let \( \mathcal{X} \) be a set of 3-connected graphs. Let \( G \) be a graph \( \left\lbrack {7.3.1}\right\rbrack \) with \( \kappa \left( G\right) \leq 2 \), and let \( {G}_{1},{G}_{2} \) be proper induced subgraphs of \( G \) such that \( G = {G}_{1} \cup {G}_{2} \) and \( \left| {{G}_{1} \cap {G}_{2}}\right| = \kappa \left( G\right) \) . If \( G \) is edge-maximal without a topological minor in \( \mathcal{X} \), then so are \( {G}_{1} \) and \( {G}_{2} \), and \( {G}_{1} \cap {G}_{2} = {K}^{2} \) . Proof. Note first that every vertex \( v \in S \mathrel{\text{:=}} V\left( {{G}_{1} \cap {G}_{2}}\right) \) has a neighbour in every component of \( {G}_{i} - S, i = 1,2 \) : otherwise \( S \smallsetminus \{ v\} \) would separate \( G \), contradicting \( \left| S\right| = \kappa \left( G\right) \) . By the maximality of \( G \), every edge \( e \) added to \( G \) lies in a \( {TX} \subseteq G + e \) with \( X \in \mathcal{X} \) . For all the choices of \( e \) considered below, the 3-connectedness of \( X \) will imply that the branch vertices of this \( {TX} \) all lie in the same \( {G}_{i} \), say in \( {G}_{1} \) . (The position of \( e \) will always be symmetrical with respect to \( {G}_{1} \) and \( {G}_{2} \), so this assumption entails no loss of generality.) Then the \( {TX} \) meets \( {G}_{2} \) at most in a path \( P \) corresponding to an edge of \( X \) . If \( S = \varnothing \), we obtain an immediate contradiction by choosing \( e \) with one end in \( {G}_{1} \) and the other in \( {G}_{2} \) . If \( S = \{ v\} \) is a singleton, let \( e \) join a neighbour \( {v}_{1} \) of \( v \) in \( {G}_{1} - S \) to a neighbour \( {v}_{2} \) of \( v \) in \( {G}_{2} - S \) (Fig. 4.4.3). Then \( P \) contains both \( v \) and the edge \( {v}_{1}{v}_{2} \) ; replacing \( {vP}{v}_{1} \) with the edge \( v{v}_{1} \), we obtain a \( {TX} \) in \( {G}_{1} \subseteq G \), a contradiction. So \( \left| S\right| = 2 \), say \( S = \{ x, y\} \) . If \( {xy} \notin G \), we let \( e \mathrel{\text{:=}} {xy} \), and in the arising \( {TX} \) replace \( e \) by an \( x - y \) path through \( {G}_{2} \) ; this gives a \( {TX} \) in \( G \) , a contradiction. Hence \( {xy} \in G \), and \( G\left\lbrack S\right\rbrack = {K}^{2} \) as claimed. It remains to show that \( {G}_{1} \) and \( {G}_{2} \) are edge-maximal without a topological minor in \( \mathcal{X} \) . So let \( {e}^{\prime } \) be an additional edge for \( {G}_{1} \), say. Replacing \( {xPy} \) with the edge \( {xy} \) if necessary, we obtain a \( {TX} \) either  Fig. 4.4.3. If \( G + e \) contains a \( {TX} \), then so does \( {G}_{1} \) or \( {G}_{2} \) in \( {G}_{1} + {e}^{\prime } \) (which shows the edge-maximality of \( {G}_{1} \), as desired) or in \( {G}_{2} \) (which contradicts \( {G}_{2} \subseteq G \) ). Lemma 4.4.5. If \( \left| G\right| \geq 4 \) and \( G \) is edge-maximal with \( T{K}^{5}, T{K}_{3,3} \nsubseteq G \) , then \( G \) is 3-connected. (4.2.11) Proof. We apply induction on \( \left| G\right| \) . For \( \left| G\right| = 4 \), we have \( G = {K}^{4} \) and the assertion holds. Now let \( \left| G\right| > 4 \), and let \( G \) be edge-maximal \( {G}_{1},{G}_{2} \) without a \( T{K}^{5} \) or \( T{K}_{3,3} \) . Suppose \( \kappa \left( G\right) \leq 2 \), and choose \( {G}_{1} \) and \( {G}_{2} \) as in Lemma 4.4.4. For \( \mathcal{X} \mathrel{\text{:=}} \left\{ {{K}^{5},{K}_{3,3}}\right\} \), the lemma says that \( {G}_{1} \cap {G}_{2} \) is \( x, y \) a \( {K}^{2} \), with vertices \( x, y \) say. By Lemmas 4.4.4,4.4.3 and the induction hypothesis, \( {G}_{1} \) and \( {G}_{2} \) are planar. For each \( i = 1,2 \) separately, choose a \( {f}_{i} \) drawing of \( {G}_{i} \), a face \( {f}_{i} \) with the edge \( {xy} \) on its boundary, and a vertex \( {z}_{i} \) \( {z}_{i} \neq x, y \) on the boundary of \( {f}_{i} \) . Let \( K \) be a \( T{K}^{5} \) or \( T{K}_{3,3} \) in the \( K \) abstract graph \( G + {z}_{1}{z}_{2} \) (Fig. 4.4.4).  Fig. 4.4.4. A \( T{K}^{5} \) or \( T{K}_{3,3} \) in \( G + {z}_{1}{z}_{2} \) If all the branch vertices of \( K \) lie in the same \( {G}_{i} \), then either \( {G}_{i} + x{z}_{i} \) or \( {G}_{i} + y{z}_{i} \) (or \( {G}_{i} \) itself, if \( {z}_{i} \) is already adjacent to \( x \) or \( y \), respectively) contains a \( T{K}^{5} \) or \( T{K}_{3,3} \) ; this contradicts Corollary 4.2.11, since these graphs are planar by the choice of \( {z}_{i} \) . Since \( G + {z}_{1}{z}_{2} \) does not contain four independent paths between \( \left( {{G}_{1} - {G}_{2}}\right) \) and \( \left( {{G}_{2} - {G}_{1}}\right) \), these subgraphs cannot both contain a branch vertex of a \( T{K}^{5} \), and cannot both contain two branch vertices of a \( T{K}_{3,3} \) . Hence \( K \) is a \( T{K}_{3,3} \) with only one branch vertex \( v \) in, say, \( {G}_{2} - {G}_{1} \) . But then also the graph \( {G}_{1} + v + \left\{ {{vx},{vy}, v{z}_{1}}\right\} \) , which is planar by the choice of \( {z}_{1} \), contains a \( T{K}_{3,3} \) . This contradicts Corollary 4.2.11. Theorem 4.4.6. (Kuratowski 1930; Wagner 1937) \( \left\lbrack \begin{array}{l} {4.5.1} \\ {12.4.3} \end{array}\right\rbrack \) The following assertions are equivalent for graphs \( G \) : (i) \( G \) is planar; (ii) \( G \) contains neither \( {K}^{5} \) nor \( {K}_{3,3} \) as a minor; (iii) \( G \) contains neither \( {K}^{5} \) nor \( {K}_{3,3} \) as a topological minor. Proof. Combine Corollary 4.2.11 with Lemmas 4.4.2, 4.4.3 and 4.4.5. (4.2.11) Corollary 4.4.7. Every maximal planar graph with at least four vertices is 3-connected. Proof. Apply Lemma 4.4.5 and Theorem 4.4.6. ## 4.5 Algebraic planarity criteria One of the most conspicuous features of a plane graph \( G \) are its facial --- facial cycles --- cycles, the cycles that bound a face. If \( G \) is 2-connected it is covered by its facial cycles, so in a sense these form a 'large' set. In fact, the set of facial cycles is large even in the sense that they generate the entire cycle space: every cycle in \( G \) is easily seen to be the sum of the facial cycles (see below). On the other hand, the facial cycles only cover \( G \) ’thinly’, as every edge lies on at most two of them. Our first aim in this section is to show that the existence of such a large yet thinly spread family of cycles is not only a conspicuous feature of planarity but lies at its very heart: it characterizes it. Let \( G = \left( {V, E}\right) \) be any graph. We call a subset \( \mathcal{F} \) of its edge space \( \mathcal{E}\left( G\right) \) simple if every edge of \( G \) lies in at most two sets of \( \mathcal{F} \) . For example, simple the cut space \( {\mathcal{C}}^{ * }\left( G\right) \) has a simple basis: according to Proposition 1.9.3 it is generated by the cuts \( E\left( v\right) \) formed by all the edges at a given vertex \( v \) , and an edge \( {xy} \in G \) lies in \( E\left( v\right) \) only for \( v = x \) and for \( v = y \) . Theorem 4.5.1. (MacLane 1937) \( \left\lbrack {4.6.3}\right\rbrack \) A graph is planar if and only if its cycle space has a simple basis. Proof. The assertion being trivial for graphs of order at most 2 , we consider a graph \( G \) of order at least 3 . If \( \kappa \left( G\right) \leq 1 \), then \( G \) is the union \( \left( {1.9.2}\right) \) of two proper induced subgraphs \( {G}_{1},{G}_{2} \) with \( \left| {{G}_{1} \cap {G}_{2}}\right| \leq 1 \) . Then \( \mathcal{C}\left( G\right) \) (4.1.1) is the direct sum of \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \), and hence has a simple basis if (4.2.2) and only if both \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \) do (proof?). Moreover, \( G \) is planar if (4.2.6) (4.4.6) and only if both \( {G}_{1} \) and \( {G}_{2} \) are: this follows at once from Kuratowski’s theorem, but also from easy geometrical considerations. The assertion for \( G \) thus follows inductively from those for \( {G}_{1} \) and \( {G}_{2} \) . For the rest of the proof, we now assume that \( G \) is 2-connected. We first assume that \( G \) is planar and choose a drawing. By Proposition 4.2.6, the face boundaries of \( G \) are cycles, so they are elements of \( \mathcal{C}\left( G\right) \) . We shall show that the face boundaries generate all the cycles in \( G \) ; then \( \mathcal{C}\left( G\right) \) has a simple basis by Lemma 4.2.2. Let \( C \subseteq G \) be any cycle, and let \( f \) be its inner face. By Lemma 4.2.2, every edge \( e \) with \( e \subseteq f \) lies on exactly two face boundaries \( G\left\lbrack {f}^{\prime }\right\rbrack \) with \( {f}^{\prime } \subseteq f \), and every edge of \( C \) lies on exactly one such face boundary. Hence the sum in \( \mathcal{C}\left( G\right) \) of all those face boundaries is exactly \( C \) . Conversely, let \( \left\{ {{C}_{1},\ldots ,{C}_{k}}\right\} \) be a simple basis of \( \mathcal{C}\left( G\right) \) . Then, for every edge \( e \in G \), also \( \mathcal{C}\left( {G - e}\right) \) has a simple basis. Indeed, if \( e \) lies in just one of the sets \( {C}_{i} \), say in \( {C}_{1} \), then \( \left\{ {{C}_{2},\ldots ,{C}_{k}}\right\} \) is a simple basis of \( \mathcal{C}\left( {G - e}\right) \) ; if \( e \) lies in two of the \( {C}_{i} \), say in \( {C}_{1} \) and \( {C}_{2} \), then \( \left\{ {{C}_{1} + {C}_{2},{C}_{3},\ldots ,{C}_{k}}\right\} \) is such a basis. (Note that the two bases are indeed subsets of \( \mathcal{C}\left( {G - e}\right) \) by Proposition 1.9.2.) Thus every subgraph of \( G \) has a cycle space with a simple basis. For our proof that \( G \) is planar, it thus suffices to show that the cycle spaces of \( {K}^{5} \) an

Lemma 4.4.4. Let \( \mathcal{X} \) be a set of 3-connected graphs. Let \( G \) be a graph with \( \kappa \left( G\right) \leq 2 \), and let \( {G}_{1},{G}_{2} \) be proper induced subgraphs of \( G \) such that \( G = {G}_{1} \cup {G}_{2} \) and \( \left| {{G}_{1} \cap {G}_{2}}\right| = \kappa \left( G\right) \) . If \( G \) is edge-maximal without a topological minor in \( \mathcal{X} \), then so are \( {G}_{1} \) and \( {G}_{2} \), and \( {G}_{1} \cap {G}_{2} = {K}^{2} \) .

Proof. Note first that every vertex \( v \in S \mathrel{\text{:=}} V\left( {{G}_{1} \cap {G}_{2}}\right) \) has a neighbour in every component of \( {G}_{i} - S, i = 1,2 \) : otherwise \( S \smallsetminus \{ v\} \) would separate \( G \), contradicting \( \left| S\right| = \kappa \left( G\right) \) . By the maximality of \( G \), every edge \( e \) added to \( G \) lies in a \( {TX} \subseteq G + e \) with \( X \in \mathcal{X} \) . For all the choices of \( e \) considered below, the 3-connectedness of \( X \) will imply that the branch vertices of this \( {TX} \) all lie in the same \( {G}_{i} \), say in \( {G}_{1} \) . (The position of \( e \) will always be symmetrical with respect to \( {G}_{1} \) and \( {G}_{2} \), so this assumption entails no loss of generality.) Then the \( {TX} \) meets \( {G}_{2} \) at most in a path \( P \) corresponding to an edge of \( X \) . If \( S = \varnothing \), we obtain an immediate contradiction by choosing \( e \) with one end in \( {G}_{1} \) and the other in \( {G}_{2} \) . If \( S = \{ v\} \) is a singleton, let \( e \) join a neighbour \( {v}_{1} \) of \( v \) in

Proposition 1.1.7. The lightcone \( {\mathcal{L}}_{0} \) is a lightlike submanifold. By taking \( V \) to be a 3-dimensional Lorentzian vector space and drawing a picture of \( {\mathcal{L}}_{0} \), one can easily convince oneself that indeed \( {\mathcal{L}}_{0} \) is lightlike; for in this case, any tangent plane to \( {\mathcal{L}}_{0} \) is a 2- dimensional vector subspace containing a generator of the cone \( {\mathcal{L}}_{0} \) and clearly such a subspace cannot contain any timelike vectors (cf. Example 1.1.8). Proof. Suppose \( v \in {\mathcal{L}}_{0} \) ; then \( g\left( {v, v}\right) = 0 \) and \( v \neq 0 \) . Let \( \mathcal{U} \) be a neighborhood of \( v \) that does not contain the origin and define \( \widetilde{g} : \mathcal{U} \rightarrow \mathbb{R} \) by \( \widetilde{g}w = g\left( {w, w}\right) \forall w \in \mathcal{U} \) . Then \( {\mathcal{L}}_{0} \cap \mathcal{U} \) is defined by \( \widetilde{g} = 0 \) . Moreover, \( d\widetilde{g} \) is nowhere zero on \( \mathcal{U} \) because \( g \) is nondegenerate (Exercise 0.0.10). Thus \( {\mathcal{L}}_{0} \) is a submanifold by the implicit function theorem. To show \( {\mathcal{L}}_{0} \) is lightlike, suppose that \( w \in {V}_{v} \) for some \( v \in {\mathcal{L}}_{0} \), and let \( {\phi }_{v} : {V}_{v} \rightarrow V \) be the canonical isomorphism of Exercise 0.0.10 . Then \( w \) is in the tangent space \( {\left( {\mathcal{L}}_{0}\right) }_{v} \) iff \( w\widetilde{g} = 0 \), iff \( g\left( {{\phi }_{v}w, v}\right) = 0 \), and iff \( g\left( {w,{\phi }_{v}{}^{-1}v}\right) = 0 \) . Thus \( {\left( {\mathcal{L}}_{0}\right) }_{v} = {\left( {\phi }_{v}{}^{-1}v\right) }^{ \bot } \subset {V}_{v} \) . But \( {\phi }_{v}{}^{-1}v \in {V}_{v} \) is lightlike because \( g\left( {{\phi }_{v}{}^{-1}v,{\phi }_{v}{}^{-1}v}\right) = g\left( {v, v}\right) = 0 \), hence \( {\left( {\mathcal{L}}_{0}\right) }_{v} \) is lightlike by Proposition 1.1.3b. Since this holds for all \( v \in {\mathcal{L}}_{0},{\mathcal{L}}_{0} \) is lightlike. EXAMPLE 1.1.8. Let \( N = 3 \) and \( \left\{ {{e}_{1},{e}_{2},{e}_{3}}\right\} \) be an orthonormal basis. Then \( v \in {\mathcal{L}}_{0} \) iff its components obey \( {\left( {v}^{3}\right) }^{2} = {\left( {v}^{1}\right) }^{2} + {\left( {v}^{2}\right) }^{2} > 0 \) . Thus \( {\mathcal{L}}_{0} \) is actually represented by a cone with the apex deleted, as shown. The timelike vectors obey \( {\left( {v}^{3}\right) }^{2} > {\left( {v}^{1}\right) }^{2} + {\left( {v}^{2}\right) }^{2} \) ; they are represented by the points inside \( {\mathcal{L}}_{0} \) and are shown as \( {\mathcal{T}}_{0}^{ + } \) and \( {\mathcal{T}}_{0}^{ - } \) .  This figure correctly suggests many features of the general case. Indeed (see Exercise 1.1.9), for an arbitrary N-dimensional Lorentzian vector space, the set \( {\mathcal{T}}_{0} \) of timelike vectors is an open submanifold with two connected components, \( {\mathcal{T}}_{0}^{ + } \) and \( {\mathcal{T}}_{0}^{ - } \), each diffeomorphic to \( {\mathbb{R}}^{N} \), and for \( N \geq 3 \) the lightcone \( {\mathcal{L}}_{0} \) also splits into two connected components, \( {\mathcal{L}}_{0}^{ + } \) and \( {\mathcal{L}}_{0}^{ - } \), each diffeomorphic to \( \mathbb{R} \times {\mathcal{S}}^{N - 2} \) . Newtonian analogue. Suppose \( N = 4 \) . Then one component of \( {\mathcal{L}}_{0} \) is like the complete history of an "information-gathering" sphere in \( {\mathbb{R}}^{3} \) which contracts with the speed of light. Again, one dimension is suppressed in the following diagram of \( {\mathcal{L}}_{0}{}^{ - } \) .  ## EXERCISE 1.1.9 Let \( e \in V \) be timelike. Use an orthonormal basis to prove the following results. (a) The set of timelike vectors, \( {\mathcal{T}}_{0} \), is an open submanifold with two connected components \( {\mathcal{T}}_{0}^{ + } = \left\{ {v \in {\mathcal{T}}_{0} \mid g\left( {e, v}\right) < 0}\right\} \) and \( {\mathcal{T}}_{0}^{ - } = \left\{ {v \in {\mathcal{T}}_{0} \mid g\left( {e, v}\right) > 0}\right\} \) ; each component is diffeomorphic to \( {\mathbb{R}}^{N} \) . (b) If \( N \geq 3 \), the lightcone \( {\mathcal{L}}_{0} \) has two connected components \( {\mathcal{L}}_{0}^{ + } = \left\{ {v \in {\mathcal{L}}_{0} \mid g\left( {e, v}\right) < 0}\right\} \) and \( {\mathcal{L}}_{0}^{ - } = \left\{ {v \in {\mathcal{L}}_{0} \mid g\left( {e, v}\right) > 0}\right\} \) , and each component is diffeomorphic to \( \mathbb{R} \times {\mathcal{S}}^{N - 2} \), where \( {\mathcal{S}}^{N - 2} \) is the unit \( \left( {N - 2}\right) \) -sphere. (c) If \( N \geq 2 \), and \( v \in {\mathcal{L}}_{0}^{ + } \cup {\mathcal{T}}_{0}^{ + }, w \in {\mathcal{L}}_{0}^{ + } \cup {\mathcal{T}}_{0}^{ + } \), then \( g\left( {v, w}\right) \) \( \leq 0 \) . Equality holds iff \( v \in {\mathcal{L}}_{0}{}^{ + } \) and \( w \) is proportional to \( v \) . ## 1 Spacetimes ## EXERCISE 1.1.10 (a) Let \( v \in V, w \in V \) be causal. Show that the Schwarz inequality now "goes the wrong way": \( \left| {g\left( {v, w}\right) }\right| \geq \left| v\right| \left| w\right| \), and equality holds iff \( v \) and \( w \) are proportional. (b) Let \( \left( {M, g}\right) \) be a Lorentzian manifold, \( X \) a timelike vector in \( {M}_{x} \), and \( \omega \in {M}_{x}^{ * } \) a timelike 1-form. Show: \( \left| {\omega X}\right| \geq \left| \omega \right| \left| X\right| \), and equality holds iff \( \omega \) and \( {aX} \) are physically equivalent for some \( a \in \mathbb{R} \) . [The norm \( \left| \omega \right| \) is taken with respect to the Lorentzian inner product \( \widehat{g}x \) on \( {M}_{x}^{ * } \) (Exercise 0.0.15b).] ## EXERCISE 1.1.11 Let \( \left( {M, g}\right) \) be a Lorentzian manifold and let \( X \in {M}_{x} \) and \( \omega \in {M}_{x}^{ * } \) be physically equivalent. Show that the causal character assigned to \( X \) by \( {gx} \) is the same as the causal character assigned to \( \omega \) by \( \widehat{g}x \), where \( \widehat{g}x \) is the Lorentzian inner product on \( {M}_{x}^{ * } \) given in Exercise 0.0.15 . ## EXERCISE 1.1.12 Let \( \left( {M, g}\right) \) be a Lorentzian manifold, \( N \) a manifold, and \( \phi : N \rightarrow M \) an immersion. \( {\phi }^{ * }g \) is called the metric induced on \( N \) by \( \phi \) iff \( {\phi }^{ * }g \) is a metric on \( N \) . Show \( {\phi }^{ * }g \) is a metric on \( N \) iff \( {\phi N} \) is timelike or spacelike. ## EXERCISE 1.1.13 Suppose \( m \in \left( {0,\infty }\right) \) . Define \( {\mathcal{T}}_{0}{}^{m} \subset V \) by \( {\mathcal{T}}_{0}{}^{m} = \left\{ {v \in V \mid g\left( {v, v}\right) = - {m}^{2}}\right\} \) . Assuming \( V \) is 4-dimensional and regarding \( \left( {V, g}\right) \) as a Lorentzian manifold, show \( {\mathcal{T}}_{0}{}^{m} \) is a spacelike 3-submanifold. ## EXERCISE 1.1.14 Show: the closure of \( {\mathcal{T}}_{0}{}^{ + } \) is the disjoint union of \( {\mathcal{T}}_{0}{}^{ + },{\mathcal{L}}_{0}{}^{ + } \), and the zero vector; the boundary of \( {\mathcal{T}}_{0}^{ + } \) is \( {\mathcal{L}}_{0}^{ + } \cup \{ 0\} \) . Some other algebraic properties of \( \left( {V, g}\right) \) are given in Optional exercises 8.1. ## 1.2 Time orientability We now consider the concepts of "past" and "future." Let \( \left( {M, g}\right) \) be a connected Lorentzian manifold, \( {TM} \) be its tangent bundle and \( \Pi : {TM} \rightarrow M \) be the projection. The causal character of \( \left( {x, X}\right) \in {TM} \) is the causal character of \( X \in {M}_{x} \) . Proposition 1.2.1. The set \( \mathcal{T} \subset {TM} \) of timelike points is an open submanifold. \( \mathcal{T} \) has either one (connected) component or two. Proof. Define \( K : {TM} \rightarrow \mathbb{R} \) by \( K\left( {x, X}\right) = g\left( {X, X}\right) \) . Then \( K \) is \( {C}^{\infty } \) . As the complete inverse image of \( \left( {-\infty ,0}\right) \) under \( K,\mathcal{T} \) is open. Let \( \mathcal{A} \) be a component of \( \mathcal{T} \) . If \( \psi : \mathcal{T} \rightarrow \mathcal{T} \) denotes the homeomorphism defined by \( \psi \left( {x, X}\right) = \) \( \left( {x, - X}\right) \), then \( \psi \mathcal{A} \) is also a component of \( \mathcal{T} \) . We will show \( \mathcal{T} = \mathcal{A} \cup \psi \mathcal{A} \) . Let \( \mathcal{B} = \mathcal{A} \cup \psi \mathcal{A},\mathcal{C} = \mathcal{T} - \mathcal{B} \) . \( \mathcal{B} \) is open and closed in \( \mathcal{T} \), and hence both \( \mathcal{B} \) and \( \mathcal{C} \) are open and closed in \( \mathcal{T} \) . It follows that \( \mathcal{B} \) and \( \mathcal{C} \) are open in \( {TM} \) . We claim \( \Pi \mathcal{B} \cap \Pi \mathcal{C} = \varnothing \) . If not, then there exist \( \left( {x, Z}\right) \in \mathcal{B} \) and \( \left( {x, Y}\right) \in \mathcal{C} \) for some \( x \in M \) . Let \( \mathcal{Y} \subset {M}_{x} \) be that one of the two components of \( {M}_{x} \cap \mathcal{T} \) in which \( \left( {x, Y}\right) \) lies (Exercise 1.1.9a); then \( \mathcal{C} \cap \mathcal{Y} \neq \varnothing \) . Since \( \mathcal{C} \) is a union of components of \( \mathcal{T} \), this implies \( \mathcal{Y} \subset \mathcal{C} \) . Now either \( \left( {x, Z}\right) \) or \( \left( {x, - Z}\right) \) is in \( \mathcal{Y} \) , while both are in \( \mathcal{B} \) by definition of \( \mathcal{B} \) . Thus \( \mathcal{B} \cap \mathcal{Y} \neq \varnothing \), and hence also \( \mathcal{Y} \subset \mathcal{B} \) because \( \mathcal{B} \) is a union of components. It follows that \( \mathcal{B} \cap \mathcal{C} \neq \varnothing \), a contradiction. We therefore have \( \Pi \mathcal{B} \cap \Pi \mathcal{C} = \varnothing \) and \( \Pi \mathcal{B} \cup \Pi \mathcal{C} = M \) . Since \( M \) is connected, \( \Pi \math

Proposition 1.1.7. The lightcone \( {\mathcal{L}}_{0} \) is a lightlike submanifold.

Suppose \( v \in {\mathcal{L}}_{0} \); then \( g\left( {v, v}\right) = 0 \) and \( v \neq 0 \). Let \( \mathcal{U} \) be a neighborhood of \( v \) that does not contain the origin and define \( \widetilde{g} : \mathcal{U} \rightarrow \mathbb{R} \) by \( \widetilde{g}w = g\left( {w, w}\right) \forall w \in \mathcal{U} \). Then \( {\mathcal{L}}_{0} \cap \mathcal{U} \) is defined by \( \widetilde{g} = 0 \). Moreover, \( d\widetilde{g} \) is nowhere zero on \( \mathcal{U} \) because \( g \) is nondegenerate (Exercise 0.0.10). Thus \( {\mathcal{L}}_{0} \) is a submanifold by the implicit function theorem. To show \( {\mathcal{L}}_{0} \) is lightlike, suppose that \( w \in {V}_{v} \) for some \( v \in {\mathcal{L}}_{0} \), and let \( {\phi }_{v} : {V}_{v} \rightarrow V \) be the canonical isomorphism of Exercise 0.0.10. Then \( w \) is in the tangent space \( {\left( {\mathcal{L}}_{0}\right) }_{v} \) iff \( w\widetilde{g} = 0 \), iff \( g\left( {{\phi }_{v}w, v}\right) = 0 \), and iff \( g\left( {w,{\phi }_{v}{}^{-1}v}\right) = 0 \). Thus \( {\left( {\mathcal{L}}_{0}\right) }_{v} = {\left( {\phi }_{v}{}^{-1}v\right) }^{ \bot } \subset {V}_{v} \). But \( {\phi }_{v}{}^{-1}v \in {V}_{v} \) is lightlike because \( g\left( {{\phi }_{v}{}^{-1}v,{\phi }_{v}{}^{-1}v}\right) = g\left( {v, v}\right) = 0 \), hence \( {\left( {\mathcal{L}}_{0}\right) }_{v} \) is lightlike by Proposition 1.1.3b. Since this holds for all \( v \in {\mathcal{L}}_{0},{\mathcal{L}}_{0} \) is lightlike.

Exercise 4.4.2 Show that \( \mathfrak{a} \) has an integral basis. Solution. Let \( \mathfrak{a} \) be an ideal of \( {\mathcal{O}}_{K} \), and let \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{n} \) be an integral basis for \( {\mathcal{O}}_{K} \) . Note that for any \( {\omega }_{i} \) in \( {\mathcal{O}}_{K},{a}_{0}{\omega }_{i} = - \left( {{\alpha }^{r} + \cdots + {a}_{1}\alpha }\right) {\omega }_{i} \in \mathfrak{a} \) . Therefore \( \mathfrak{a} \) has finite index in \( {\mathcal{O}}_{K} \) and \( \mathfrak{a} \subseteq {\mathcal{O}}_{K} = \mathbb{Z}{\omega }_{1} + \mathbb{Z}{\omega }_{2} + \cdots + \mathbb{Z}{\omega }_{n} \) has maximal rank. Then since \( \mathfrak{a} \) is a submodule of \( {\mathcal{O}}_{K} \), by Theorem 4.2.2 there exists an integral basis for \( \mathfrak{a} \) . Exercise 4.4.3 Show that if \( \mathfrak{a} \) is a nonzero ideal in \( {\mathcal{O}}_{K} \), then \( \mathfrak{a} \) has finite index in \( {\mathcal{O}}_{K} \) . Solution. Surely, if \( {\mathcal{O}}_{K} = \mathbb{Z}{\omega }_{1} + \mathbb{Z}{\omega }_{2} + \cdots + \mathbb{Z}{\omega }_{n} \), then by the preceding two exercises we can pick a rational integer \( a \) such that \[ a{\mathcal{O}}_{K} = a\mathbb{Z}{\omega }_{1} + a\mathbb{Z}{\omega }_{2} + \cdots + a\mathbb{Z}{\omega }_{n} \subset \mathfrak{a} \subset {\mathcal{O}}_{K}. \] But \( a{\mathcal{O}}_{K} \) obviously has index \( {a}^{n} \) in \( {\mathcal{O}}_{K} \) . Thus, the index of \( \mathfrak{a} \) in \( {\mathcal{O}}_{K} \) must be finite. Exercise 4.4.4 Show that every nonzero prime ideal in \( {\mathcal{O}}_{K} \) contains exactly one integer prime. Solution. If \( \wp \) is a prime ideal of \( {\mathcal{O}}_{K} \), then certainly it contains an integer, from Exercise 4.4.1. By the definition of a prime ideal, if \( {ab} \in \wp \), either \( a \in \wp \) or \( b \in \wp \) . So \( \wp \) must contain some rational prime. Now, if \( \wp \) contained two distinct rational primes \( p, q \), say, then it would necessarily contain their greatest common denominator which is 1 . But this contradicts the assumption of nontriviality. So every prime ideal of \( {\mathcal{O}}_{K} \) contains exactly one integer prime. Exercise 4.4.5 Let \( \mathfrak{a} \) be an integral ideal with basis \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) . Show that \[ {\left\lbrack \det \left( {\alpha }_{i}^{\left( j\right) }\right) \right\rbrack }^{2} = {\left( N\mathfrak{a}\right) }^{2}{d}_{K} \] Solution. Since \( \mathfrak{a} \) is a submodule of index \( N\mathfrak{a} \) in \( {\mathcal{O}}_{K} \), this is immediate from Exercise 4.2.8. ## 4.5 Supplementary Problems Exercise 4.5.1 Let \( K \) be an algebraic number field. Show that \( {d}_{K} \in \mathbb{Z} \) . Solution. By definition \[ {d}_{K} = \det {\left( {\omega }_{i}^{\left( j\right) }\right) }^{2} = \det \left( {\operatorname{Tr}\left( {{\omega }_{i}{\omega }_{j}}\right) }\right) \] where \( {\omega }_{1},\ldots ,{\omega }_{n} \) is an integral basis of \( {\mathcal{O}}_{K} \) . Since \( \operatorname{Tr}\left( {{\omega }_{i}{\omega }_{j}}\right) \in \mathbb{Z} \), the determinant is an integer. Exercise 4.5.2 Let \( K/\mathbb{Q} \) be an algebraic number field of degree \( n \) . Show that \( {d}_{K} \equiv 0 \) or 1 (mod 4). This is known as Stickelberger’s criterion. Solution. Let \( {\omega }_{1},\ldots ,{\omega }_{n} \) be an integral basis of \( {\mathcal{O}}_{K} \) . By definition, \[ {d}_{K} = \det {\left( {\sigma }_{i}\left( {\omega }_{j}\right) \right) }^{2} \] where \( {\sigma }_{1},\ldots ,{\sigma }_{n} \) are the distinct embeddings of \( K \) into \( \overline{\mathbb{Q}} \) . Now write \[ \det \left( {{\sigma }_{i}\left( {\omega }_{j}\right) }\right) = P - N \] where \( P \) is the contribution arising from the even permutations and \( N \) the odd permutations in the definition of the determinant. Then \[ {d}_{K} = {\left( P - N\right) }^{2} = {\left( P + N\right) }^{2} - {4PN}. \] Since \( {\sigma }_{i}\left( {P + N}\right) = P + N \), and \( {\sigma }_{i}\left( {PN}\right) = {PN} \) we see that \( P + N \) and \( {PN} \) are integers. Reducing mod 4 gives the result. Exercise 4.5.3 Let \( f\left( x\right) = {x}^{n} + {a}_{n - 1}{x}^{n - 1} + \cdots + {a}_{1}x + {a}_{0} \) with \( {a}_{i} \in \mathbb{Z} \) be the minimal polynomial of \( \theta \) . Let \( K = \mathbb{Q}\left( \theta \right) \) . If for each prime \( p \) such that \( {p}^{2} \mid {d}_{K/\mathbb{Q}}\left( \theta \right) \) we have \( f\left( x\right) \) Eisensteinian with respect to \( p \), show that \( {\mathcal{O}}_{K} = \mathbb{Z}\left\lbrack \theta \right\rbrack \) . Solution. By Example 4.3.1, the index of \( \theta \) is not divisible by \( p \) for any prime \( p \) satisfying \( {p}^{2} \mid {d}_{K/\mathbb{Q}}\left( \theta \right) \) . By Exercise 4.2.8, \[ {d}_{K/\mathbb{Q}}\left( \theta \right) = {m}^{2}{d}_{K} \] where \( m = \left\lbrack {{\mathcal{O}}_{K} : \mathbb{Z}\left\lbrack \theta \right\rbrack }\right\rbrack \) . Hence \( m = 1 \) . Exercise 4.5.4 If the minimal polynomial of \( \alpha \) is \( f\left( x\right) = {x}^{n} + {ax} + b \), show that for \( K = \mathbb{Q}\left( \alpha \right) \) , \[ {d}_{K/\mathbb{Q}}\left( \alpha \right) = {\left( -1\right) }^{\left( \begin{matrix} n \\ 2 \end{matrix}\right) }\left( {{n}^{n}{b}^{n - 1} + {a}^{n}{\left( 1 - n\right) }^{n - 1}}\right) . \] Solution. By Exercise 4.3.3, \[ {d}_{K/\mathbb{Q}}\left( \alpha \right) = {\left( -1\right) }^{\left( \begin{matrix} n \\ 2 \end{matrix}\right) }\mathop{\prod }\limits_{{i = 1}}^{n}{f}^{\prime }\left( {\alpha }^{\left( i\right) }\right) , \] where \( {\alpha }^{\left( 1\right) },\ldots ,{\alpha }^{\left( n\right) } \) are the conjugates of \( \alpha \) . Now \[ {f}^{\prime }\left( x\right) = n{x}^{n - 1} + a \] \[ = \frac{1}{x}\left( {n{x}^{n} + {ax}}\right) \] so that \[ {f}^{\prime }\left( {\alpha }^{\left( i\right) }\right) = \frac{\left( -n\left( a{\alpha }^{\left( i\right) } + b\right) + a{\alpha }^{\left( i\right) }\right) }{{\alpha }^{\left( i\right) }}. \] Hence \[ \mathop{\prod }\limits_{{i = 1}}^{n}{f}^{\prime }\left( {\alpha }^{\left( i\right) }\right) = {\left( -1\right) }^{n}{b}^{-1}\mathop{\prod }\limits_{{i = 1}}^{n}\left( {a\left( {1 - n}\right) {\alpha }^{\left( i\right) } - {nb}}\right) \] \[ = {b}^{-1}{a}^{n}{\left( 1 - n\right) }^{n}f\left( \frac{nb}{a\left( {1 - n}\right) }\right) . \] Exercise 4.5.5 Determine an integral basis for \( K = \mathbb{Q}\left( \theta \right) \) where \( {\theta }^{3} + {2\theta } + 1 = 0 \) . Solution. By applying the previous exercise, the discriminant of \( \theta \) is -59, which is squarefree. Therefore \( {\mathcal{O}}_{K} = \mathbb{Z}\left\lbrack \theta \right\rbrack \) . Exercise 4.5.6 (Dedekind) Let \( K = \mathbb{Q}\left( \theta \right) \) where \( {\theta }^{3} - {\theta }^{2} - {2\theta } - 8 = 0 \) . (a) Show that \( f\left( x\right) = {x}^{3} - {x}^{2} - {2x} - 8 \) is irreducible over \( \mathbb{Q} \) . (b) Consider \( \beta = \left( {{\theta }^{2} + \theta }\right) /2 \) . Show that \( {\beta }^{3} - 3{\beta }^{2} - {10\beta } - 8 = 0 \) . Hence \( \beta \) is integral. (c) Show that \( {d}_{K/\mathbb{Q}}\left( \theta \right) = - 4\left( {503}\right) \), and \( {d}_{K/\mathbb{Q}}\left( {1,\theta ,\beta }\right) = - {503} \) . Deduce that \( 1,\theta ,\beta \) is a \( \mathbb{Z} \) -basis of \( {\mathcal{O}}_{K} \) . (d) Show that every integer \( x \) of \( K \) has an even discriminant. (e) Deduce that \( {\mathcal{O}}_{K} \) has no integral basis of the form \( \mathbb{Z}\left\lbrack \alpha \right\rbrack \) . Solution. Note that if (a) is not true, then \( f \) has a linear factor and by the rational root theorem, this factor must be of the form \( x - a \) where \( a \mid 8 \) . A systematic check rules out this possibility. (b) can be checked directly. (c) This is easy to deduce from the formula \[ {d}_{K/\mathbb{Q}}\left( {1,\theta ,\beta }\right) = \frac{1}{4}{d}_{K/\mathbb{Q}}\left( \theta \right) \] as a simple computation shows. For (d), write \( x = A + {B\theta } + {C\beta }, A, B, C \in \mathbb{Z} \) . Since \[ {\beta }^{2} = 6 + {2\theta } + {3\beta } \] \[ {\theta }^{2} = {2\beta } - \theta \] \[ {\theta \beta } = {2\beta } + 4 \] we find \[ {x}^{2} = \left( {{a}^{2} + 6{C}^{2} + {8BC}}\right) + \theta \left( {2{C}^{2} - {B}^{2} + {2AB}}\right) + \beta \left( {2{B}^{2} + 3{C}^{2} + {2AC} + {4BC}}\right) \] so that \[ {d}_{K/\mathbb{Q}}\left( {1, x,{x}^{2}}\right) \equiv - {503}{\left( BC\right) }^{2}{\left( 3C + B\right) }^{2}\;\left( {\;\operatorname{mod}\;2}\right) , \] which is an even number in all cases. By (d), \( {d}_{K/\mathbb{Q}}\left( \alpha \right) \) is even and hence is not equal to -503, which proves (e). Exercise 4.5.7 Let \( m = {p}^{a} \), with \( p \) prime and \( K = \mathbb{Q}\left( {\zeta }_{m}\right) \) . Show that \[ {\left( 1 - {\zeta }_{m}\right) }^{\varphi \left( m\right) } = p{\mathcal{O}}_{K} \] Solution. First note that \[ \frac{{x}^{m} - 1}{{x}^{m/p} - 1} = \mathop{\prod }\limits_{\substack{{1 \leq b < m} \\ {\left( {b, m}\right) = 1} }}\left( {x - {\zeta }_{m}^{b}}\right) \] so that taking the limit as \( x \) goes to 1 of both sides gives \[ p = \mathop{\prod }\limits_{\substack{{1 \leq b < m} \\ {\left( {b, m}\right) = 1} }}\left( {1 - {\zeta }_{m}^{b}}\right) \] \[ = {\left( 1 - {\zeta }_{m}\right) }^{\varphi \left( m\right) }\mathop{\prod }\limits_{\substack{{1 \leq b < m} \\ {\left( {b, m}\right) = 1} }}\frac{1 - {\zeta }_{m}^{b}}{1 - {\zeta }_{m}}. \] This latter quantity is a unit since \[ \frac{1 - {\zeta }_{m}}{1 - {\zeta }_{m}^{b}} = \frac{1 - {\zeta }_{m}^{ab}}{1 - {\zeta }_{m}^{b}} = 1 + {\zeta }_{m}^{b} + \cdots + {\zeta }_{m}^{b\left( {a - 1}\right) } \] for any \( a \) satisfying \( {ab} \equiv 1\left( {\;\operatorname{mod}\;m}\right) \) . Exercise 4.5.8 Let \( m = {p}^{a} \), with \( p \) prime, and \( K = \mathbb{Q}\left( {\zeta }_{m

Exercise 4.4.2 Show that \( \mathfrak{a} \) has an integral basis.

Solution. Let \( \mathfrak{a} \) be an ideal of \( {\mathcal{O}}_{K} \), and let \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{n} \) be an integral basis for \( {\mathcal{O}}_{K} \). Note that for any \( {\omega }_{i} \) in \( {\mathcal{O}}_{K},{a}_{0}{\omega }_{i} = - \left( {{\alpha }^{r} + \cdots + {a}_{1}\alpha }\right) {\omega }_{i} \in \mathfrak{a} \). Therefore \( \mathfrak{a} \) has finite index in \( {\mathcal{O}}_{K} \) and \( \mathfrak{a} \subseteq {\mathcal{O}}_{K} = \mathbb{Z}{\omega }_{1} + \mathbb{Z}{\omega }_{2} + \cdots + \mathbb{Z}{\omega }_{n} \) has maximal rank. Then since \( \mathfrak{a} \) is a submodule of \( {\mathcal{O}}_{K} \), by Theorem 4.2.2 there exists an integral basis for \( \mathfrak{a} \).

Exercise 8.4.4. Prove that the group \[ B = \left\{ {\left. \left( \begin{array}{ll} a & b \\ & 1 \end{array}\right) \right| \;a, b \in \mathbb{R}, a > 0}\right\} , \] which is also called the ’ \( {ax} + b \) ’ group to reflect its natural action on \[ \left\{ {\left. \left( \begin{array}{l} x \\ 1 \end{array}\right) \right| \;x \in \mathbb{R}}\right\} \] is amenable but not unimodular. ## 8.5 Mean Ergodic Theorem for Amenable Groups Følner sequences permit ergodic averages to be formed, and the mean and pointwise ergodic theorems hold under suitable conditions for measure-preserving actions of amenable groups. In the theorem below we deal with integration of functions on the group taking values in the Hilbert space \( {L}_{\mu }^{2} \) (see Sect. A.3). The reader may find it a helpful exercise to specialize the proof to the case of countable discrete amenable groups, in which Haar measure is cardinality, the elements of a Følner sequence are finite sets, and the integrals appearing are simply finite sums. We will also be using the induced unitary representation* of \( G \) defined by \[ {U}_{g}\left( f\right) \left( x\right) = f\left( {{g}^{-1} \cdot x}\right) \] for all \( x \in X \) . As every element of \( G \) is assumed to preserve the measure \( \mu \) in its action on \( X \), we know that \( {U}_{g} : {L}_{\mu }^{2} \rightarrow {L}_{\mu }^{2} \) is unitary. Moreover, if \( g, h \in G \) , then by definition \[ {U}_{h}\left( {{U}_{g}\left( f\right) }\right) \left( x\right) = {U}_{g}\left( f\right) \left( {{h}^{-1} \cdot x}\right) = f\left( {{g}^{-1}{h}^{-1} \cdot x}\right) = f\left( {{\left( hg\right) }^{-1} \cdot x}\right) = {U}_{hg}\left( f\right) \left( x\right) , \] which shows that \( g \mapsto {U}_{g} \) for \( g \in G \) defines an action of \( G \) on \( {L}_{\mu }^{2} \) . (The inverse in the definition of \( {U}_{g} \) is used to ensure that this is indeed an action.) Theorem 8.13. Let \( G \) be a \( \sigma \) -locally compact amenable group with left Haar measure \( {m}_{G} \) acting continuously on \( X \), and let \( \mu \) be a \( G \) -invariant Borel probability measure on \( X \) . Let \( {P}_{G} \) be the orthogonal projection onto the closed subspace \[ I = \left\{ {f \in {L}_{\mu }^{2}\left( X\right) \mid {U}_{g}f = f\text{ for all }g \in G}\right\} \subseteq {L}_{\mu }^{2}\left( X\right) . \] Then, for any Følner sequence \( \left( {F}_{n}\right) \) and \( f \in {L}_{\mu }^{2}\left( X\right) \) , \[ \frac{1}{{m}_{G}\left( {F}_{n}\right) }{\int }_{{F}_{n}}{U}_{{g}^{-1}}f\mathrm{\;d}{m}_{G}\left( g\right) \underset{{L}_{\mu }^{2}}{ \rightarrow }{P}_{G}f. \] Thus the action is ergodic if and only if \[ \frac{1}{{m}_{G}\left( {F}_{n}\right) }{\int }_{{F}_{n}}{U}_{{g}^{-1}}f\mathrm{\;d}{m}_{G}\left( g\right) \underset{{L}_{\mu }^{2}}{ \rightarrow }{\int }_{X}f\mathrm{\;d}\mu \] for all \( f \in {L}_{\mu }^{2}\left( X\right) \) (or for all \( f \) in a dense subset of \( {L}_{\mu }^{2}\left( X\right) \) ). As will become clear, it is important that we average the expression \[ {U}_{{g}^{-1}}f\left( x\right) = f\left( {g \cdot x}\right) \] instead of the expression \( {U}_{g}f\left( x\right) \) . * For now we will just use the action of \( {U}_{g} \) for \( g \in G \) on \( {L}_{\mu }^{2}\left( X\right) \) defined here; a more formal definition of the notion of unitary representation will be given in Sect. 11.3. Proof of Theorem 8.13. Let \( u \) be a function of the form \[ u\left( x\right) = v\left( {h \cdot x}\right) - v\left( x\right) \] for some \( v \in {L}_{\mu }^{2}\left( X\right) \) and \( h \in G \), that is \( u = {U}_{{h}^{-1}}v - v \) . Then \[ {\int }_{{F}_{n}}{U}_{{g}^{-1}}{U}_{{h}^{-1}}v\mathrm{\;d}{m}_{G}\left( g\right) = {\int }_{{F}_{n}}{U}_{{\left( hg\right) }^{-1}}v\mathrm{\;d}{m}_{G}\left( g\right) = {\int }_{h{F}_{n}}{U}_{{g}^{-1}}v\mathrm{\;d}{m}_{G}\left( g\right) , \] and so \[ {\begin{Vmatrix}\frac{1}{{m}_{G}\left( {F}_{n}\right) }{\int }_{{F}_{n}}{U}_{{g}^{-1}}u\mathrm{\;d}{m}_{G}\left( g\right) \end{Vmatrix}}_{2} = \parallel \frac{1}{{m}_{G}\left( {F}_{n}\right) }\left( {{\int }_{h{F}_{n}}{U}_{{g}^{-1}}v\mathrm{\;d}{m}_{G}\left( g\right) }\right. \] \[ - {\left. \left. {\int }_{{F}_{n}}{U}_{{g}^{-1}}v\mathrm{\;d}{m}_{G}\left( g\right) \right) \right| }_{2} \] \[ \leq \frac{1}{{m}_{G}\left( {F}_{n}\right) }{\int }_{{F}_{n}\bigtriangleup h{F}_{n}}\begin{Vmatrix}{{U}_{{g}^{-1}}v}\end{Vmatrix}\mathrm{d}{m}_{G}\left( g\right) \rightarrow 0 \] as \( n \rightarrow \infty \), by (8.13) and Sect. B.7. It follows that the same holds for any function in the \( {L}_{\mu }^{2} \) -closure \( V \) of the space of finite linear combinations of such functions. Just as in the proof of the mean ergodic theorem (Theorem 2.21), if \( u \bot V \) then, for every \( v \in {L}_{\mu }^{2} \) , \[ \left\langle {{U}_{g}u, v}\right\rangle = \left\langle {u,{U}_{{g}^{-1}}v - v}\right\rangle + \langle u, v\rangle = \langle u, v\rangle \] so \( u \in I \) . Thus \( {L}_{\mu }^{2} = V \oplus I \), showing the first part of the theorem. As discussed on page 233, the \( G \) -action is ergodic if and only if \( I = \mathbb{C} \), the constant functions, or equivalently if and only if \( {P}_{G}f = {\int }_{X}f\mathrm{\;d}\mu \), completing the proof. Just as the mean ergodic theorem (Theorem 2.21) readily implies a mean ergodic theorem in \( {L}^{1} \) for single transformations (Corollary 2.22), Theorem 8.13 implies an \( {L}^{1} \) theorem. See Sect. B. 7 for an explanation of the meaning of the integral arising. Corollary 8.14. Let \( G \) be a locally compact amenable group with left Haar measure \( {m}_{G} \) acting continuously on \( X \), and let \( \mu \) be a \( G \) -invariant Borel probability measure on \( X \) . Then, for any Følner sequence \( \left( {F}_{n}\right) \) and \( f \in {L}_{\mu }^{1}\left( X\right) \) , \[ \frac{1}{{m}_{G}\left( {F}_{n}\right) }{\int }_{{F}_{n}}f \circ g\mathrm{\;d}{m}_{G}\left( g\right) \rightarrow E\left( {f \mid \mathcal{E}}\right) \] in \( {L}_{\mu }^{1} \), where \( \mathcal{E} \) is the \( \sigma \) -algebra of \( G \) -invariant sets. ## Exercises for Sect. 8.5 Exercise 8.5.1. Emulate the proof of Corollary 2.22 to deduce Corollary 8.14 from Theorem 8.13. ## 8.6 Pointwise Ergodic Theorems and Polynomial Growth While the mean ergodic theorem (Theorem 2.21) extends easily to the setting of amenable group actions both in its statement and in its proof, extending the pointwise ergodic theorem (Theorem 2.30) is much more involved \( {}^{\left( {83}\right) } \) . The general pointwise ergodic theorem for amenable groups is due to Linden-strauss [233]; a condition is needed on the averaging Følner sequence used (but every amenable group has Følner sequences satisfying the condition). That some condition on the sequence is needed is already visible for single transformations ( \( \mathbb{Z} \) -actions): del Junco and Rosenblatt [67] show that for any non-trivial measure-preserving system the pointwise ergodic theorem does not hold along the Følner sequence defined by \( {F}_{n} = \left\lbrack {{n}^{2},{n}^{2} + n}\right) \cap \mathbb{Z} \) . ## 8.6.1 Flows We start by showing how Theorem 2.30 extends to the case of continuous time. As this is convenient, we state and prove the theorem in the measurable context. A flow is a family \( \left\{ {{T}_{t} \mid t \in \mathbb{R}}\right\} \) of measurable transformations of the probability space \( \left( {X,\mathcal{B},\mu }\right) \) satisfying the identity \( {T}_{s}{T}_{t} = {T}_{s + t} \) for all \( s, t \in \mathbb{R} \), and with \( {T}_{0} = {I}_{X} \) . The flow is measure-preserving if \( {T}_{t} \) preserves \( \mu \) for each \( t \in \mathbb{R} \), and is measurable (as a flow) if the map \( \left( {x, t}\right) \mapsto {T}_{t}\left( x\right) \) is a measurable map from \( \left( {X \times \mathbb{R},\mathcal{B} \otimes {\mathcal{B}}_{\mathbb{R}}}\right) \) to \( \left( {X,\mathcal{B}}\right) \) . Similarly, a semi-flow is an action of the semigroup \( {\mathbb{R}}_{ \geq 0} \) . The pointwise ergodic theorem for (semi-)flows is a direct corollary of Theorem 2.30. Corollary 8.15. Let \( T \) be a measurable and measure-preserving (semi-)flow on the probability space \( \left( {X,\mathcal{B},\mu }\right) \) . Then, for any \( f \in {L}_{\mu }^{1} \), there is a measurable set of full measure on which \[ \frac{1}{s}{\int }_{0}^{s}f\left( {{T}_{s}x}\right) \mathrm{d}s \rightarrow E\left( {f \mid \mathcal{E}}\right) \left( x\right) \] converges everywhere and in \( {L}_{\mu }^{1} \) to the expectation with respect to the \( \sigma \) - algebra \( \mathcal{E} = \left\{ {B \in \mathcal{B} \mid \mu \left( {B\bigtriangleup {T}_{t}^{-1}B}\right) = 0\text{for all}t \in \mathbb{R}}\right\} \) . Proof. The function \( \left( {x, s}\right) \mapsto f\left( {{T}_{s}\left( x\right) }\right) \) is integrable on \( X \times \left\lbrack {0, s}\right\rbrack \) for any non-negative \( s \) by Fubini’s theorem (Theorem A.13). Thus the integral \( {\int }_{0}^{s}f\left( {{T}_{t}x}\right) \mathrm{d}t \) is well-defined for almost every \( x \in X \) . In particular, \( F\left( x\right) = {\int }_{0}^{1}f\left( {{T}_{t}x}\right) \mathrm{d}t \) is well-defined for almost every \( x \), and therefore defines a function in \( {L}_{\mu }^{1} \) . Now \[ {\int }_{0}^{n}f\left( {{T}_{t}x}\right) \mathrm{d}t = \mathop{\sum }\limits_{{j = 0}}^{{n - 1}}F\left( {{T}_{1}^{j}x}\right) \] so, by Theorem 2.30, the averages \[ \frac{1}{N}{\int }_{0}^{N}f\left( {{T}_{t}x}\right) \mathrm{d}t \] converge almost everywhere as \( N \rightarrow \infty \) . Moreover, Theorem 2.30 applied to \( {F}_{\mathrm{{abs}}} = {\int }_{0}^{1}\left| {f\left( {{T}_{s}x}\right) }\right| \mathrm{d}s \) also implies (by taking the difference between \( {\mathrm{A}}_{N}\left( {F}_{\mathrm{{abs}}}\right) \) and \( \frac{N + 1}{N}{\mathrm{\;A}}_{N + 1}\left( {F}_{\mathrm{{abs}}}\right) \), which conver

Exercise 8.4.4. Prove that the group \[ B = \left\{ {\left. \left( \begin{array}{ll} a & b \\ & 1 \end{array}\right) \right| \;a, b \in \mathbb{R}, a > 0}\right\} , \] which is also called the ’ \( {ax} + b \) ’ group to reflect its natural action on \[ \left\{ {\left. \left( \begin{array}{l} x \\ 1 \end{array}\right) \right| \;x \in \mathbb{R}}\right\} \] is amenable but not unimodular.

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Proposition 18.13. If \( K \) is a compact subset of \( \mathbb{C} \) then the algebra \( \mathcal{P}\left( K\right) \) coincides with the restriction to \( K \) of the subalgebra \( \mathcal{P}\left( \widehat{K}\right) \) of \( \mathcal{C}\left( \widehat{K}\right) \) . Thus the functions in \( \mathcal{P}\left( K\right) \) are not only locally analytic on \( {K}^{ \circ } \), but can be extended so as to be locally analytic on the interior of \( \widehat{K} \) . Proof. A sequence \( \left\{ {p}_{n}\right\} \) of polynomials converging uniformly on \( K \) automatically converges uniformly on the outer boundary of \( K \) . But then, by the maximum modulus principle (Ex. 5M), \( \left\{ {p}_{n}\right\} \) converges uniformly on \( \widetilde{K} \) , and the limit is therefore differentiable at each point of \( {\widehat{K}}^{ \circ } \) (see Problem 11U). The foregoing discussion shows that for a compact subset \( K \) of \( \mathbb{C} \), the algebra \( \mathcal{P}\left( K\right) \) may fail to be equal to \( \mathcal{C}\left( K\right) \) for two quite different reasons, either because \( {K}^{ \circ } \neq \varnothing \), or because \( K \) has one or more holes. Example G. Let us take for \( K \) the unit circle \( Z \) . Then \( \widehat{K} \) is just the closed unit disc \( {D}^{ - } \) of Example \( \mathrm{F} \), so \( \mathcal{P}\left( Z\right) \) consists of the restrictions to \( Z \) of all of the functions in \( \mathcal{P}\left( {D}^{ - }\right) = \mathcal{A}\left( {D}^{ - }\right) \) . We observe that the mapping \( f \rightarrow f \mid Z \) is an isometric Banach algebra isomorphism of the disc algebra \( \mathcal{P}\left( {D}^{ - }\right) \) onto \( \mathcal{P}\left( Z\right) \) by the maximum modulus principle (Ex. 5M). (The algebra \( \mathcal{P}\left( Z\right) \) is also frequently called the disc algebra.) Example H. Let \( K \) be any compact subset of \( \mathbb{C} \) such that \( {K}^{ \circ } \neq \varnothing \) . Then, just as in the preceding example, the mapping \( f \rightarrow f \mid \left( {\partial K}\right) \) is an isometric Banach algebra isomorphism of \( \mathcal{A}\left( K\right) \) onto a subalgebra of \( \mathcal{C}\left( {\partial K}\right) \), so any bounded linear functional \( \varphi \) on \( \mathcal{A}\left( K\right) \) may be regarded as a functional on a subspace of \( \mathcal{C}\left( {\partial K}\right) \), which may then be extended to a bounded linear functional on \( \mathcal{C}\left( {\partial K}\right) \) by the Hahn-Banach theorem (Th. 14.3). Hence, by the Riesz representation theorem (Th. 18.3), there exists a complex Borel measure \( \xi \) on \( \partial K \) such that \[ \varphi \left( f\right) = {\int }_{\partial K}{fd\xi },\;f \in \mathcal{A}\left( K\right) . \] (If \( \mathcal{A}\left( K\right) \) is replaced by \( \mathcal{P}\left( K\right) \), the boundary \( \partial K \) may be replaced by the outer boundary of \( K \) .) Suppose, in particular, we take for \( K \) the closed unit disc \( {D}^{ - } \), so that \( \partial K \) is the unit circle \( Z \) . Then for each point \( \alpha \) of \( {D}^{ - } \) evaluation at \( \alpha \) is a bounded linear functional on \( \mathcal{A}\left( {D}^{ - }\right) \), so there exists a complex Borel measure \( {\xi }_{\alpha } \) on \( Z \) such that \[ f\left( \alpha \right) = {\int }_{Z}{fd}{\xi }_{\alpha },\;f \in \mathcal{A}\left( {D}^{ - }\right) . \] Thus, for example, for \( \alpha = 0 \) we have \[ f\left( 0\right) = \frac{1}{2\pi i}{\int }_{Z}\frac{f\left( \xi \right) }{\xi }{d\xi } = \frac{1}{2\pi }{\int }_{0}^{2\pi }f\left( {e}^{it}\right) {dt} = \frac{1}{2\pi }{\int }_{Z}{fd\theta } \] (where \( \theta \) denotes arc-length measure; see Example \( 8\mathrm{\;F} \) ), so we may take \( {\xi }_{0} \) to be \( \theta /{2\pi } \) . (This use of the Cauchy integral formula is readily justified because of the uniform continuity of \( f \) on \( {D}^{ - } \) .) Example I. Let \( r \) and \( R \) be radii such that \( 0 < r < R \), let \( A \) denote the annular domain \( \{ \lambda \in \mathbb{C} : r < \left| \lambda \right| < R\} \), and let \( K = {A}^{ - } \) . Then the circle \( {C}_{R} = \) \( \{ \lambda \in \mathbb{C} : \left| \lambda \right| = R\} \) is the outer boundary of \( K \) and \( \widehat{K} \) is the closed disc \( {D}_{R}^{ - } = \) \( \{ \lambda \in \mathbb{C} : \left| \lambda \right| \leq R\} \) . Thus every function \( f \) in \( \mathcal{P}\left( K\right) \) is the restriction to \( K \) of a function \( g \) in the algebra \( \mathcal{A}\left( {D}_{R}^{ - }\right) = \mathcal{P}\left( {D}_{R}^{ - }\right) \) . Next let us consider the algebra \( \mathcal{A}\left( K\right) \) . If \( f \in \mathcal{A}\left( K\right) \) then \( f \) possesses a Laurent expansion \[ f\left( \lambda \right) = \mathop{\sum }\limits_{{n = - \infty }}^{{+\infty }}{\alpha }_{n}{\lambda }^{n},\;\lambda \in A \] (Prop. 5.9), and we may write \( f = {f}_{1} + {f}_{2} \) where \[ {f}_{1}\left( \lambda \right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\alpha }_{n}{\lambda }^{n}\;\text{ and }\;{f}_{2}\left( \lambda \right) = \mathop{\sum }\limits_{{n = - 1}}^{{-\infty }}{\alpha }_{n}{\lambda }^{n}. \] The function \( {f}_{1} \) can be extended to the closed disc \( {D}_{R}^{ - } \) by continuity (since \( f \) can be and \( {f}_{2} \) is already continuous in a neighborhood of \( {C}_{R} \) ), and there exists a sequence \( \left\{ {p}_{n}\right\} \) of polynomials \( {p}_{n}\left( \lambda \right) = {\beta }_{0}^{\left( n\right) } + \cdots + {\beta }_{n}^{\left( n\right) }{\lambda }^{n} \) converging uniformly to \( {f}_{1} \) on \( {D}_{R}^{ - } \), and therefore on \( K \) (Ex. F). Similarly, \( {f}_{2} \) can be extended by continuity to the closed region \( \mathbb{C} \smallsetminus {D}_{r} = \{ \lambda \in \mathbb{C} : \left| \lambda \right| \geq r\} \), and an argument exactly like the one in Example \( \mathrm{F} \) shows that there exists a sequence \( \left\{ {q}_{n}\right\} \) of functions of the form \[ {q}_{n}\left( \lambda \right) = \frac{{\beta }_{-1}^{\left( n\right) }}{\lambda } + \cdots + \frac{{\beta }_{-n}^{\left( n\right) }}{{\lambda }^{n}} \] converging uniformly to \( {f}_{2} \) on \( \mathbb{C} \smallsetminus {D}_{r} \), and therefore on \( K \) . Thus, summarizing, we see that there exists a sequence of rational functions of the form \[ {r}_{n}\left( \lambda \right) = \frac{{\beta }_{-n}^{\left( n\right) }}{{\lambda }^{n}} + \cdots + {\beta }_{0}^{\left( n\right) } + {\beta }_{1}^{\left( n\right) }\lambda + \cdots + {\beta }_{n}^{\left( n\right) }{\lambda }^{n} \] converging uniformly to \( f \) on \( K \) . The foregoing example suggests the introduction of yet one more Banach algebra. Definition. If \( K \) is a compact subset of \( \mathbb{C} \) we shall denote by \( \mathcal{R}\left( K\right) \) the closure in \( \mathcal{C}\left( K\right) \) of the algebra of all rational functions with poles off \( K \) . It is clear that \( \mathcal{R}\left( K\right) \) is a closed subalgebra of \( \mathcal{C}\left( K\right) \) and that \[ \mathcal{P}\left( K\right) \subset \mathcal{R}\left( K\right) \subset \mathcal{A}\left( K\right) \] for any compact set \( K \) of complex numbers. Example I shows that \( \mathcal{R}\left( K\right) \) may coincide with \( \mathcal{A}\left( K\right) \) when \( K \) has a hole. But also in Example I every function in \( \mathcal{A}\left( K\right) \) is approximable by means of special rational functions with poles only at 0 (and \( \infty \) ). As a matter of fact, a similar restriction on the poles of approximating rational functions can be effected whenever, as in Example I, the functions in \( \mathcal{R}\left( K\right) \) can be uniformly approximated on \( K \) by functions that are locally analytic on some neighborhood of \( K \) . Proposition 18.14. Let \( K \) be a nonempty compact subset of \( \mathbb{C} \), let \( P \) be a subset of the Riemann sphere \( \widehat{\mathbb{C}} \) (Prob. \( {3W} \) ) with the property that \( P \) contains at least one point of each connected component of \( \widehat{\mathbb{C}} \smallsetminus K \), and let \( f \) be a locally analytic function on some open neighborhood \( U \) of \( K \) . Then there exists a sequence \( \left\{ {r}_{n}\right\} \) of rational functions converging uniformly to fon \( K \) such that each \( {r}_{n} \) has poles only in \( P \) . Proof. Let us denote by \( \mathcal{L} \) the linear submanifold of \( \mathcal{C}\left( K\right) \) consisting of (the restrictions to \( K \) of) those rational functions having their poles in \( P \) . We are to show that \( f \) belongs to \( {\mathcal{L}}^{ - } \), and by Proposition 14.10 (a consequence of the Hahn-Banach theorem) it suffices to show that if a bounded linear functional \( \varphi \) on \( \mathcal{C}\left( K\right) \) annihilates \( \mathcal{L} \), then \( \varphi \left( f\right) = 0 \) . By virtue of Theorem 18.3 this amounts to showing that if \( \xi \) is a complex Borel measure on \( K \) such that \( {\int }_{K}{rd\xi } = 0 \) for every rational function \( r \) with poles in \( P \), then \( {\int }_{K}{fd\xi } = 0 \) . Suppose, accordingly, that \( {\int }_{K}{rd\xi } = 0 \) for every rational function \( r \) with poles in \( P \) . Let \( V \) be a component of \( \widehat{\mathbb{C}} \smallsetminus K \), suppose \( \alpha \) is a point of \( V \cap P \) such that \( \alpha \neq \infty \), and let \( d = d\left( {\alpha, K}\right) \) . Clearly \( d > 0 \), and for each \( \lambda \) such that \( \left| {\lambda - \alpha }\right| < d \) the power series \[ \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{\left( \lambda - \alpha \right) }^{n}}{{\left( \zeta - \alpha \right) }^{n + 1}} \] (8) converges to \( 1/\left( {\zeta - \lambda }\right) \) uniformly (in \( \zeta \) ) on \( K \) . Hence the Cauchy transform \[ h\left( \lambda \right) = {\int }_{K}\frac{{d\xi }\left( \zeta \right) }{\zeta -

Proposition 18.13. If \( K \) is a compact subset of \( \mathbb{C} \) then the algebra \( \mathcal{P}\left( K\right) \) coincides with the restriction to \( K \) of the subalgebra \( \mathcal{P}\left( \widehat{K}\right) \) of \( \mathcal{C}\left( \widehat{K}\right) \) . Thus the functions in \( \mathcal{P}\left( K\right) \) are not only locally analytic on \( {K}^{ \circ } \), but can be extended so as to be locally analytic on the interior of \( \widehat{K} \) .

A sequence \( \left\{ {p}_{n}\right\} \) of polynomials converging uniformly on \( K \) automatically converges uniformly on the outer boundary of \( K \) . But then, by the maximum modulus principle (Ex. 5M), \( \left\{ {p}_{n}\right\} \) converges uniformly on \( \widetilde{K} \) , and the limit is therefore differentiable at each point of \( {\widehat{K}}^{ \circ } \) (see Problem 11U).

Theorem 11.4.11. The action of \( {W}_{G} \) on \( \mathfrak{h} \) coincides with the action of \( W\left( {\mathfrak{g},\mathfrak{h}}\right) \) . Furthermore, every coset in \( {W}_{G} \) has a representative from \( U \) . Proof. For \( \alpha \in \Phi \) and \( {X}_{\alpha } \) as in (7.42), set \[ {u}_{\alpha } = \frac{1}{2}\left( {{X}_{\alpha } - {X}_{-\alpha }}\right) \;\text{ and }\;{v}_{\alpha } = \frac{1}{2\mathrm{i}}\left( {{X}_{\alpha } + {X}_{-\alpha }}\right) . \] (11.13) Then \( {X}_{\alpha } = {u}_{\alpha } + \mathrm{i}{v}_{\alpha } \), and \( {u}_{\alpha },{v}_{\alpha } \in \mathfrak{u} \) . We calculate the action of ad \( {u}_{\alpha } \) on \( h \in \mathfrak{h} \) as follows: \[ \left\lbrack {{u}_{\alpha }, h}\right\rbrack = - \frac{1}{2}\langle \alpha, h\rangle \left( {{X}_{\alpha } + {X}_{-\alpha }}\right) = - \mathrm{i}\langle \alpha, h\rangle {v}_{\alpha }, \] (11.14) \[ \left\lbrack {{u}_{\alpha },{v}_{\alpha }}\right\rbrack = \frac{1}{4\mathrm{i}}\left\lbrack {{X}_{\alpha } - {X}_{-\alpha },{X}_{\alpha } + {X}_{-\alpha }}\right\rbrack = \frac{1}{2\mathrm{i}}\left\lbrack {{X}_{\alpha },{X}_{-\alpha }}\right\rbrack = \frac{1}{2\mathrm{i}}{H}_{\alpha }. \] (11.15) From (11.14) we see that for all \( s \in \mathbb{C} \) , \[ \exp \left( {s\operatorname{ad}{u}_{\alpha }}\right) h = h\;\text{ if }\langle \alpha, h\rangle = 0. \] (11.16) Taking \( h = {H}_{\alpha } \) in (11.14), we obtain \( \operatorname{ad}\left( {u}_{\alpha }\right) {H}_{\alpha } = \mathrm{i}\parallel \alpha {\parallel }^{2}{v}_{\alpha } \), and hence by (11.15) we have \[ {\left( \operatorname{ad}{u}_{\alpha }\right) }^{2}{H}_{\alpha } = - {r}^{2}{H}_{\alpha } \] where \( r = \parallel \alpha \parallel /\sqrt{2} \) . Continuing in this way, we calculate that \[ \exp \left( {s\operatorname{ad}{u}_{\alpha }}\right) {H}_{\alpha } = \cos \left( {rs}\right) {H}_{\alpha } - \left( {2\mathrm{i}/r}\right) \sin \left( {rs}\right) {v}_{\alpha } \] for all \( s \in \mathbb{C} \) . In particular, when \( s = \pi /r \) the second term vanishes and we have \[ \exp \left( {\left( {\pi /r}\right) \operatorname{ad}{u}_{\alpha }}\right) {H}_{\alpha } = - {H}_{\alpha }. \] (11.17) Thus by (11.16) and (11.17) we see that the element \( {g}_{\alpha } = \exp \left( {\left( {\pi /r}\right) {u}_{\alpha }}\right) \in U \) acts on \( \mathfrak{h} \) by the reflection \( {s}_{\alpha } \) . This proves that every element in the algebraic Weyl group \( W\left( {\mathfrak{g},\mathfrak{h}}\right) \) can be implemented by the adjoint action of an element \( k \in {\operatorname{Norm}}_{G}\left( H\right) \cap U \) . To prove the converse, fix a set \( {\Phi }^{ + } \) of positive roots and let \[ \rho = \frac{1}{2}\mathop{\sum }\limits_{{\alpha \in {\Phi }^{ + }}}\alpha \] If \( g \in {\operatorname{Norm}}_{G}\left( H\right) \), then \( \operatorname{Ad}{\left( g\right) }^{t}{\Phi }^{ + } \subset \Phi \) is another set of positive roots, so by Theorem 3.1.9 there is an element \( s \in W\left( {\mathfrak{g},\mathfrak{h}}\right) \) such that \( s\operatorname{Ad}{\left( g\right) }^{t}{\Phi }^{ + } = {\Phi }^{ + } \) . Hence by what has just been proved, there exists \( k \in {\operatorname{Norm}}_{G}\left( H\right) \cap U \) such that \( {\left. \operatorname{Ad}\left( k\right) \right| }_{\mathfrak{h}} = s \), so \( \operatorname{Ad}{\left( kg\right) }^{t}{\Phi }^{ + } = {\Phi }^{ + } \) . Thus \( {kg} \) fixes \( \rho \), which means that \( \operatorname{Ad}\left( {kg}\right) {H}_{\rho } = {H}_{\rho } \) . The element \( {H}_{\rho } \) satisfies \( \left\langle {\alpha ,{H}_{\rho }}\right\rangle \neq 0 \) for all \( \alpha \in \Phi \) by Lemma 3.1.21. Hence \[ \mathfrak{h} = \left\{ {X \in \mathfrak{g} : \left\lbrack {{H}_{\rho }, X}\right\rbrack = 0}\right\} . \] (11.18) We claim that the one-parameter subgroup \[ \Gamma = \left\{ {\exp \left( {t{H}_{\rho }}\right) : t \in \mathbb{C}}\right\} \subset H \] is dense in \( H \) (in the Lie group topology). Indeed, the closure (in the Lie group topology) of \( \Gamma \) in \( G \) is a closed Lie subgroup of \( H \) that has Lie algebra \( \mathfrak{h} \) by (11.18) and Theorem 1.3.8, hence coincides with \( H \), since \( H \) is connected. But \( {kg} \) commutes with the elements of \( \Gamma \), so the semisimple and unipotent factors of \( {kg} \) in its Jordan-Chevalley decomposition also commute with \( \Gamma \), and hence with \( H \) . Since the unipotent factor is of the form \( \exp X \) with \( X \) nilpotent, it follows from (11.18) that \( X \in \mathfrak{h} \), and hence \( X = 0 \) . Thus \( {kg} \) is semisimple and commutes with the elements of \( H \), so \( {kg} \in H \), since \( H \) is a maximal algebraic torus. This completes the proof of the theorem. Remark 11.4.12. For the classical groups, Theorem 11.4.11 can be easily proved on a case-by-case basis using the descriptions of \( {W}_{G} \) in Section 3.1.1; see Goodman-Wallach [56, Lemma 7.4.3]. Corollary 11.4.13. The natural inclusion map \( {\operatorname{Norm}}_{U}\left( T\right) /T \rightarrow {\operatorname{Norm}}_{G}\left( H\right) /H \) is an isomorphism. Proof. This follows from Theorem 11.4.11 and Proposition 7.3.2. Define the regular elements in the maximal torus \( H \) as \[ {H}^{\prime } = \left\{ {h \in H : {h}^{\alpha } \neq 1\text{ for all }\alpha \in \Phi }\right\} . \] We note that \( \left( \star \right) \;h \in H \) is regular if and only if \( {\left( {\operatorname{Cent}}_{G}\left( h\right) \right) }^{ \circ } = H \) . Indeed, let \( M = {\operatorname{Cent}}_{G}\left( h\right) \) . We have \( X \in \operatorname{Lie}\left( M\right) \) if and only if \[ I = \exp \left( {-{tX}}\right) h\exp \left( {tX}\right) {h}^{-1} = \exp \left( {-{tX}}\right) \exp \left( {t\operatorname{Ad}\left( h\right) X}\right) \;\text{ for all }t \in \mathbb{C}. \] Differentiating this equation at \( t = 0 \), we find that \( X \in \operatorname{Ker}\left( {\operatorname{Ad}\left( h\right) - I}\right) \) . If \( {h}^{\alpha } = 1 \) for some root \( \alpha \), then the one-parameter unipotent group \( \exp \left( {\mathfrak{g}\alpha }\right) \) is contained in \( M \) . Thus if \( h \) is not regular, then \( {M}^{ \circ } \) is strictly larger than \( H \) . Conversely, suppose that \( h \in {H}^{\prime } \) . Then \( \operatorname{Ker}\left( {\operatorname{Ad}\left( h\right) - I}\right) = \mathfrak{h} \) . Hence we see that \( \operatorname{Lie}\left( M\right) = \mathfrak{h} \) . Hence \( {M}^{ \circ } = H \) . This proves \( \left( \star \right) \) . Remark 11.4.14. The group \( {\operatorname{Cent}}_{G}\left( h\right) \) is not necessarily \( H \) when \( h \in {H}^{\prime } \) . For example, take \( G = \mathbf{{PSL}}\left( {2,\mathbb{C}}\right) = \mathbf{{SL}}\left( {2,\mathbb{C}}\right) /\{ \pm I\}, H \) the diagonal matrices modulo \( \pm I \) , and \( h = \pm \operatorname{diag}\left\lbrack {\mathrm{i}, - \mathrm{i}}\right\rbrack \) . Then \( w = \pm \left\lbrack \begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}\right\rbrack \) commutes with \( h \) but is not in \( H \) . If \( h \in {H}^{\prime } \), then by \( \left( \star \right) \) we see that \( {\operatorname{Cent}}_{G}\left( h\right) \subset {\operatorname{Norm}}_{G}\left( H\right) \) . Hence if \( {\operatorname{Cent}}_{G}\left( h\right) \neq H \) , then there exists \( 1 \neq w \in {W}_{G} \) such that \( {wh}{w}^{-1} = h \) . This leads us to define the strongly regular elements in \( H \) as \[ {H}^{\prime \prime } = \left\{ {h \in H : {h}^{\alpha } \neq 1\text{ and }{wh}{w}^{-1} \neq h\text{ for all }\alpha \in \Phi \text{ and }1 \neq w \in {W}_{G}}\right\} . \] (11.19) Clearly \( {H}^{\prime \prime } \) is an open, dense subset of \( H \) . From the remarks preceeding (11.19) we conclude that \( \left( {\star \star }\right) \;{\operatorname{Cent}}_{G}\left( h\right) = H \) for every \( h \in {H}^{\prime \prime }. \) Lemma 11.4.15. Define a map \( \Psi : \left( {G/H}\right) \times H \rightarrow G \) by \( \Psi \left( {{gH}, h}\right) = {gh}{g}^{-1} \) . If \( g \in G \) and \( h \in {H}^{\prime \prime } \), then \[ {\Psi }^{-1}\left( {{gh}{g}^{-1}}\right) = \left\{ {\left( {{gwH},{w}^{-1}{hw}}\right) : w \in {W}_{G}}\right\} , \] and this set has cardinality \( \left| {W}_{G}\right| \) . Proof. Let \( h \in {H}^{\prime \prime } \) . Suppose \( {g}_{1} \in G \) and \( {h}_{1} \in H \) satisfy \( {g}_{1}{h}_{1}{g}_{1}^{-1} = {gh}{g}^{-1} \) . Set \( w = \) \( {g}^{-1}{g}_{1} \) . Then \( w{h}_{1} = {hw} \) . Given any \( {h}_{2} \in H \), we have \[ {hw}{h}_{2}{w}^{-1} = w{h}_{1}{h}_{2}{w}^{-1} = w{h}_{2}{h}_{1}{w}^{-1} = w{h}_{2}{w}^{-1}h. \] Hence \( w{h}_{2}{w}^{-1} \in {\operatorname{Cent}}_{G}\left( h\right) = H \) by \( \left( {\star \star }\right) \) . This shows that \( w \in {\operatorname{Norm}}_{G}\left( H\right) \) and \( \left( {{g}_{1},{h}_{1}}\right) = \left( {{wg},{w}^{-1}{hw}}\right) \) . Furthermore, \( {g}_{1}h{g}_{1}^{-1} = {gh}{g}^{-1} \) if and only if \( {g}^{-1}{g}_{1} \in H \) . Hence \( w \) is uniquely determined as an element of \( {W}_{G} \) . Define \( {G}^{\prime } = \left\{ {{gh}{g}^{-1} \in H : g \in G, h \in {H}^{\prime }}\right\} \) . We call the elements of \( {G}^{\prime } \) the regular semisimple elements in \( G \) . Fix a set \( {\Phi }^{ + } \) of positive roots, and let \( B = H{N}^{ + } \) be the corresponding Borel subgroup. Given \( \alpha \in \mathcal{X}\left( H\right) \), we extend \( \alpha \) to a character of \( B \) by setting \( {b}^{\alpha } = {h}^{\alpha } \) for \( b = {hn} \) with \( h \in H \) and \( n \in {N}^{ + } \) . Then the regular semisimple elements in \( B \) have the following explicit characterization: Lemma 11.4.16. An element \( b \) is in \( B \cap {G}^{\prime } \) if and only if \( {b}^{\alpha } \neq 1 \) for all \( \alpha \in \Phi \) . Thus \( B \cap {G}^{\prime } = {H}^{\prime }{N}^{ + } \) is open and Zariski dense in \( B \) . Proof. Wri

Theorem 11.4.11. The action of \( {W}_{G} \) on \( \mathfrak{h} \) coincides with the action of \( W\left( {\mathfrak{g},\mathfrak{h}}\right) \) . Furthermore, every coset in \( {W}_{G} \) has a representative from \( U \) .

For \( \alpha \in \Phi \) and \( {X}_{\alpha } \) as in (7.42), set \[ {u}_{\alpha } = \frac{1}{2}\left( {{X}_{\alpha } - {X}_{-\alpha }}\right) \;\text{ and }\;{v}_{\alpha } = \frac{1}{2\mathrm{i}}\left( {{X}_{\alpha } + {X}_{-\alpha }}\right) . \] Then \( {X}_{\alpha } = {u}_{\alpha } + \mathrm{i}{v}_{\alpha } \), and \( {u}_{\alpha },{v}_{\alpha } \in \mathfrak{u} \) . We calculate the action of ad \( {u}_{\alpha } \) on \( h \in \mathfrak{h} \) as follows: \[ \left\lbrack {{u}_{\alpha }, h}\right\rbrack = - \frac{1}{2}\langle \alpha, h\rangle \left( {{X}_{\alpha } + {X}_{-\alpha }}\right) = - \mathrm{i}\langle \alpha, h\rangle {v}_{\alpha }, \] \[ \left\lbrack {{u}_{\alpha },{v}_{\alpha }}\right\rbrack = \frac{1}{4\mathrm{i}}\left\lbrack {{X}_{\alpha } - {X}_{-\alpha },{X}_{\alpha } + {X}_{-\alpha }}\right\rbrack = \frac{1}{2\mathrm{i}}\left\lbrack {{X}_{\alpha },{X}_{-\alpha }}\right\rbrack = \frac{1}{2\mathrm{i}}{H}_{\alpha }. \] From (11.14) we see that for all \( s \in \mathbb{C} \) , \[ \exp \left( {s\operatorname{ad}{u}_{\alpha }}\right) h = h\;\text{ if }\langle \alpha, h\rangle = 0. \] Taking \( h = {H}_{\alpha } \) in (11.14), we obtain \( \operatorname{ad}\left( {u}_{\alpha }\right) {H}_{\alpha } = \mathrm{i}\parallel \alpha {\parallel }^{2}{v}_{\alpha } \), and hence by (11.15) we have \[ {\left( \operatorname{ad}{u}_{\alpha }\right) }^{2}{H}_{\alpha } = - {r}^{2}{H}_{\alpha } \] where \( r = \parallel \alpha \parallel /\sqrt{2} \) . Continuing in this way, we calculate that \[ \exp \left( {s\operatorname{ad}{u}_{\alpha }}\right) {H}_{\alpha } = \cos \left( {rs}\right) {H}_{\alpha } - \left( {2\mathrm{i}/r}\right) \sin \left( {rs}\right) {v}_{\alpha } \] for all \( s \in \mathbb{C} \) . In particular, when \( s = \pi /r \) the second term vanishes and we have \[ exp((π/r)aduα)) Hα=−Hα. \] Thus by (11.16) and (11.17), we see that the element gα=exp((π/r)uα))∈U acts on ℎ by the reflection sα. This proves that every element in the algebraic Weyl group W((g,ℎ)) can be implemented by the adjoint action of an element k∈NormG(H)∩U. To prove the converse, fix a set Φ+ of positive roots and let ρ=∑α∈Φ+ α/2 If g∈NormG(H), then Ad(g^t )Φ+⊂Φ is another set of positive roots, so by Theorem 3.1.9 there is an element s∈W((g,ℎ)) such that sAd(g^t )Φ+=Φ+. Hence by what has just been proved, there exists k∈NormG(H)∩U such that Ad|k|ℎ=s, so Ad(kg)^t Φ+=Φ+. Thus kg fixes ρ, which means that Ad(kg)) Hρ=Hρ The element Hρ satisfies <α, Hρ>≠0 for all α∈Φ by Lemma 3.1.21 Hence ℎ={ X∈g : [ Hρ , X]=0}. We claim that the one-parameter subgroup Γ={ exp ( t Hρ ): t∈C})⊂H is dense in H (in the Lie group topology). Indeed, the closure (in the Lie group topology) of Γ in G is a closed Lie subgroup of H that has Lie algebra ℎ by (

Example 1.83. (Type \( {\mathrm{D}}_{n} \) ) Let \( W \) be the subgroup of the signed permutation group consisting of elements that change an even number of signs (Example 1.13). Then \( \mathcal{H} \) consists of the hyperplanes \( {x}_{i} - {x}_{j} = 0 \) and \( {x}_{i} + {x}_{j} = 0\left( {i \neq j}\right) \) . To figure out what the chambers look like, consider two coordinates, say \( {x}_{1} \) and \( {x}_{2} \) . From the fact that \( {x}_{1} \) is comparable to both \( {x}_{2} \) and \( - {x}_{2} \) on any given chamber \( C \), one can deduce that one of the coordinates is bigger than the other in absolute value and that this coordinate has a constant sign. In other words, we have an inequality of the form \( \epsilon {x}_{1} > \left| {x}_{2}\right| \) or \( \epsilon {x}_{2} > \left| {x}_{1}\right| \) on \( C \), where \( \epsilon = \pm 1 \) . It follows that there are \( {2}^{n - 1}n \) ! chambers, each defined by inequalities of the form \[ {\epsilon }_{1}{x}_{\pi \left( 1\right) } > {\epsilon }_{2}{x}_{\pi \left( 2\right) } > \cdots > {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > \left| {x}_{\pi \left( n\right) }\right| \] (1.21) with \( {\epsilon }_{i} \in \{ \pm 1\} \) and \( \pi \in {S}_{n} \) . Note that the last inequality is equivalent to two linear inequalities, \( {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > {x}_{\pi \left( n\right) } \) and \( {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > - {x}_{\pi \left( n\right) } \), so we have \( n \) linear inequalities in all. As fundamental chamber we take \[ {x}_{1} > {x}_{2} > \cdots > {x}_{n - 1} > \left| {x}_{n}\right| \] with walls \( {x}_{1} = {x}_{2},{x}_{2} = {x}_{3},\ldots ,{x}_{n - 1} = {x}_{n} \), and \( {x}_{n - 1} = - {x}_{n} \) . Further analysis is left to the interested reader. Example 1.84. This final example is intended to provide some geometric intuition. Several statements will be made without proof, and the reader is advised not to worry too much about this. Let \( W \) be the reflection group of type \( {\mathrm{H}}_{3} \), i.e., the group of symmetries of a regular dodecahedron in \( V \mathrel{\text{:=}} {\mathbb{R}}^{3} \) . It is convenient to restrict the action of \( W \) to the unit sphere \( {S}^{2} \) and to think of \( W \) as a group of isometries of this sphere. As such, it is the group of symmetries of the regular tessellation of the sphere obtained by radially projecting the faces of the dodecahedron onto the sphere. Let \( P \) be one of the 12 spherical pentagons that occur in this tessellation. It has interior angles \( {2\pi }/3 \), since there are 3 pentagons at each vertex. The circles of symmetry of this tessellation (corresponding to the planes of symmetry of the dodecahedron) barycentrically subdivide \( P \), thereby cutting it into 10 spherical triangles. A typical such triangle \( T \) has angles \( \pi /2,\pi /3 \) , and \( \pi /5 \) . The angle \( \pi /5 = {2\pi }/{10} \) occurs at the center of \( P \) ; the angle \( \pi /3 \) , which is half of the interior angle \( {2\pi }/3 \) of \( P \), occurs at a vertex of \( P \) ; and the angle \( \pi /2 \) occurs at the midpoint of an edge of \( P \), where the line from the center of \( P \) perpendicularly bisects that edge. See Figure 1.11.* Finally, a typical chamber \( C \) in \( V \) is simply the cone over such a triangle \( T \) . There are \( {12} \cdot {10} = {120} \) such chambers, so \( \left| W\right| = {120} \) . Thus the dodecahedral group \( W \) is * Figure 1.11 first appeared in Klein-Fricke [145, p. 106] and is reprinted from a digital image provided by the Cornell University Library's Historic Monograph Collection.  Fig. 1.11. The dodecahedral tessellation, barycentrically subdivided. a group of order 120 generated by 3 reflections. The calculation of the angles of \( T \) above makes it easy to compute the orders of the pairwise products of the generating reflections. One has, for a suitable numbering \( {s}_{1},{s}_{2},{s}_{3} \) of these reflections, \[ {\left( {s}_{1}{s}_{2}\right) }^{3} = {\left( {s}_{2}{s}_{3}\right) }^{5} = {\left( {s}_{1}{s}_{3}\right) }^{2} = 1. \] ## Exercises 1.85. Recall from the discussion near the end of Section 1.5.1 that we can distinguish various types of adjacency. Spell out what that means in Examples 1.81,1.82, and 1.83. (For the \( {\mathrm{A}}_{n - 1} \) case, see Example 1.51.) 1.86. Find \( {w}_{0} \) and the induced involution of \( S \) in Examples 1.81,1.82, and 1.83, where \( {w}_{0} \) is the element of maximal length (Section 1.5.2). 1.87. In Example 1.84, \( W \) is a familiar group of order 120 . Which one is it? ## 1.5.4 The Chambers Are Simplicial Let \( \left( {W, V}\right) \) be a finite reflection group, and let the notation be as in Section 1.5.1. Thus we have a fundamental chamber \( C \) with walls \( {H}_{s}\left( {s \in S}\right) \) . Let \( {e}_{s} \) be the unit normal to \( {H}_{s} \) pointing to the side of \( {H}_{s} \) containing \( C \) . The Gram matrix of \( C \) is the matrix of inner products \( \left\langle {{e}_{s},{e}_{t}}\right\rangle \), whose rows and columns are indexed by \( S \) (see Section 1.4.5). The reader who has worked through the examples in Section 1.5.3 will not be surprised by the following computation of the Gram matrix: Theorem 1.88. With the notation above, we have \[ \left\langle {{e}_{s},{e}_{t}}\right\rangle = - \cos \frac{\pi }{m\left( {s, t}\right) } \] (1.22) for \( s, t \in S \), where \( m\left( {s, t}\right) \) is the order of \( {st} \) . In particular, \( \left\langle {{e}_{s},{e}_{t}}\right\rangle \leq 0 \) for \( s \neq t \) . Consequently, \( C \) is a simplicial cone if \( \left( {W, V}\right) \) is essential. The proof will make use of the following lemma: Lemma 1.89. Given \( s \neq t \) in \( S \), let \( {W}^{\prime } \) be the group generated by \( s \) and \( t \) . Then \( {W}^{\prime } \) is a rank-2 reflection group, and \( C \) is contained in a \( {W}^{\prime } \) -chamber \( {C}^{\prime } \) having \( {H}_{s} \) and \( {H}_{t} \) as its walls. Proof. We have \( {V}^{{W}^{\prime }} = {H}_{s} \cap {H}_{t} = {\left( \mathbb{R}{e}_{s} \oplus \mathbb{R}{e}_{t}\right) }^{ \bot } \), so \( \left( {{W}^{\prime }, V}\right) \) has rank 2 . Let \( {\mathcal{H}}^{\prime } \subseteq \mathcal{H} \) be the set of hyperplanes of the form \( {w}^{\prime }{H}_{s} \) or \( {w}^{\prime }{H}_{t} \) with \( {w}^{\prime } \in {W}^{\prime } \) . Then \( {\mathcal{H}}^{\prime } \) is \( {W}^{\prime } \) -invariant, and the reflections with respect to the elements of \( {\mathcal{H}}^{\prime } \) are in \( {W}^{\prime } \) and generate it. Hence \( {\mathcal{H}}^{\prime } \) is the set of \( {W}^{\prime } \) -walls, i.e., the set of hyperplanes that define the \( {W}^{\prime } \) -cells. Since \( C \) is convex and is disjoint from all the elements of \( {\mathcal{H}}^{\prime } \), it is contained in a \( {W}^{\prime } \) -chamber \( {C}^{\prime } \) . The rank calculation shows that \( {C}^{\prime } \) has two walls. To see that these are \( {H}_{s} \) and \( {H}_{t} \) , note that \( \overline{{C}^{\prime }} \cap {H}_{s} \supseteq \bar{C} \cap {H}_{s} \), which has nonempty interior in \( {H}_{s} \), and similarly for \( {H}_{t} \) . So \( {H}_{s} \) and \( {H}_{t} \) are walls of \( {C}^{\prime } \) by Proposition 1.33. Proof of the theorem. The last assertion follows from Proposition 1.37, so we need only prove the first assertion. We may assume \( s \neq t \) . Let \( {W}^{\prime } \) and \( {C}^{\prime } \) be as in the lemma. Then \( \left\langle {{e}_{s}, - }\right\rangle \) and \( \left\langle {{e}_{t}, - }\right\rangle \) are positive on \( C \subseteq {C}^{\prime } \), so \( {e}_{s} \) and \( {e}_{t} \) are the canonical unit normals to the walls of \( {C}^{\prime } \) . The inner product formula (1.22) now follows from Example 1.80. One immediate consequence of the theorem is the following criterion for reducibility: Corollary 1.90. Assume that \( \left( {W, V}\right) \) is essential. Then \( \left( {W, V}\right) \) is reducible if and only if there is a partition of \( S \) into (nonempty) subsets \( {S}^{\prime },{S}^{\prime \prime } \) such that \( m\left( {s, t}\right) = 2 \) for all \( s \in {S}^{\prime } \) and \( t \in {S}^{\prime \prime } \) . Proof. Suppose there is such a partition. Let \( {W}^{\prime } \) and \( {W}^{\prime \prime } \) be the subgroups \( \left\langle {S}^{\prime }\right\rangle \) and \( \left\langle {S}^{\prime \prime }\right\rangle \), and let \( {V}^{\prime } \) (respectively \( {V}^{\prime \prime } \) ) be the subspace of \( V \) spanned by the \( {e}_{s} \) with \( s \in {S}^{\prime } \) (respectively \( s \in {S}^{\prime \prime } \) ). Then we have an orthogonal decomposition \( V = {V}^{\prime } \oplus {V}^{\prime \prime } \), and \( W \) can be identified with \( {W}^{\prime } \times {W}^{\prime \prime } \) acting on this direct sum. Thus \( \left( {W, V}\right) \) is reducible. The converse is equally easy and is left to the reader. The next corollary is more interesting. Recall from Section 1.5.2 that \( W \) has a unique longest element \( {w}_{0} \) and that \( {w}_{0} \) normalizes \( S \) . Corollary 1.91. Assume that \( \left( {W, V}\right) \) is essential and irreducible. (1) \( {w}_{0} \) is the only nontrivial element of \( W \) that normalizes \( S \) . (2) The center of \( W \) is trivial unless \( W \) contains -1, where -1 denotes \( - {\mathrm{{id}}}_{V} \) . In this case \( {w}_{0} = - 1 \) and the center is \( \{ \pm 1\} = \left\{ {1,{w}_{0}}\right\} \) . Proof. (1) We will give two proofs, one algebraic and one geometric. Both are instructive. Algebraic proof. Suppose \( w \in W \) normalizes \( S \), and set \( {s}^{\prime } \mathrel{\text{:=}} {ws}{w}^{-1} \) for any \( s \in S \) . Since \( w{e}_{s} \) is a u

Example 1.83. (Type \( {\mathrm{D}}_{n} \)) Let \( W \) be the subgroup of the signed permutation group consisting of elements that change an even number of signs (Example 1.13). Then \( \mathcal{H} \) consists of the hyperplanes \( {x}_{i} - {x}_{j} = 0 \) and \( {x}_{i} + {x}_{j} = 0\left( {i \neq j}\right) \). To figure out what the chambers look like, consider two coordinates, say \( {x}_{1} \) and \( {x}_{2} \). From the fact that \( {x}_{1} \) is comparable to both \( {x}_{2} \) and \( - {x}_{2} \) on any given chamber \( C \), one can deduce that one of the coordinates is bigger than the other in absolute value and that this coordinate has a constant sign. In other words, we have an inequality of the form \( \epsilon {x}_{1} > \left| {x}_{2}\right| \) or \( \epsilon {x}_{2} > \left| {x}_{1}\right| \) on \( C \), where \( \epsilon = \pm 1 \). It follows that there are \( {2}^{n - 1}n \) ! chambers, each defined by inequalities of the form \[ {\epsilon }_{1}{x}_{\pi \left( 1\right) } > {\epsilon }_{2}{x}_{\pi \left( 2\right) } > \cdots > {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > \left| {x}_{\pi \left( n\right) }\right| \] with \( {\epsilon }_{i} \in \{ \pm 1\} \) and \( \pi \in {S}_{n} \). Note that the last inequality is equivalent to two linear inequalities, \( {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > {x}_{\pi \left( n\right) } \) and \( {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > - {x}_{\pi \left( n\right) } \), so we have \( n \) linear inequalities in all. As fundamental chamber we take \[ {x}_{1} > {x}_{2} > \cdots > {x}_{n - 1} > \left| {x}_{n}\right| \] with walls \( {x}_{1} = {x}_{2},{x}_{2} = {x}_{3},\ldots ,{x}_{n - 1} = {x}_{n} \), and \( {x}_{n - 1} = - {x}_{n} \). Further analysis is left to the interested reader.

The proof process for Example 1.83 is not provided in the text block. Therefore, the proof process section is "null".

Theorem 12.5.1 An internal set \( A \) is hyperfinite with internal cardinality \( N \) if and only if there is an internal bijection \( f : \{ 1,\ldots, N\} \rightarrow A \) . Proof. Let \( A = \left\lbrack {A}_{n}\right\rbrack \) . If \( A \) is hyperfinite with internal cardinality \( N = \) \( \left\lbrack {N}_{n}\right\rbrack \), then we may suppose that for each \( n \in \mathbb{N},{A}_{n} \) is a finite set of cardinality \( {N}_{n} \) . Thus there is a bijection \( {f}_{n} : \left\{ {1,\ldots ,{N}_{n}}\right\} \rightarrow {A}_{n} \) . Let \( f = \) \( \left\lbrack {f}_{n}\right\rbrack \) . Then \( f \) is an internal function with domain \( \{ 1,\ldots, N\} \) that is injective \( \left( {{12.2}\left( 4\right) }\right) \) and has range \( A\left( {{12.2}\left( 1\right) }\right) \) . Conversely, suppose that \( f = \left\lbrack {f}_{n}\right\rbrack \) is an internal bijection from \( \{ 1,\ldots, N\} \) onto \( A \) . Then \[ \left\lbrack {\operatorname{dom}{f}_{n}}\right\rbrack = \operatorname{dom}\left\lbrack {f}_{n}\right\rbrack = \{ 1,\ldots, N\} = \left\lbrack \left\{ {1,\ldots ,{N}_{n}}\right\} \right\rbrack , \] so for \( \mathcal{F} \) -almost all \( n \) , \[ \operatorname{dom}{f}_{n} = \left\{ {1,\ldots ,{N}_{n}}\right\} \] (i) Also, as \( A \) is the image of \( \{ 1,\ldots, N\} \) under \( \left\lbrack {f}_{n}\right\rbrack \), Exercise 12.2(1) implies that \( A = \left\lbrack {{f}_{n}\left( \left\{ {1,\ldots ,{N}_{n}}\right\} \right) }\right\rbrack \), so \[ {f}_{n}\left( \left\{ {1,\ldots ,{N}_{n}}\right\} \right) = {A}_{n} \] (ii) for \( \mathcal{F} \) -almost all \( n \) . Finally, by \( {12.2}\left( 4\right) \) , \[ {f}_{n}\text{is injective} \] (iii) for \( \mathcal{F} \) -almost all \( n \) . Then the set \( J \) of those \( n \in \mathbb{N} \) satisfying (i)-(iii) must belong to \( \mathcal{F} \) . But for \( n \in J,{A}_{n} \) is finite of cardinality \( {N}_{n} \) . Hence \( A \) is hyperfinite of internal cardinality \( N \) . An important feature of this result is that it gives a characterisation of hyperfinite sets that makes no reference to the ultrafilter \( \mathcal{F} \), but requires only the hypernatural numbers \( {}^{ * }\mathbb{N} \) and the notion of an internal function. This approach will be revisited in Section 13.17. ## 12.6 Hyperfinite Pigeonhole Principle One classical way to distinguish the finite from the infinite is to characterise the infinite sets as those that are equinumerous with a proper subset of themselves. Thus \( A \) is infinite iff there is an injection \( f : A \rightarrow A \) whose range \( f\left( A\right) \) is a proper subset of \( A \) . Equivalently, \( A \) is finite iff every injection \( f : A \rightarrow A \) mapping \( A \) into itself is surjective, i.e., has \( f\left( A\right) = A \) (this latter statement is known as the pigeonhole principle). Correspondingly, we have the following characterisation of hyperfiniteness. Theorem 12.6.1 An internal set \( A = \left\lbrack {A}_{n}\right\rbrack \) is hyperfinite if and only if every injective internal function \( f \) whose domain includes \( A \) and has \( f\left( A\right) \subseteq A \) must in fact have \( f\left( A\right) = A \) . Proof. Suppose \( A \) is hyperfinite. Let \( f = \left\lbrack {f}_{n}\right\rbrack \) be an internal injective function with \( A \subseteq \operatorname{dom}f \) and \( f\left( A\right) \subseteq A \) . Then each of the following is true for \( \mathcal{F} \) -almost all \( n \in \mathbb{N} \) : \[ {A}_{n}\text{is finite,} \] \[ {A}_{n} \subseteq \operatorname{dom}{f}_{n} \] \[ {f}_{n}\left( {A}_{n}\right) \subseteq {A}_{n},\;\text{ cf. }{12.2}\left( 1\right) ,{11.2}\left( 2\right) \] \( {f}_{n} \) is injective. Thus the set \( J \) of those \( n \in \mathbb{N} \) satisfying all of these conditions must belong to \( \mathcal{F} \) . But \[ J \subseteq \left\{ {n \in \mathbb{N} : {f}_{n}\left( {A}_{n}\right) = {A}_{n}}\right\} \] by the standard pigeonhole principle, and so \( f\left( A\right) = \left\lbrack {{f}_{n}\left( {A}_{n}\right) }\right\rbrack = \left\lbrack {A}_{n}\right\rbrack = A \) . For the converse, suppose \( A \) is not hyperfinite. It follows that \[ {J}^{\prime } = \left\{ {n \in \mathbb{N} : {A}_{n}\text{ is infinite }}\right\} \in \mathcal{F}. \] But for each \( n \in {J}^{\prime } \) there is an injective function \( {f}_{n} : {A}_{n} \rightarrow {A}_{n} \) and some \( {r}_{n} \in {A}_{n} - {f}_{n}\left( {A}_{n}\right) \) . Let \( f = \left\lbrack {f}_{n}\right\rbrack \) . This makes \( f \) an internal function with domain \( A \) that is injective \( \left( {{12.2}\left( 4\right) }\right) \) and has \( f\left( A\right) = \left\lbrack {{f}_{n}\left( {A}_{n}\right) }\right\rbrack \subseteq A \), while \( \left\lbrack {r}_{n}\right\rbrack \in A - f\left( A\right) \) and so \( f\left( A\right) \neq A \) . Here now is an example of a noninternal function: \[ f\left( n\right) = \left\{ \begin{array}{ll} {2n} & \text{ if }n \in \mathbb{N}, \\ n & \text{ if }n \notin \mathbb{N} \end{array}\right. \] This function maps the internal set * \( N \) injectively into, but not onto, itself. Hence by the hyperfinite pigeonhole principle (Theorem 12.6.1), \( f \) cannot be internal. ## 12.7 Integrals as Hyperfinite Sums The operation of forming the sum of finitely many numbers can be extended to hyperfinitely many. More generally, we can define the sum over a hyperfinite set of the values of an internal function. To see this, if \( X \) is a finite set, let the symbol \[ \mathop{\sum }\limits_{{x \in X}}g\left( x\right) \] denote the sum of the members of \( \{ g\left( x\right) : x \in X\} \) . Then if \( A = \left\lbrack {A}_{n}\right\rbrack \) is a hyperfinite set included in the domain of an internal function \( f = \left\lbrack {f}_{n}\right\rbrack \) , define \( \mathop{\sum }\limits_{{x \in A}}f\left( x\right) \) to be the hyperreal number \( \left\lbrack {r}_{n}\right\rbrack \) given by \[ {r}_{n} = \mathop{\sum }\limits_{{x \in {A}_{n}}}{f}_{n}\left( x\right) \] This makes sense because for \( \mathcal{F} \) -almost all \( n \) we have \( {A}_{n} \) a finite subset of \( \operatorname{dom}{f}_{n} \) . Thus \[ \mathop{\sum }\limits_{{x \in \left\lbrack {A}_{n}\right\rbrack }}\left\lbrack {f}_{n}\right\rbrack \left( x\right) = \left\lbrack {\;\mathop{\sum }\limits_{{x \in {A}_{n}}}{f}_{n}\left( x\right) }\right\rbrack . \] This operation has many of the properties familiar from finite summations: if \( f, g \) are internal functions, \( A, B \) are hyperfinite sets, and \( c \in * \mathbb{R} \), then: \[ \text{-}\mathop{\sum }\limits_{{x \in A}}{cf}\left( x\right) = c\left( {\mathop{\sum }\limits_{{x \in A}}f\left( x\right) }\right) \text{.} \] \[ \text{-}\mathop{\sum }\limits_{{x \in A}}f\left( x\right) + g\left( x\right) = \mathop{\sum }\limits_{{x \in A}}f\left( x\right) + \mathop{\sum }\limits_{{x \in A}}g\left( x\right) \text{.} \] - \( \mathop{\sum }\limits_{{x \in A \cup B}}f\left( x\right) = \mathop{\sum }\limits_{{x \in A}}f\left( x\right) + \mathop{\sum }\limits_{{x \in B}}f\left( x\right) \; \) if \( A \) and \( B \) are disjoint. \[ \text{-}\mathop{\sum }\limits_{{x \in A}}f\left( x\right) \leq \mathop{\sum }\limits_{{x \in A}}g\left( x\right) \;\text{if}f\left( x\right) \leq g\left( x\right) \text{on}A\text{.} \] These are analogues of familiar properties of integrals (cf. Section 9.3). We will now see that standard integrals can be realised as hyperfinite Riemann sums over hyperfinite partitions. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be an integrable function on the closed interval \( \left\lbrack {a, b}\right\rbrack \subseteq \) \( \mathbb{R} \) . Take a positive infinitesimal \( {\Delta x} = \left\lbrack {\varepsilon }_{n}\right\rbrack \) . Then for each \( n \in \mathbb{N} \) we may assume that \( {\varepsilon }_{n} \) is a positive real number less than \( b - a \) . Let \( {P}_{n} \cup \{ b\} \) be the finite partition of \( \left\lbrack {a, b}\right\rbrack \) into subintervals of width \( {\varepsilon }_{n} \) . Thus if \( {P}_{n} \) is of size \( {N}_{n} \), we have the description \[ {P}_{n} = \left\{ {a + k{\varepsilon }_{n} : k \in \mathbb{Z}\text{ and }0 \leq k < {N}_{n}}\right\} . \] Let \( P \) be the hyperfinite set \( \left\lbrack {P}_{n}\right\rbrack \), of internal size \( N = \left\lbrack {N}_{n}\right\rbrack \in {}^{ * }\mathbb{N} \) . Then in fact, \[ P = \{ a + {k\Delta x} : k \in * \mathbb{Z}\text{ and }0 \leq k < N\} , \] so \( P \cup \{ b\} \) is a hyperfinite partition of \( \left\lbrack {a, b}\right\rbrack \) into subintervals of infinitesimal width \( {\Delta x} \) . Now, the original function \( f \) lifts to the internal function \( \left\lbrack {f}_{n}\right\rbrack : {}^{ * }\left\lbrack {a, b}\right\rbrack \rightarrow \) * \( \mathbb{R} \) determined by the constant sequence of functions \( {f}_{n} = f \) . We continue to use the symbol " \( f \) " for this extended function. Its domain includes \( P \) , so the hyperfinite sum \( \mathop{\sum }\limits_{{x \in P}}f\left( x\right) \) is specified as the hyperreal number \[ \left\lbrack \left\langle {\mathop{\sum }\limits_{{x \in {P}_{1}}}f\left( x\right) ,\mathop{\sum }\limits_{{x \in {P}_{2}}}f\left( x\right) ,\ldots }\right\rangle \right\rbrack . \] Thus \[ \mathop{\sum }\limits_{{x \in P}}f\left( x\right) = \left\lbrack {\mathop{\sum }\limits_{{x \in {P}_{n}}}f\left( x\right) }\right\rbrack \] The ordinary Riemann sum for the real partition \( {P}_{n} \cup \{ b\} \) was defined in Section 9.1 as the number \[ \left. {{S}_{a}^{b}(f,{\varepsilon }_{n}}\right) = \mathop{\sum }\limits_{{x \in {P}_{n}}}f\left( x\right) {\varepsilon }_{n} = \left( {\mathop{\sum }\limits_{{x \in {P}_{n}}}f\left( x\right) }\right) {\varepsilon }_{n}. \] But the sequence of numbers \( \left\langle {{S}_{a}^{b}\left( {f,{\varepsilon }_{n}}\right) : n \in \mathbb{N}}\right\rangle \)

Theorem 12.5.1 An internal set \( A \) is hyperfinite with internal cardinality \( N \) if and only if there is an internal bijection \( f : \{ 1,\ldots, N\} \rightarrow A \).

Let \( A = \left\lbrack {A}_{n}\right\rbrack \). If \( A \) is hyperfinite with internal cardinality \( N = \left\lbrack {N}_{n}\right\rbrack \), then we may suppose that for each \( n \in \mathbb{N}, {A}_{n} \) is a finite set of cardinality \( {N}_{n} \). Thus there is a bijection \( {f}_{n} : \left\{ {1,\ldots ,{N}_{n}}\right\} \rightarrow {A}_{n} \). Let \( f = \left\lbrack {f}_{n}\right\rbrack \). Then \( f \) is an internal function with domain \( \{ 1,\ldots, N\} \) that is injective (by 12.2(4)) and has range \( A \) (by 12.2(1)). Conversely, suppose that \( f = \left\lbrack {f}_{n}\right\rbrack \) is an internal bijection from \( \{ 1,\ldots, N\} \) onto \( A \). Then \[ \left\lbrack {\operatorname{dom}{f}_{n}}\right\rbrack = \operatorname{dom}\left\lbrack {f}_{n}\right\rbrack = \{ 1,\ldots, N\} = \left\lbrack \left\{ {1,\ldots ,{N}_{n}}\right\} \right\rbrack , \] so for \( \mathcal{F} \)-almost all \( n \), \[ \operatorname{dom}{f}_{n} = \left\{ {1,\ldots ,{N}_{n}}\right\} \] (i) Also, as \( A \) is the image of \( \{ 1,\ldots, N\} \) under \( \left\lbrack {f}_{n}\right\rbrack \), Exercise 12.2(1) implies that \( A = \left\lbrack {{f}_{n}\left( \left\{ {1,\ldots ,{N}_{n}}\right\} \right) }\right\rbrack \), so \[ {f}_{n}\left( \left\{ {1,\ldots ,{N}_{n}}\right\} \right) = {A}_{n} \] (ii) for \( \mathcal{F} \)-almost all \( n \). Finally, by 12.2(4), \[ {f}_{n}\text{ is injective} \] (iii) for \( \mathcal{F} \)-almost all \( n \). Then the set \( J \) of those \( n \in \mathbb{N} \) satisfying (i)-(iii) must belong to \( \mathcal{F} \). But for \( n \in J, {A}_{n} \) is finite of cardinality \( {N}_{n} \). Hence \( A \) is hyperfinite of internal cardinality \( N \).

Proposition 23.41 Let \( {\delta }_{P} \) be a fixed square root of \( {\mathcal{K}}_{P} \) . For any vector field \( X \) lying in \( P \), there is a unique linear operator \( {\nabla }_{X} \) mapping sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P} \), such that \[ {\nabla }_{X}\left( {f{s}_{1}}\right) = X\left( f\right) {s}_{1} + f{\nabla }_{X}{s}_{1} \] (23.27) \[ {\nabla }_{X}\left( {{s}_{1} \otimes {s}_{2}}\right) = \left( {{\nabla }_{X}{s}_{1}}\right) \otimes {s}_{2} + {s}_{1} \otimes \left( {{\nabla }_{X}{s}_{2}}\right) \] (23.28) for all smooth functions \( f \) and all sections \( {s}_{1} \) and \( {s}_{2} \) of \( {\delta }_{P} \) . On the left-hand side of (23.28), \( {\nabla }_{X} \) is the partial connection on \( {\mathcal{K}}_{P} \) given by (23.26). If \( X \) is a vector field on \( N \) that preserves \( P \), then there is a unique linear operator \( {\mathcal{L}}_{X} \), mapping sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P} \) such that \[ {\mathcal{L}}_{X}\left( {f{s}_{1}}\right) = X\left( f\right) {s}_{1} + f{\mathcal{L}}_{X}{s}_{1} \] \[ {\mathcal{L}}_{X}\left( {{s}_{1} \otimes {s}_{2}}\right) = \left( {{\mathcal{L}}_{X}{s}_{1}}\right) \otimes {s}_{2} + {s}_{1} \otimes \left( {{\mathcal{L}}_{X}{s}_{2}}\right) \] for all smooth functions \( f \) and all sections \( {s}_{1} \) and \( {s}_{2} \) of \( {\delta }_{P} \) . Both of these constructions extend naturally from sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P}^{\mathbb{C}} \) . We may then say that a section \( s \) of \( {\delta }_{P}^{\mathbb{C}} \) is polarized if \( {\nabla }_{X}s = 0 \) for every smooth vector field \( X \) lying in \( P \) . Proof. If \( V \) is a one-dimensional vector space, then the map \( \otimes : V \times V \rightarrow \) \( V \otimes V \) is commutative: \( u \otimes v = v \otimes u \) for all \( u, v \in V \) . Furthermore, if \( {u}_{0} \) is a nonzero element of \( V \), then the map \( u \mapsto u \otimes {u}_{0} \) is an invertible linear map of \( V \) to \( V \otimes V \) . Suppose \( {s}_{0} \) is a local nonvanishing section of \( {\delta }_{P} \) . Applying (23.28) with \( {s}_{1} = {s}_{2} = {s}_{0} \), we want \[ 2\left( {{\nabla }_{X}{s}_{0}}\right) \otimes {s}_{0} = {\nabla }_{X}\left( {{s}_{0} \otimes {s}_{0}}\right) . \] (23.29) Since the operation of tensoring with \( {s}_{0} \) is invertible, there is a unique section " \( {\nabla }_{X}{s}_{0} \) " of \( {\delta }_{P} \) for which (23.29) holds. Locally, any section \( s \) of \( {\delta }_{P} \) can be written as \( s = g{s}_{0} \) for a unique function \( g \) . We then define \( {\nabla }_{X}s \) by \[ {\nabla }_{X}s = X\left( g\right) {s}_{0} + g{\nabla }_{X}{s}_{0} \] (23.30) in which case,(23.27) is easily seen to hold. If \( {s}_{1} = {g}_{1}{s}_{0} \) and \( {s}_{2} = {g}_{2}{s}_{0} \) , then using (23.29) and the symmetry of the tensor product, it is easy to verify that (23.28) holds, with both sides of the equation equal to \[ X\left( {{g}_{1}{g}_{2}}\right) {\nabla }_{X}\left( {{s}_{0} \otimes {s}_{0}}\right) . \] Uniqueness of \( {\nabla }_{X} \) holds because both (23.29) and (23.30) are required by the definition of \( {\nabla }_{X} \) . The action of \( {\nabla }_{X} \) extends to sections of \( {\delta }_{P}^{\mathbb{C}} \), by writing such sections as complex-valued functions times \( {s}_{0} \) . The analysis of the Lie derivative is similar and is omitted. ## 23.6.4 The Half-Form Hilbert Space We continue to assume that the leaf space \( \Xi \) of \( P \) is an orientable manifold, and that we have chosen an orientation on \( \Xi \) . We assume that we have chosen a square root \( {\delta }_{P} \) of \( {\mathcal{K}}_{P} \), as in Sect. 23.6.3. If \( L \) is a prequantum line bundle over \( N \), we now form the tensor product bundle \( L \otimes {\delta }_{P}^{\mathbb{C}} \) . Given two sections \( {s}_{1} \) and \( {s}_{2} \) of \( L \otimes {\delta }_{P}^{\mathbb{C}} \), we decompose them locally as \( {s}_{j} = {\mu }_{j} \otimes {\nu }_{j} \) , where \( {\mu }_{j} \) is a section of \( L \) and \( {\nu }_{j} \) is a section of \( {\delta }_{P}^{\mathbb{C}} \), and where, say, the \( {\mu }_{j} \) ’s are taken to be nonvanishing. Then we can combine these sections to form the quantity \[ \left( {{s}_{1},{s}_{2}}\right) \mathrel{\text{:=}} \left( {{\mu }_{1},{\mu }_{2}}\right) \overline{{\nu }_{1}} \otimes {\nu }_{2} \] (23.31) where \( \left( {{\mu }_{1},{\mu }_{2}}\right) \) is the pointwise inner product given by the Hermitian structure on \( L \) . Since \( \left( {{\mu }_{1},{\mu }_{2}}\right) \) is a scalar-valued function and \( \overline{{\nu }_{1}} \otimes {\nu }_{2} \) is a section of \( {\mathcal{K}}_{P}^{\mathbb{C}} \), the quantity \( \left( {{s}_{1},{s}_{2}}\right) \) is a section of \( {\mathcal{K}}_{P}^{\mathbb{C}} \) . Any other decomposition of \( {s}_{j} \) as the tensor product of a nonvanishing section of a \( L \) and a section of \( {\delta }_{P} \) is of the form \( \left( {f{\mu }_{j}}\right) \otimes \left( {{\nu }_{j}/f}\right) \) for some nonvanishing function \( f \), and the value of \( \left( {{s}_{1},{s}_{2}}\right) \) is the same as for the original decomposition. Since it is independent of the choice of local decomposition, \( \left( {{s}_{1},{s}_{2}}\right) \) is actually defined globally. Given the connection on \( L \) and the partial connection (23.41) on \( {\delta }_{P}^{\mathbb{C}} \), we can form a partial connection on \( L \otimes {\delta }_{P}^{\mathbb{C}} \) with the following property. For any vector field \( X \) lying in \( P \), and any section \( s \) of \( L \otimes {\delta }_{P}^{\mathbb{C}} \), if we decompose \( s \) locally as \( s = \mu \otimes \nu \), where \( \mu \) is a nonvanishing section of \( L \) and \( \nu \) is a section of \( {\delta }_{P} \), then \[ {\nabla }_{X}\left( s\right) = \left( {{\nabla }_{X}\mu }\right) \otimes \nu + \mu \otimes \left( {{\nabla }_{X}\nu }\right) \] (23.32) The reader may verify that if \( \mu \otimes \nu \) is replaced by \( \left( {f\mu }\right) \otimes \left( {\nu /f}\right) \) for some nonvanishing function \( f \), the value of \( {\nabla }_{X}\left( s\right) \) is unchanged. Thus, as with the quantity \( \left( {{s}_{1},{s}_{2}}\right) \) in (23.31), \( {\nabla }_{X}\left( s\right) \) is defined globally. We then define a section \( s \) of \( L \otimes {\delta }_{P}^{\mathbb{C}} \) to be polarized if \( {\nabla }_{X}s = 0 \) for each vector field \( X \) lying in \( P \) . If \( {s}_{1} \) and \( {s}_{2} \) are polarized sections of \( L \otimes {\delta }_{P}^{\mathrm{C}} \), then the section \( \left( {{s}_{1},{s}_{2}}\right) \) in (23.31) is easily seen to be a polarized section of \( {\mathcal{K}}_{P} \) . As in the case without half-forms there is an obstruction to the existence of globally defined polarized sections of \( L \otimes {\delta }_{P}^{\mathbb{C}} \) . We say that a leaf \( R \) is Bohr-Sommerfeld (in the half-form sense, with respect to a particular choice of \( {\delta }_{P} \) ) if there exists a nonzero section \( s \) of \( L \otimes {\delta }_{P}^{\mathbb{C}} \) defined over \( R \) such that \( {\nabla }_{X}s = 0 \) for each tangent vector to \( R \) . As in the case without half-forms, if the leaves are topologically nontrivial, the Bohr-Sommerfeld leaves will in general be a discrete set in the space of all leaves. The Bohr-Sommerfeld leaves in the half-form sense need not be the same as the Bohr-Sommerfeld leaves in the sense of Definition 23.27. In the setting of Example 23.29, for instance, the canonical bundle \( {\mathcal{K}}_{P} \) is trivial, but the square-root bundle \( {\delta }_{P} \) may be chosen to be nontrivial, by putting in a twist by 180 degrees over each copy of \( {S}^{1} \) . (That is to say, we think of \( {S}^{1} \) as the interval \( \left\lbrack {0,{2\pi }}\right\rbrack \) with the ends identified, and we attach a copy of \( \mathbb{R} \) to each point. But when identifying the fiber at \( {2\pi } \) with the fiber at 0 , we use the negative of the identity map.) As Exercise 9 shows, in this example, the Bohr-Sommerfeld leaves are the sets of the form \( \{ x\} \times {S}^{1} \) , where \( x/\hslash = n + 1/2 \) for some integer \( n \) . Definition 23.42 For any purely real polarization \( P \) and any square root \( {\delta }_{P} \) of \( {\mathcal{K}}_{P} \), the half-form space is the space of smooth, polarized sections of \( L \otimes {\delta }_{P}^{\mathbb{C}} \) . For a polarized section \( s \) of \( L \otimes {\delta }_{P}^{\mathbb{C}} \), define the norm of \( s \) by \[ \parallel s{\parallel }^{2} = {\int }_{\Xi }\widetilde{\left( s, s\right) } \] (23.33) where \( \left( {s, s}\right) \) is as in (23.31) and where \( \left( \widetilde{s, s}\right) \) is the \( n \) -form on \( \Xi \) given by Proposition 23.37. If \( {s}_{1} \) and \( {s}_{2} \) are elements of the half-form space with \( \begin{Vmatrix}{s}_{1}\end{Vmatrix} < \infty \) and \( \begin{Vmatrix}{s}_{2}\end{Vmatrix} < \infty \), define the inner product of \( {s}_{1} \) and \( {s}_{2} \) by \[ \left\langle {{s}_{1},{s}_{2}}\right\rangle = {\int }_{\Xi }\left( \overset{⏜}{{s}_{1},{s}_{2}}\right) \] The half-form Hilbert space is the completion with respect to the norm (23.33) of the space of polarized sections \( s \) for which \( \parallel s{\parallel }^{2} < \infty \) . The integral of \( n \) -forms on \( \Xi \) is taken with respect to the chosen orientation on \( \Xi \) . We can always decompose \( s \) locally as \( s = \mu \otimes \nu \) with \( \nu \) being a section of \( {\delta }_{P} \) (as opposed to \( {\delta }_{P}^{\mathbb{C}} \) ) and \( \mu \) being a section of \( L \) . Then \[ \left( {s, s}\right) = \left( {\mu ,\mu }\right) \nu \otimes \nu \] from which we see that \( \left( {s, s}\right) \) is a non-negative section of \( {\mathcal{K}}_{P} \) (Definition 23.40). (Recall that we ha

Proposition 23.41 Let \( {\delta }_{P} \) be a fixed square root of \( {\mathcal{K}}_{P} \). For any vector field \( X \) lying in \( P \), there is a unique linear operator \( {\nabla }_{X} \) mapping sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P} \), such that \[ {\nabla }_{X}\left( {f{s}_{1}}\right) = X\left( f\right) {s}_{1} + f{\nabla }_{X}{s}_{1} \] \[ {\nabla }_{X}\left( {{s}_{1} \otimes {s}_{2}}\right) = \left( {{\nabla }_{X}{s}_{1}}\right) \otimes {s}_{2} + {s}_{1} \otimes \left( {{\nabla }_{X}{s}_{2}}\right) \] for all smooth functions \( f \) and all sections \( {s}_{1} \) and \( {s}_{2} \) of \( {\delta }_{P} \). On the left-hand side of (23.28), \( {\nabla }_{X} \) is the partial connection on \( {\mathcal{K}}_{P} \) given by (23.26). If \( X \) is a vector field on \( N \) that preserves \( P \), then there is a unique linear operator \( {\mathcal{L}}_{X} \), mapping sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P} \) such that \[ {\mathcal{L}}_{X}\left( {f{s}_{1}}\right) = X\left( f\right) {s}_{1} + f{\mathcal{L}}_{X}{s}_{1} \] \[ {\mathcal{L}}_{X}\left( {{s}_{1} \otimes {s}_{2}}\right) = \left( {{\mathcal{L}}_{X}{s}_{1}}\right) \otimes {s}_{2} + {s}_{1} \otimes \left( {{\mathcal{L}}_{X}{s}_{2}}\right) \] for all smooth functions \( f \) and all sections \( {s}_{1} \) and \( {s}_{2} \) of \( {\delta }_{P} \). Both of these constructions extend naturally from sections of \( {\delta }_{P} \) to sections of \( {\delta }_{P}^{\mathbb{C}} \). We may then say that a section \( s \) of \( {\delta }_{P}^{\mathbb{C}} \) is polarized if \( {\nabla }_{X}s = 0 \) for every smooth vector field \( X \) lying in \( P \).

Proof. If \( V \) is a one-dimensional vector space, then the map \( \otimes : V \times V \rightarrow V \otimes V \) is commutative: \( u \otimes v = v \otimes u \) for all \( u, v \in V \). Furthermore, if \( {u}_{0} \) is a nonzero element of \( V \), then the map \( u \mapsto u \otimes {u}_{0} \) is an invertible linear map of \( V \) to \( V \otimes V \). Suppose \( {s}_{0} \) is a local nonvanishing section of \( {\delta }_{P} \). Applying (23.28) with \( {s}_{1} = {s}_{2} = {s}_{0} \), we want \[ 2\left( {{\nabla }_{X}{s}_{0}}\right) \otimes {s}_{0} = {\nabla }_{X}\left( {{s}_{0} \otimes {s}_{0}}\right) . \] Since the operation of tensoring with \( {s}_{0

Theorem 7. Let \( M \) and \( {M}^{\prime } \) be totally disconnected compact sets in \( {\mathbf{R}}^{2} \), and let \( f \) be a homeomorphism \( M \leftrightarrow {M}^{\prime } \) . Then \( f \) has an extension \( F : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \) . Proof. (1) Let \( A \) and \( {A}^{\prime } \) be 2-cells containing \( M \) and \( {M}^{\prime } \) respectively in their interiors. Let \( {N}_{1} \) be a frame of \( M \), lying in Int \( A \), and lying in a sufficiently small neighborhood of \( M \) so that every component of \( {N}_{1} \) has diameter \( < 1 \) . (Theorem 6.) Then the sets \( f\left( {M \cap C}\right) \), where \( C \) is a component of \( {N}_{1} \), are disjoint and compact. Let \( L \) be a frame of \( {M}^{\prime } \), lying in Int \( {A}^{\prime } \), with components \( {C}^{\prime } \) of sufficiently small diameter so that no \( {C}^{\prime } \) intersects two different sets \( f\left( {M \cap C}\right) \) . By repeated applications of Theorem 2 we get a frame \( {N}_{1}^{\prime } \) of \( {M}^{\prime } \), lying in \( \operatorname{Int}{A}^{\prime } \), such that each set \( f\left( {M \cap C}\right) \) is the intersection of \( {M}^{\prime } \) and a component of \( {N}_{1}^{\prime } \) . Now there is a homeomorphism \( {f}_{0} : {\mathbf{R}}^{2} - \operatorname{Int}A \leftrightarrow {\mathbf{R}}^{2} - \operatorname{Int}{A}^{\prime } \) . Let \( {E}_{1} = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{1},{E}_{1}^{\prime } = \) \( {\mathbf{R}}^{2} \) - Int \( {N}_{1}^{\prime } \) . By Theorem \( 1,{f}_{0} \) can be extended so as to give a homeomorphism \( {f}_{1} : {E}_{1} \leftrightarrow {E}_{1}^{\prime } \), such that if \( D \) and \( {D}^{\prime } \) are components of \( {N}_{1} \) and \( {N}_{1}^{\prime } \) , with \( {f}_{1}\left( {\operatorname{Bd}D}\right) = \operatorname{Bd}{D}^{\prime } \), then \( f\left( {M \cap D}\right) = {M}^{\prime } \cap {D}^{\prime } \) . (2) Suppose that we have given a frame \( {N}_{{2i} - 1} \) of \( M \), a frame \( {N}_{{2i} - 1}^{\prime } \) of \( {M}^{\prime } \), and a homeomorphism \( {f}_{{2i} - 1} : {E}_{{2i} - 1} \leftrightarrow {E}_{{2i} - 1}^{\prime } \), where \[ {E}_{{2i} - 1} = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{{2i} - 1},\;{E}_{{2i} - 1}^{\prime } = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{{2i} - 1}^{\prime }. \] Suppose that the components of \( {N}_{{2i} - 1} \) have diameter less than \( 1/\left( {{2i} - 1}\right) \) . For each component \( A \) of \( {N}_{{2i} - 1} \), let \( {A}^{\prime } \) be the component of \( {N}_{{2i} - 1}^{\prime } \) bounded by \( {f}_{{2i} - 1}\left( {\operatorname{Bd}A}\right) \) . Suppose (as an induction hypothesis) that for each such \( A,{A}^{\prime } \) we have \( f\left( {M \cap A}\right) = {M}^{\prime } \cap {A}^{\prime } \) . Let \( {N}_{2i}^{\prime } \) be a frame of \( {M}^{\prime } \), lying in Int \( {N}_{{2i} - 1}^{\prime } \), and lying in a sufficiently small neighborhood of \( {M}^{\prime } \) so that each component of \( {N}_{2i}^{\prime } \) has diameter less than \( 1/{2i} \) . (Theorem 6.) Then there is a frame \( {N}_{2i} \) of \( M \), lying in Int \( {N}_{{2i} - 1} \) , such that for each component \( {D}^{\prime } \) of \( {N}_{2i}^{\prime },{f}^{-1}\left( {{M}^{\prime } \cap {D}^{\prime }}\right) = M \cap D \), where \( D \) is a component of \( {N}_{2i} \) . (The construction of \( {N}_{2i} \) is like that of \( {N}_{1}^{\prime } \) . We work with the sets \( {M}^{\prime } \cap {A}^{\prime }\left( {A}^{\prime }\right. \) a component of \( \left. {N}_{{2i} - 1}^{\prime }\right) \) one at a time. For each such \( {A}^{\prime } \), let \( A \) be the component of \( {N}_{{2i} - 1} \) such that \( \operatorname{Bd}{A}^{\prime } = {f}_{{2i} - 1}\left( {\operatorname{Bd}A}\right) \) . In the construction of \( {N}_{1}^{\prime } \), described in (1), we use \( {f}^{-1},{M}^{\prime } \cap {A}^{\prime }, M \cap A,{A}^{\prime } \) , and \( A \) in place of \( f, M,{M}^{\prime }, A \) and \( {A}^{\prime } \) respectively.) Now extend \( {f}_{{2i} - 1} \) to get \[ {f}_{2i} : {E}_{2i} \leftrightarrow {E}_{2i}^{\prime } \] where \[ {E}_{2i} = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{2i},\;{E}_{2i}^{\prime } = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{2i}^{\prime }, \] such that if \( D \) and \( {D}^{\prime } \) are components of \( {N}_{2i} \) and \( {N}_{2i}^{\prime } \), with \( {f}_{2i}\left( {\operatorname{Bd}D}\right) = \) Bd \( {D}^{\prime } \), then \( f\left( {M \cap D}\right) = {M}^{\prime } \cap {D}^{\prime } \) . (The construction of \( {f}_{2i} \) is like that of \( \left. {f}_{1}\right) \) . Thus, when we pass from \( {N}_{{2i} - 1},{N}_{{2i} - 1}^{\prime },{f}_{{2i} - 1} \) to \( {N}_{2i},{N}_{2i}^{\prime },{f}_{2i} \), the induction hypothesis is preserved. (3) The recursive step from \( {N}_{2i},{N}_{2i}^{\prime },{f}_{2i} \) to \( {N}_{{2i} + 1},{N}_{{2i} + 1}^{\prime },{f}_{{2i} + 1} \) is entirely similar, and in fact the whole situation is logically symmetric. Thus we have sequences \( {N}_{1},{N}_{2},\ldots ,{N}_{1}^{\prime },{N}_{2}^{\prime },\ldots ,{f}_{1},{f}_{2},\ldots \) such that: (a) \( {N}_{i} \) is a frame of \( M \), and \( {N}_{i}^{\prime } \) is a frame of \( {M}^{\prime } \) ; (b) \( {N}_{i + 1} \subset \operatorname{Int}{N}_{i} \) and \( {N}_{i + 1}^{\prime } \subset \operatorname{Int}{N}_{i}^{\prime } \) ; (c) Each component of \( {N}_{{2i} - 1} \) (or \( {N}_{2i}^{\prime } \) ) has diameter less than \( 1/\left( {{2i} - 1}\right) \) (or \( 1/{2i}) \) ; (d) Each \( {f}_{i} \) is a homeomorphism \( {E}_{i} \leftrightarrow {E}_{i}^{\prime } \), where \[ {E}_{i} = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{i},\;{E}_{i}^{\prime } = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{i}^{\prime }; \] (e) For each \( i,{f}_{i + 1} \) is an extension of \( {f}_{i} \) . Now let \[ F = f \cup \mathop{\bigcup }\limits_{{i = 1}}^{\infty }{f}_{i} \] By (e), \( F \) is a well-defined function. Since \[ {\mathbf{R}}^{2} = M \cup \mathop{\bigcup }\limits_{{i = 1}}^{\infty }{E}_{i} = {M}^{\prime } \cup \mathop{\bigcup }\limits_{{i = 1}}^{\infty }{E}_{i}^{\prime }, \] \( F \) is a bijection \( {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \) . It remains to show that \( F \) and \( {F}^{-1} \) are continuous. Given \( P \in {\mathbf{R}}^{2} - M, Q = F\left( P\right) \), we have \( Q \in {\mathbf{R}}^{2} - {M}^{\prime } \) . Given an open set \( U \) containing \( Q \), we may suppose that \( U \subset \operatorname{Int}{E}_{i}^{\prime } \) for some \( i \) . Since \( {f}_{i} \) is a homeomorphism, some neighborhood of \( P \) is mapped into \( U \) by \( F \) . If \( P \in M \), then \( Q = F\left( P\right) = f\left( P\right) \in {M}^{\prime } \), and \( Q \) has arbitrarily small neighborhoods which are components \( {D}^{\prime } \) of sets \( {N}_{i}^{\prime } \) . It is now easy to check that \( {D}^{\prime } = F\left( D\right) \) for some component \( D \) of \( {N}_{i} \) . Thus \( F \) maps small neighborhoods of \( P \) onto small neighborhoods of \( Q \) . Therefore \( F \) is continuous. The continuity of \( {F}^{-1} \) can be shown similarly. (Again, the situation is logically symmetric.) ## PROBLEM SET 13 Prove or disprove: 1. Every Cantor set in \( {\mathbf{R}}^{2} \) lies in an arc in \( {\mathbf{R}}^{2} \) . 2. Let \( C \) be the middle-third Cantor set in \( \left\lbrack {0,1}\right\rbrack \), and let \( D \) be any countable dense set in \( C \) . Then there is an arc \( A \) in \( {\mathbf{R}}^{2} \) such that (1) the end-points of \( A \) lie in \( D \) and (2) the other points of \( D \) are the end-points of the components of \( A - D \) . 3. (The Moore-Kline theorem.) Every totally disconnected compact set in \( {\mathbf{R}}^{2} \) lies in an arc in \( {\mathbf{R}}^{2} \) . 4. Every totally disconnected compact set in \( {\mathbf{R}}^{2} \) lies in a Cantor set in \( {\mathbf{R}}^{2} \) . 5. Let \( A \) be an arc in \( {\mathbf{R}}^{3} \) . Then every point of \( A \) is arcwise accessible from \( {\mathbf{R}}^{3} - A \) . 6. Let \( S \) be a 2-sphere in \( {\mathbf{R}}^{3} \), and let \( P \in S \) . Then there is a plane \( E \) such that \( P \in E \) and \( S \cap E \) contains no 2-manifold. 7. Let \( A \) be an arc in \( {\mathbf{R}}^{3} \), and let \( P \in A \) . Then there is a plane \( E \) such that \( P \in E \) and \( A \cap E \) is totally disconnected. 8. Every totally disconnected compact metric space \( M \) is imbeddable in \( \mathbf{R} \) . That is, there is a homeomorphism \( f : M \leftrightarrow {M}^{\prime } \subset \mathbf{R} \) . Geometric topology in dimensions 2 and 3 The theorem stated in Problem 3 was extended (by R. L. Moore and J. R. Kline) so as to apply to every compact set \( M \) in which each component is an arc whose interior is open in the space \( M \) . Thus every compact set in \( {\mathbf{R}}^{2} \) lies in an arc in \( {\mathbf{R}}^{2} \) unless it obviously cannot. The proof is technical, but Theorem 10.8 is helpful. The first proof appeared in [MK]. The fundamental group (summary) 14 This section is a brief account of elementary definitions and theorems. Let \( \left\lbrack {X,\mathcal{O}}\right\rbrack \) be a topological space, and suppose that \( X \) is pathwise connected, in the sense defined at the beginning of Section 1. Topological generality will not concern us in the sequel: \( X \) will always be a polyhedron in a Cartesian space, or an open subset of such a space, or at least a space homeomorphic to one of these. Let \( {P}_{0} \in X \), and let \( \operatorname{CP}\left( {X,{P}_{0}}\right) \) be the set of all closed paths \[ p : \left\lbrack {0,1}\right\rbrack \rightarrow X,\;0 \mapsto {P}_{0},\;1 \mapsto {P}_{0}. \] \( {P}_{0} \) will be called the base point. In \( \mathrm{{CP}}\left( {X,{P}_{0}}\right) \) we multiply paths by shrinking them and laying them end to end. That is, \[ {pq}\left( t\right) = \left\{ \begin{arr

Theorem 7. Let \( M \) and \( {M}^{\prime } \) be totally disconnected compact sets in \( {\mathbf{R}}^{2} \), and let \( f \) be a homeomorphism \( M \leftrightarrow {M}^{\prime } \). Then \( f \) has an extension \( F : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \).

(1) Let \( A \) and \( {A}^{\prime } \) be 2-cells containing \( M \) and \( {M}^{\prime } \) respectively in their interiors. Let \( {N}_{1} \) be a frame of \( M \), lying in Int \( A \), and lying in a sufficiently small neighborhood of \( M \) so that every component of \( {N}_{1} \) has diameter \( < 1 \). (Theorem 6.) Then the sets \( f\left( {M \cap C}\right) \), where \( C \) is a component of \( {N}_{1} \), are disjoint and compact. Let \( L \) be a frame of \( {M}^{\prime } \), lying in Int \( {A}^{\prime } \), with components \( {C}^{\prime } \) of sufficiently small diameter so that no \( {C}^{\prime } \) intersects two different sets \( f\left( {M \cap C}\right) \). By repeated applications of Theorem 2 we get a frame \( {N}_{1}^{\prime } \) of \( {M}^{\prime } \), lying in \( \operatorname{Int}{A}^{\prime } \), such that each set \( f\left( {M \cap C}\right) \) is the intersection of \( {M}^{\prime } \) and a component of \( {N}_{1}^{\prime } \). Now there is a homeomorphism \( {f}_{0} : {\mathbf{R}}^{2} - \operatorname{Int}A \leftrightarrow {\mathbf{R}}^{2} - \operatorname{Int}{A}^{\prime } \). Let \( {E}_{1} = {\mathbf{R}}^{2} - \operatorname{Int}{N}_{1},{E}_{1}^{\prime } = {\mathbf{R}}^{2} - Int {N}_{1}^{\prime } \). By Theorem 1, \( {f}_{0} \) can be extended so as to give a homeomorphism \( {f}_{1} : {E}_{1} \leftrightarrow {E}_{1}^{\prime } \), such that if \( D \) and \( {D}^{\prime } \) are components of \( {N}_{1} \) and \( {N}_{1}^{\prime } \), with \( {f}_{1}\left( {\operatorname{Bd}D}\right) = \operatorname{Bd}{D}^{\prime } \), then \( f\left( {M \cap D}\right) = {M}^{\prime } \cap {D}^{\prime } \). (2) Suppose that we have given a frame \( {N}_{{2i} - 1} \) of \( M \), a frame

Lemma 7.4. If \( {h}_{\mathfrak{p}} \) is the natural homomorphism \[ {h}_{\mathfrak{p}} : \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \rightarrow \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}, \] then \( {h}_{\mathfrak{p}}{}^{-1} \) induces a natural lattice-embedding \( \mathfrak{a} \rightarrow {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{a}\right) \) of \( \left( {\mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) , \subset ,\cap , + }\right) \) into \( \left( {\mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) , \subset ,\cap , + }\right) \) . Proof. Let \( {\mathfrak{a}}_{1} \neq {\mathfrak{a}}_{2} \) be distinct ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \) ; say \( p \in {\mathfrak{a}}_{1} \) but \( p \notin {\mathfrak{a}}_{2} \) . Then for any \( q \in \left\{ {{h}_{\mathfrak{p}}{}^{-1}\left( p\right) }\right\} \), we have that \( q \in {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) \) and \( q \notin {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{2}\right) \) -that is, \( {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) \neq {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{2}\right) \) . Hence the mapping is 1:1 on the set of ideals. Next, for any two ideals \( {\mathfrak{a}}_{1},{\mathfrak{a}}_{2} \) of \( \left. {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) /\mathfrak{p} \), we have \[ {h}_{\mathfrak{p}}{}^{-1}\left( {{\mathfrak{a}}_{1} \cap {\mathfrak{a}}_{2}}\right) = {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) \cap {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{2}\right) \] (11) \[ {h}_{\mathfrak{p}}{}^{-1}\left( {{\mathfrak{a}}_{1} + {\mathfrak{a}}_{2}}\right) = {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) + {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{2}\right) \] (12) these follow at once from the definitions of \( {h}_{\mathfrak{p}}{}^{-1}, \cap \), and + . Hence the embedding preserves the lattice structure. For an analogous \( \mathcal{J} \) -result, we need to define closed ideal of \[ \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} = {R}_{V} \] Since \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \) is a subset of \[ \mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \] it would seem natural to define the closed ideals of \( {R}_{V} \) by means of Lemma 7.4-that is, to be simply the ideals corresponding under \( {h}_{\mathfrak{p}}{}^{-1} \) to the closed ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) . But there is a problem: \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \) can be represented in many different ways as a quotient ring of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{m}}\right\rbrack \) , for some \( m \) . We would have to show this definition of closed ideal is independent of this ring's representation as a quotient ring. However, since the closed ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) are precisely intersections of prime ideals (Theorem 4.9), one possible definition is Definition 7.5. An ideal in \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \) is closed if it is the intersection of some set of prime ideals in \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \) . For any ideal \( \mathfrak{a} \subset \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \), the map \( \mathfrak{a} \rightarrow \sqrt{\mathfrak{a}} = \mathop{\bigcap }\limits_{{\mathfrak{P} \supset \mathfrak{a}}}\mathfrak{P} \) is a closure map. Thus we see (from Lemma 2.6) that for closed ideals \( {\mathfrak{c}}_{1},{\mathfrak{c}}_{2} \subset \) \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \), defining + by \( {\mathfrak{c}}_{1} + {\mathfrak{c}}_{2} = \sqrt{{\mathfrak{c}}_{1} + {\mathfrak{c}}_{2}} \), makes the set \( \mathcal{J} \) of closed ideals into a lattice \( \left( {\mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) ,\cap , + }\right) \) . That these closed ideals do indeed correspond under \( {h}_{\mathrm{p}}{}^{-1} \) to the closed ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) containing \( \mathfrak{p} \) is shown by Lemma 7.6. \( {h}_{\mathfrak{p}}{}^{-1} \) defines a natural lattice-embedding of \[ \left( {\mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) ,\cap , + }\right) \] into \( \left( {\mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) ,\cap , + }\right) \) . Proof. Clearly \( {h}_{\mathfrak{p}}{}^{-1} \) defines a set-injection of \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) \) into \( \mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \) . That this injection is actually into \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \) , i.e., that \( {h}_{\mathfrak{p}}{}^{-1} \) embeds \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) \) into \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \), may be seen as follows: First note that for any ideal \( \mathfrak{a} \supset \mathfrak{p} \) in \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack ,{h}_{\mathfrak{p}}{}^{-1} \) induces a 1:1-onto map \( \mathfrak{P} \rightarrow {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{P}\right) \) from the set of prime ideals \( \mathfrak{P} \) of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \) containing \( \mathfrak{a}/\mathfrak{p} \), to the set of prime ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) containing a. (Note that \( {h}_{\mathfrak{p}}{}^{-1} \) preserves primality of ideals in \[ \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \] as does \( {h}_{\mathfrak{p}} \) for those ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) containing \( \mathfrak{p} \) .) Now let \( \mathfrak{c} \) be any ideal in \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) \) ; then \[ {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{c}\right) = {h}_{\mathfrak{p}}{}^{-1}\left( {\mathop{\bigcap }\limits_{{\mathfrak{P} \supset \mathfrak{c}}}\mathfrak{P}}\right) = \mathop{\bigcap }\limits_{{\mathfrak{P} \supset \mathfrak{c}}}\left( {{h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{P}\right) }\right) = \mathop{\bigcap }\limits_{{{\mathfrak{P}}^{\prime } \supset {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{c}\right) }}{\mathfrak{P}}^{\prime } \] \[ = \sqrt{{h}_{\mathfrak{p}}^{-1}\left( \mathfrak{c}\right) } \in \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) . \] It is easy to see that this embedding preserves lattice structure, for \( {h}_{\mathrm{p}}{}^{-1} \) preserves intersection (from (11)); it also preserves + , that is, for closed ideals \( {\mathrm{c}}_{1},{\mathrm{c}}_{2} \) \[ {h}_{\mathfrak{p}}{}^{-1}\left( \sqrt{{\mathfrak{c}}_{1} + {\mathfrak{c}}_{2}}\right) = \sqrt{{h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{c}}_{1}\right) + {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{c}}_{2}\right) }. \] This follows since \( {h}_{\mathrm{p}}{}^{-1} \) preserves sum (from (12)) and radical. Now that we have shown that \( {h}_{\mathfrak{p}}{}^{-1} \) induces lattice embeddings, it is natural to ask if it likewise preserves decomposition of ideals into irreducibles. It does indeed. Since any homomorphic image of a Noetherian ring is Noetherian, the p.o. set \( \mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) \) satisfies the a.c.c., so a fortiori \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) \) does too. Hence any element in either of these sets has an irredundant decomposition into irreducibles. This decomposition is unique in the case of \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) \) since it is distributive. It is isomorphic to a sublattice of \( \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \), which itself is isomorphic to the distributive lattice of subvarieties \( \mathcal{V}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) \) . Now if \( \mathfrak{a} \subset \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \) is irreducible, then \( {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{a}\right) \subset \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) is too; this is obvious from the definition of irreducibility. Therefore if \( \mathfrak{a} = {\mathfrak{a}}_{1} \cap \ldots \cap {\mathfrak{a}}_{r} \) is a decomposition of \( \mathfrak{a} \in \mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) \) into irreducibles, then \[ {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{a}\right) = {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) \cap \ldots \cap {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{r}\right) \] is a decomposition of \( {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{a}\right) \) into irreducibles in \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \) . And if \( \mathfrak{a} \in \mathcal{J}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{

Lemma 7.4. If \( {h}_{\mathfrak{p}} \) is the natural homomorphism \[ {h}_{\mathfrak{p}} : \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack \rightarrow \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}, \] then \( {h}_{\mathfrak{p}}{}^{-1} \) induces a natural lattice-embedding \( \mathfrak{a} \rightarrow {h}_{\mathfrak{p}}{}^{-1}\left( \mathfrak{a}\right) \) of \( \left( {\mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p}}\right) , \subset ,\cap , + }\right) \) into \( \left( {\mathcal{I}\left( {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) , \subset ,\cap , + }\right) \) .

Proof. Let \( {\mathfrak{a}}_{1} \neq {\mathfrak{a}}_{2} \) be distinct ideals of \( \mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack /\mathfrak{p} \) ; say \( p \in {\mathfrak{a}}_{1} \) but \( p \notin {\mathfrak{a}}_{2} \) . Then for any \( q \in \left\{ {{h}_{\mathfrak{p}}{}^{-1}\left( p\right) }\right\} \), we have that \( q \in {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) \) and \( q \notin {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{2}\right) \) -that is, \( {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) \neq {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{2}\right) \) . Hence the mapping is 1:1 on the set of ideals. Next, for any two ideals \( {\mathfrak{a}}_{1},{\mathfrak{a}}_{2} \) of \( \left. {\mathbb{C}\left\lbrack {{X}_{1},\ldots ,{X}_{n}}\right\rbrack }\right) /\mathfrak{p} \), we have \[ {h}_{\mathfrak{p}}{}^{-1}\left( {{\mathfrak{a}}_{1} \cap {\mathfrak{a}}_{2}}\right) = {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{1}\right) \cap {h}_{\mathfrak{p}}{}^{-1}\left( {\mathfrak{a}}_{2}\right) \] \[ {h}_{\mathfrak{p}}{}^{-1}\left( {{\mathfrak{a}}_{1} + {\mathfrak{a}}_{2}}\right) = {h}_{\mathfrak

Theorem 5.1.15. Singular homology satisfies Axiom 3. Proof. Immediate from the definition of the boundary map on singular cubes and from the definition of the induced map on singular cubes as composition. Theorem 5.1.16. Singular homology satisfies Axiom 4. Proof. We have defined \( {C}_{n}\left( {X, A}\right) = {C}_{n}\left( X\right) /{C}_{n}\left( A\right) \) . Thus for every \( n \), we have a short exact sequence \[ 0 \rightarrow {C}_{n}\left( A\right) \rightarrow {C}_{n}\left( X\right) \rightarrow {C}_{n}\left( {X, A}\right) \rightarrow 0. \] In other words, we have a short exact sequence of chain complexes \[ 0 \rightarrow {C}_{ * }\left( A\right) \rightarrow {C}_{ * }\left( X\right) \rightarrow {C}_{ * }\left( {X, A}\right) \rightarrow 0. \] But then we have a long exact sequence in homology by Theorem A.2.10. Theorem 5.1.17. Singular homology satisfies Axiom 5. Proof. For simplicity we consider the case of homotopic maps of spaces \( f : X \rightarrow Y \) and \( g : X \rightarrow Y \) (rather than maps of pairs). Then by definition, setting \( {f}_{0} = f \) and \( {f}_{1} = g \), there is a map \( F : X \times I \rightarrow Y \) with \( F\left( {x,0}\right) = {f}_{0}\left( x\right) \) and \( F\left( {x,1}\right) = {f}_{1}\left( x\right) \) . Define a map \( \widetilde{F} : {C}_{n}\left( X\right) \rightarrow {C}_{n + 1}\left( Y\right) \) as follows. Let \( \Phi : {I}^{n} \rightarrow X \) be a singular \( n \) -cube. Then \( \widetilde{F}\Phi : {I}^{n + 1} \rightarrow Y \) is defined by \[ \widetilde{F}\mathbf{\Phi }\left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) = F\left( {\mathbf{\Phi }\left( {{x}_{1},\ldots ,{x}_{n}}\right) ,{x}_{n + 1}}\right) . \] Then it is routine (but lengthy) to check that \( F \) provides a chain homotopy between \( {f}_{ * } : {C}_{ * }\left( X\right) \rightarrow {C}_{ * }\left( Y\right) \) and \( {g}_{ * } : {C}_{ * }\left( X\right) \rightarrow {C}_{ * }\left( Y\right) \), so that \( {f}_{ * } = {g}_{ * } : {H}_{ * }\left( X\right) \rightarrow \) \( {H}_{ * }\left( Y\right) \) by Lemma A.2.9. ## Theorem 5.1.18. Singular homology satisfies Axiom 6. We shall not prove that singular homology satisfies Axiom 6, the excision axiom. This proof is quite involved. We merely give a quick sketch of the basic idea. Consider a singular chain, say \( c \), the singular 1-cube which is the illustrated path  This chain is too large, and we subdivide it as illustrated.  Then we show \( c \) is homologous to \( {c}_{1} + {c}_{2} \) with \( {c}_{1} \) (resp. \( {c}_{2} \) ) the left (resp. right) subpaths in \( {C}_{ * }\left( {X, A}\right) \) . But \( {c}_{1} \) is in the image of the inclusion \( {C}_{ * }\left( {X - U, A - U}\right) \rightarrow \) \( {C}_{ * }\left( {X, A}\right) \) . Let us now compute the singular homology of a point. Theorem 5.1.19. Let \( X \) be the space consisting of a single point. Then \( {H}_{0}\left( X\right) \cong \mathbb{Z} \) and \( {H}_{i}\left( X\right) = 0 \) for \( i \neq 0 \) . Thus singular homology satisfies the dimension axiom, Axiom 7, and has coefficient group \( \mathbb{Z} \) . Proof. Let \( \Phi : {I}^{0} \rightarrow X \) be the unique map. Then \( {C}_{0}\left( X\right) \) is the free abelian group generated by \( \Phi \) . On the other hand, for any \( i > 0,\Phi : {I}^{i} \rightarrow X \) is a degenerate \( i \) -cube. Hence \( {C}_{i}\left( X\right) = \{ 0\} \) for \( i > 0 \) . Thus \( {C}_{ * }\left( X\right) \) is the chain complex \[ \cdots \rightarrow 0 \rightarrow 0 \rightarrow \mathbb{Z} \rightarrow 0 \rightarrow 0 \rightarrow \cdots \] with homology as stated. Remark 5.1.20. It is to make Theorem 5.1.19 hold that we must use \( {C}_{i}\left( X\right) = \) \( {Q}_{i}\left( X\right) /{D}_{i}\left( X\right) \) rather than work with \( {Q}_{i}\left( X\right) \) itself. Let \( X \) be a point. Then in \( {Q}_{i}\left( X\right) \) we have, for each \( i \), the unique map \( \Phi : {I}^{i} \rightarrow X \), and \( {\partial }_{i}^{Q}\Phi = 0 \) as the front and back faces cancel each other out. Then \( {Q}_{ * }\left( X\right) \) is the chain complex \[ \cdots \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0 \rightarrow 0 \rightarrow \cdots \] with each boundary map being the 0 map, and this chain complex has homology \( \mathbb{Z} \) in every nonnegative dimension. However, it is not just that this complex gives the "wrong" answer. Rather, it is that it gives the wrong answer for a stupid reason, the presence of all these geometrically meaningless singular cubes. So we divide out by them to get a geometrically meaningful theory. As in Chap. 4, we once and for all establish an isomorphism between \( {H}_{0}\left( *\right) \) and \( \mathbb{Z} \) by choosing a generator \( {1}_{ * } \in {H}_{0}\left( *\right) \) which we identify with \( 1 \in \mathbb{Z} \) . Definition 5.1.21. The class \( {1}_{ * } \in {H}_{0}\left( *\right) \) is the homology class represented by the unique map \( \Phi : {I}^{0} \rightarrow * \) . Then for any space \( p \) consisting of a simple point, the class \( {1}_{p} \in {H}_{0}\left( p\right) \) is \( {1}_{p} = {f}_{0}\left( {1}_{ * }\right) \) where \( f : * \rightarrow p \) is the unique map. (More simply, \( {I}_{p} \in {H}_{0}\left( p\right) \) is the homology class represented by the unique map \( \Phi : {I}^{0} \rightarrow p \) .) \( \diamond \) Let us make a definition and a pair of observations. Definition 5.1.22. Let \( X \) be a space and let \( c \in {C}_{n}\left( X\right) \) be a singular chain. Choose a representative of \( c \) of the form \( \mathop{\sum }\limits_{{i = 0}}^{N}{m}_{i}{\Phi }_{i} \) where \( {m}_{i} \neq 0 \) for each \( i \) and \( {\Phi }_{i} \) is a nondegenerate singular \( n \) -cube for each \( i \) . The support of \( c \), supp \( \left( c\right) \), is defined to be \( \varnothing \) if \( N = 0 \), and otherwise \( \operatorname{supp}\left( c\right) = \mathop{\bigcup }\limits_{{i = 0}}^{N}{\Phi }_{i}\left( {I}^{n}\right) \subseteq X \) . Lemma 5.1.23. For any singular chain \( c,\operatorname{supp}\left( {\partial c}\right) \subseteq \operatorname{supp}\left( c\right) \) . Theorem 5.1.24. For any singular chain \( c \) , \( \operatorname{supp}\left( c\right) \) is a compact subset of \( X \) . Proof. For any \( \Phi : {I}^{n} \rightarrow X,\Phi \left( {I}^{n}\right) \) is a compact subset of \( X \) as it is the continuous image of a compact set. Then for any singular chain \( c \), supp \( \left( c\right) \) is a finite union of compact sets and hence is compact. Corollary 5.1.25. Let \( X \) be a union of components, \( X = \mathop{\bigcup }\limits_{{i \in I}}{X}_{i} \) . Then for any \( n \) , \( {H}_{n}\left( X\right) = {\bigoplus }_{i \in I}{H}_{n}\left( {X}_{i}\right) \) Proof. This follows for any generalized homology theory from Lemma 3.2.1 if there are only finitely many components. But for singular homology theory, if \( c \in {C}_{n}\left( X\right) \) is any chain, then \( \operatorname{supp}\left( c\right) \) is compact, by Theorem 5.1.24, so is contained in \( \mathop{\bigcup }\limits_{{i \in J}}{X}_{i} \) for some finite subset \( J \) of \( I \) . Thus \( {C}_{n}\left( X\right) = {\bigoplus }_{i \in I}{C}_{n}\left( {X}_{i}\right) \) . But clearly \( \partial : {C}_{n}\left( {X}_{i}\right) \rightarrow {C}_{n - 1}\left( {X}_{i - 1}\right) \) for any \( i \), so as chain complexes \( {C}_{ * }\left( X\right) = {\bigoplus }_{i \in I}{C}_{ * }\left( {X}_{i}\right) \) and hence \( {H}_{ * }\left( X\right) = {\bigoplus }_{i \in I}{H}_{ * }\left( {X}_{i}\right) \) . We record the following result for future use. Lemma 5.1.26. (1) For any space \( X \), the group of singular \( n \) -chains \( {C}_{n}\left( X\right) \) is isomorphic to the free abelian group with basis the non-degenerate n-cubes. (2) For any pair \( \left( {X, A}\right) \), the group \( {C}_{n}\left( {X, A}\right) \) is isomorphic to the free abelian group with basis those non-degenerate \( n \) -cubes whose support is not contained in \( A \) . (3) The short exact sequence \[ 0 \rightarrow {C}_{n}\left( A\right) \rightarrow {C}_{n}\left( X\right) \rightarrow {C}_{n}\left( {X, A}\right) \rightarrow 0 \] splits, and hence \( {C}_{n}\left( X\right) \) is isomorphic to \( {C}_{n}\left( A\right) \oplus {C}_{n}\left( {X, A}\right) \) . Proof. This follows easily once we recall that \( {C}_{n}\left( X\right) = {Q}_{n}\left( X\right) /{D}_{n}\left( X\right) \) where \( {Q}_{n}\left( X\right) \) is the free abelian group on all \( n \) -cubes and \( {D}_{n}\left( X\right) \) is the free abelian group on the degenerate \( n \) -cubes We also record the following definition. Definition 5.1.27. A space \( X \) is of finite type if \( {H}_{n}\left( X\right) \) is finitely generated for each \( n \) . ## 5.2 The Geometric Meaning of \( {H}_{0} \) and \( {H}_{1} \) In this section we see the geometric content of the singular homology groups \( {H}_{0}\left( X\right) \) and \( {H}_{1}\left( X\right) \) . Theorem 5.2.1. Let \( X \) be a space. Then \( {H}_{0}\left( X\right) \) is isomorphic to the free abelian group on the path components of \( X \) . Proof. We assume \( X \) nonempty. We have already seen in Corollary 5.1.25 that if \( X = {X}_{1} \cup {X}_{2} \cup \cdots \) is a union of path components, then \( {H}_{i}\left( X\right) = {\bigoplus }_{k}{H}_{i}\left( {X}_{k}\right) \) . Thus it satisfies to prove the theorem in case \( X \) is path connected, so we make that assumption. A singular 0 -simplex of \( X \) is \( f\left( *\right) = x \) for some \( x \in X \), so we may identify \( {C}_{0}\left( X\right) \) with the free abelian group on the points of \( X,{C}_{0}\left( X\right) = \left\{ {\mathop{\sum }\limits_{i}{n}_{i}{x

(Theorem 5.1.15. Singular homology satisfies Axiom 3.)

Immediate from the definition of the boundary map on singular cubes and from the definition of the induced map on singular cubes as composition.

Exercise 9.1.9 Prove that \[ \pi \left( {x, z}\right) = {xV}\left( z\right) + O\left( {x{\left( \log z\right) }^{2}\exp \left( {-\frac{\log x}{\log z}}\right) }\right) , \] where \[ V\left( z\right) = \mathop{\prod }\limits_{{p \leq z}}\left( {1 - \frac{1}{p}}\right) \] and \( z = z\left( x\right) \rightarrow \infty \) as \( x \rightarrow \infty \) . Exercise 9.1.10 Prove that \[ \pi \left( x\right) \ll \frac{x}{\log x}\log \log x \] by setting \( \log z = \epsilon \log x/\log \log x \), for some sufficiently small \( \epsilon \), in the previous exercise. Exercise 9.1.11 For any \( A > 0 \), show that \[ \pi \left( {x,{\left( \log x\right) }^{A}}\right) \sim \frac{x{e}^{-\gamma }}{A\log \log x} \] as \( x \rightarrow \infty \) . The estimate of Exercise 9.1.9 for \( \pi \left( x\right) \) will be seen to be as good as the one obtained by the elementary Brun sieve of the next section. Let \( A \) be any set of natural numbers and let \( \mathcal{P} \) be a set of primes. To each prime \( p \in \mathcal{P} \), let there be \( \omega \left( p\right) \) distinguished residue classes \( {\;\operatorname{mod}\;p} \) . Let \( {A}_{p} \) denote the set of elements of \( A \) belonging to at least one of these distinguished classes \( {\;\operatorname{mod}\;p} \) . For any square-free number \( d \) composed of primes \( p \in \mathcal{P} \), let \[ {A}_{d} = { \cap }_{p \mid d}{A}_{p} \] We denote by \( S\left( {A,\mathcal{P}, z}\right) \) the number of elements of \[ A \smallsetminus { \cup }_{p \in \mathcal{P}, p \leq z}{A}_{p} \] Let \( \omega \left( d\right) = \mathop{\prod }\limits_{{p \mid d}}\omega \left( p\right) \), and \( P\left( z\right) = \mathop{\prod }\limits_{{p \leq z, p \in \mathcal{P}}}p \) . Exercise 9.1.12 Suppose that \[ \mathop{\sum }\limits_{\substack{{p \leq z} \\ {p \in \mathcal{P}} }}\frac{\omega \left( p\right) \log p}{p} \leq \kappa \log z + O\left( 1\right) \] Show that \[ {F}_{\omega }\left( {t, z}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{\substack{{d \leq t} \\ {d \mid P\left( z\right) } }}\omega \left( d\right) \] is bounded by \( O\left( {t{\left( \log z\right) }^{\kappa }\exp \left( {-\frac{\log t}{\log z}}\right) }\right) \) . Exercise 9.1.13 Let \( C \) be a constant. With the same hypothesis as in the previous exercise, show that \[ \mathop{\sum }\limits_{\substack{{d \mid P\left( z\right) } \\ {d > {Cx}} }}\frac{\omega \left( d\right) }{d} = O\left( {{\left( \log z\right) }^{\kappa + 1}\exp \left( {-\frac{\log x}{\log z}}\right) }\right) . \] We are now ready to prove our version of the sieve of Eratosthenes. We follow [MS]. We suppose there is an \( X \) such that \[ \left| {A}_{d}\right| = \frac{{X\omega }\left( d\right) }{d} + {R}_{d} \] with \( {R}_{d} = O\left( {\omega \left( d\right) }\right) \) . We also assume \[ \mathop{\sum }\limits_{\substack{{p \leq z} \\ {p \in \mathcal{P}} }}\frac{\omega \left( p\right) \log p}{p} \leq \kappa \log z + O\left( 1\right) \] and set \[ W\left( z\right) = \mathop{\prod }\limits_{\substack{{p \leq z} \\ {p \in \mathcal{P}} }}\left( {1 - \frac{\omega \left( p\right) }{p}}\right) \] Exercise 9.1.14 (Sieve of Eratosthenes) Suppose there is a constant \( C > 0 \) such that \( \left| {A}_{d}\right| = 0 \) for \( d > {Cx} \) . Then \[ S\left( {A,\mathcal{P}, z}\right) = {XW}\left( z\right) + O\left( {x{\left( \log z\right) }^{\kappa + 1}\exp \left( {-\frac{\log x}{\log z}}\right) }\right) . \] ## 9.2 Brun's Elementary Sieve By comparing coeffients of \( {x}^{r} \) on both sides of the identity \[ {\left( 1 - x\right) }^{-1}{\left( 1 - x\right) }^{\nu } = {\left( 1 - x\right) }^{\nu - 1} \] we deduce \[ \mathop{\sum }\limits_{{k \leq r}}{\left( -1\right) }^{k}\left( \begin{array}{l} \nu \\ k \end{array}\right) = {\left( -1\right) }^{r}\left( \begin{matrix} \nu - 1 \\ r \end{matrix}\right) \] This implies that \[ \mathop{\sum }\limits_{\substack{{d \mid n} \\ {\nu \left( d\right) \leq r} }}\mu \left( d\right) = {\left( -1\right) }^{r}\left( \begin{matrix} \nu \left( n\right) - 1 \\ r \end{matrix}\right) \] where \( \nu \left( n\right) \) is the number of prime factors of \( n \) . This observation is the basis of Brun's elementary sieve. Namely, let \[ {\mu }_{r}\left( d\right) = \left\{ \begin{array}{lll} \mu \left( d\right) & \text{ if } & \nu \left( d\right) \leq r \\ 0 & \text{ if } & \nu \left( d\right) > r \end{array}\right. \] Then setting \[ {\psi }_{r}\left( n\right) = \mathop{\sum }\limits_{{d \mid n}}{\mu }_{r}\left( d\right) \] we find that if \( r \) is even, \( \mathop{\sum }\limits_{{d \mid n}}\mu \left( d\right) \leq {\psi }_{r}\left( n\right) \) and if \( r \) is odd, \( \mathop{\sum }\limits_{{d \mid n}}\mu \left( d\right) \) \( \geq {\psi }_{r}\left( n\right) \) . Thus \[ \mathop{\sum }\limits_{{d \mid n}}\mu \left( d\right) = {\psi }_{r}\left( n\right) + O\left( {\mathop{\sum }\limits_{\substack{{d \mid n} \\ {\nu \left( d\right) = r + 1} }}\left| {\mu \left( d\right) }\right| }\right) \] Exercise 9.2.1 Show that for \( r \) even, \[ \pi \left( {x, z}\right) \leq x\mathop{\sum }\limits_{{d \mid {P}_{z}}}\frac{{\mu }_{r}\left( d\right) }{d} + O\left( {z}^{r}\right) \] We now turn our attention to \[ \mathop{\sum }\limits_{{d \mid {P}_{z}}}\frac{{\mu }_{r}\left( d\right) }{d} \] By Möbius inversion, \[ {\mu }_{r}\left( d\right) = \mathop{\sum }\limits_{{\delta \mid d}}\mu \left( {d/\delta }\right) {\psi }_{r}\left( \delta \right) \] so that \[ \mathop{\sum }\limits_{{d \mid {P}_{z}}}\frac{{\mu }_{r}\left( d\right) }{d} = \mathop{\sum }\limits_{{d \mid {P}_{z}}}\frac{1}{d}\mathop{\sum }\limits_{{\delta \mid d}}\mu \left( {d/\delta }\right) {\psi }_{r}\left( \delta \right) \] \[ = \mathop{\sum }\limits_{{\delta \mid {P}_{z}}}\frac{{\psi }_{r}\left( \delta \right) }{\delta }\mathop{\sum }\limits_{{d \mid {P}_{z}/\delta }}\frac{\mu \left( d\right) }{d} \] \[ = V\left( z\right) \mathop{\sum }\limits_{{\delta \mid {P}_{z}}}\frac{{\psi }_{r}\left( \delta \right) }{\phi \left( \delta \right) } \] where \( V\left( z\right) \) is as in the previous section and \( \phi \) denotes Euler’s function. Let us note that \[ \mathop{\sum }\limits_{{d \mid {P}_{z}}}\frac{{\mu }_{r}\left( d\right) }{d} = V\left( z\right) + V\left( z\right) \mathop{\sum }\limits_{\substack{{\delta \mid {P}_{z}} \\ {\delta > 1} }}\frac{{\psi }_{r}\left( \delta \right) }{\phi \left( \delta \right) }. \] We now want to estimate the last sum. Observe that \[ {\psi }_{r}\left( \delta \right) \leq \left( \begin{matrix} \nu \left( \delta \right) - 1 \\ r \end{matrix}\right) \] so that the sum under consideration is bounded by \[ \mathop{\sum }\limits_{\substack{{\delta \mid {P}_{z}} \\ {\delta > 1} }}\left( \begin{matrix} \nu \left( \delta \right) - 1 \\ r \end{matrix}\right) \frac{1}{\phi \left( \delta \right) } \leq \mathop{\sum }\limits_{{r \leq m \leq \pi \left( z\right) }}\left( \begin{matrix} m \\ r \end{matrix}\right) \frac{1}{m!}{\left( \mathop{\sum }\limits_{{p \leq z}}\frac{1}{p - 1}\right) }^{m} \] \[ \leq \frac{1}{r!}{\left( \log \log z + {c}_{1}\right) }^{r}\exp \left( {\log \log z + {c}_{1}}\right) \] where we have utilized the elementary estimate \[ \mathop{\sum }\limits_{{p \leq z}}\frac{1}{p} < \log \log z + {c}_{1} \] for some constant \( {c}_{1} \) . Since \( {e}^{r} \geq \frac{{r}^{r}}{r!} \), we can write \( 1/r! \leq {\left( e/r\right) }^{r} \), and thus \[ V\left( z\right) \mathop{\sum }\limits_{\substack{{\delta \mid {P}_{z}} \\ {\delta > 1} }}\frac{{\psi }_{r}\left( \delta \right) }{\phi \left( \delta \right) } \leq {c}_{2}\exp \left( {r - r\log r + r\log \Lambda }\right) \] where \( \Lambda = \log \log z + {c}_{1} \), and we have used the estimate \[ V\left( z\right) \ll \frac{1}{\log z} \] The idea is to choose \( r \) so that the \( r\log r \) term dominates. This will minimize the error term. Indeed, choosing \( r \) to be the nearest even integer to \( \alpha \log x/\log z \), with \( \alpha < 1 \), gives an error term of \[ O\left( {x\exp \left( {-{c}_{3}\frac{\log x}{\log z}}\right) }\right) \] for some constant \( {c}_{3} \), and we impose \[ \frac{\alpha \log x}{\log z} > 2\left( {\log \log z + {c}_{1}}\right) \] to ensure that the error term is sufficiently small. This proves the following theorem: Theorem 9.2.2 There is a constant \( {c}_{4} > 0 \) such that for \[ \log z < {c}_{4}\log x/\log \log x \] we have \[ \pi \left( {x, z}\right) \leq {xV}\left( z\right) + O\left( {x\exp \left( {-{c}_{3}\frac{\log x}{\log z}}\right) }\right) . \] Remark. Observe that this is comparable to the estimate obtained earlier by using the sieve of Eratosthenes combined with the careful application of Rankin's trick (Exercises 9.1.8 and 9.1.9). Also note that Theorem 9.2.2 gives us the upper bound \[ \pi \left( x\right) \ll \frac{x}{\log x}\left( {\log \log x}\right) \] which is comparable to the estimate we obtained in Exercise 9.1.10. Brun used his method described above to deduce that the number of primes \( p \leq x \) such that \( p + 2 \) is also prime is bounded by \[ \ll \frac{x}{{\left( \log x\right) }^{2}}{\left( \log \log x\right) }^{2}. \] From this, it is easy to deduce by partial summation that \[ \mathop{\sum }\limits^{\prime }\frac{1}{p} < \infty \] where \( p \) is such that \( p + 2 \) is prime, a result that created a sensation at the time it was proved by Brun. Let \( A \) be a finite set of natural numbers, \( \mathcal{P} \) a set of primes. For square-free \( d \) composed of primes from \( \mathcal{P} \), let \( {A}_{d} \) be the set of elements of \( A \) divisible by \( d \) . For some \( \omega \left( d\right) \) multiplicative, suppose \[ \left| {A}_{d}\right| = \frac{\omega \left( d\right) }{d}\left| A\right| + {R}_{d} \] Let \( S\left( {A,\mathcal{P}, z}\right) \) denote the number of elements of \( A \) coprime to \[ P\left( z\right) = \mathop{\prod }\limits_{\substack{{p \leq z} \\ {p \in \mathcal{P}} }}p \] As above \[ S\lef

Exercise 9.1.9 Prove that \[ \pi \left( {x, z}\right) = {xV}\left( z\right) + O\left( {x{\left( \log z\right) }^{2}\exp \left( {-\frac{\log x}{\log z}}\right) }\right) , \] where \[ V\left( z\right) = \mathop{\prod }\limits_{{p \leq z}}\left( {1 - \frac{1}{p}}\right) \] and \( z = z\left( x\right) \rightarrow \infty \) as \( x \rightarrow \infty \) .

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Corollary 3.10. Every convex function that is continuous on an open convex subset \( U \) of a normed space is locally Lipschitzian on \( U \) . Now let us turn to some links between convex functions and continuous affine functions. Hereinafter we say that a convex function is closed if it is lower semicontinuous and either it is identically equal to \( - \infty \) (in which case we denote it by \( \left. {-{ \sim }^{X}}\right) \) or it takes its values in \( {\mathbb{R}}_{\infty } \mathrel{\text{:=}} \mathbb{R} \cup \{ + \infty \} \) . Recall that \( f \in {\overline{\mathbb{R}}}^{X} \) is proper if \( f \) does not take the value \( - \infty \) and if it is not the constant function \( + {\infty }^{X} \) . Then its epigraph is a proper subset of \( X \times \mathbb{R} \) (i.e., is nonempty and different from the whole space). We observed that a lower semicontinuous convex function assuming the value \( - \infty \) cannot take a finite value (Exercise 1 of Sect. 1.4.1). Thus a lower semicontinuous convex function \( f \in {\overline{\mathbb{R}}}^{X} \) taking a finite value is either proper or \( + {\infty }^{X} \) . Note that given a closed convex subset \( C \) of \( X \), the function given by \( f\left( x\right) = - \infty \) for \( x \in C \) , \( f\left( x\right) = + \infty \) for \( x \in X \smallsetminus C \) is an example of a lower semicontinuous convex function that is not closed and not proper. If \( f \) is the supremum of a nonempty family of continuous affine functions, then \( f \) is either \( + {\infty }^{X} \) or a closed proper convex function. In both cases, and in the case of \( f = - {\infty }^{X} \) (which corresponds to the empty family), it is a closed convex function. A remarkable converse holds. Theorem 3.11. Every closed convex function is the supremum of a family of continuous affine functions (the ones it majorizes). If \( f \) is proper, this family is nonempty. Clearly, if \( f = + {\infty }^{X} \), one can take the family of all continuous affine functions on \( X \), while if \( f = - {\infty }^{X} \) one takes the empty family. The following lemma is the first step of the proof of this result for the case \( f \neq - {\infty }^{X} \) . Lemma 3.12. For every lower semicontinuous convex function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) there exists a continuous affine function \( g \) such that \( g \leq f \) . Moreover, if \( w \in \operatorname{dom}f \) and \( r < f\left( w\right) \), we may require that \( g\left( w\right) > r \) . Proof. The case \( f = + {\infty }^{X} \) is obvious. Let us suppose \( f \neq + {\infty }^{X} \), so that the epigraph \( {E}_{f} \) of \( f \) is nonempty. Let \( w \in \operatorname{dom}f \) and \( r < f\left( w\right) \) . The Hahn-Banach theorem allows us to separate the compact set \( \{ \left( {w, r}\right) \} \) from the closed convex set \( {E}_{f} \) : there exist \( \left( {h, c}\right) \in {X}^{ * } \times \mathbb{R} = {\left( X \times \mathbb{R}\right) }^{ * } \) and \( b \in \mathbb{R} \) such that \[ \forall \left( {x, s}\right) \in {E}_{f},\;\langle h, x\rangle + {cs} > b > \langle h, w\rangle + {cr}. \] (3.1) Taking \( x = w, s > f\left( w\right) > r \), we see that \( c > 0 \) . Dividing both sides of the first inequality by \( c \), we get \[ s > - {c}^{-1}h\left( x\right) + {c}^{-1}b\;\forall x \in \operatorname{dom}f,\forall s \geq f\left( x\right) . \] It follows that \( f \geq g \) for \( g \) given by \( g\left( x\right) \mathrel{\text{:=}} - {c}^{-1}h\left( x\right) + {c}^{-1}b \) . Moreover, the second inequality in relation (3.1) can be written \( g\left( w\right) > r \) . Now let us prove Theorem 3.11. Again, the cases \( f = + {\infty }^{X}, f = - {\infty }^{X} \) being obvious, we may suppose \( \operatorname{dom}f \neq \varnothing \) . Let \( w \in X \) and \( r < f\left( w\right) \) . If \( w \in \operatorname{dom}f \), the preceding lemma provides us with a continuous affine function \( g \leq f \) with \( g\left( w\right) > r \) . Now let us consider the case \( w \in X \smallsetminus \operatorname{dom}f \) . Separating \( \{ \left( {w, r}\right) \} \) from \( {E}_{f} \), we get some \( \left( {h, c}\right) \in {\left( X \times \mathbb{R}\right) }^{ * } \) and \( b \in \mathbb{R} \) such that relation (3.1) holds. Taking \( x \in \operatorname{dom}f \) and \( s \) large, we see that \( c \geq 0 \) . If \( c > 0 \), we can conclude as in the preceding proof. If \( c = 0 \), observing that \( b - h\left( w\right) > 0 \), taking a continuous affine function \( k \) such that \( k \leq f \) (such a function exists, by the lemma) and setting \[ g \mathrel{\text{:=}} k + n\left( {b - h}\right) , \] with \( n > {\left( b - h\left( w\right) \right) }^{-1}\left( {r - k\left( w\right) }\right) \), we see that \( g\left( w\right) > r \) and \( g \leq f \), since \( k \leq f \) and \( b - h\left( x\right) \leq 0 \) for \( x \in \operatorname{dom}f \) by relation (3.1) with \( c = 0 \) . Since lower semicontinuity is stable by the operation of taking suprema, one can deduce Theorem 3.6 from Theorem 3.11. ## 3.1.1 Supplement: Another Proof of the Robinson-Ursescu Theorem We are in a position to prove the Robinson-Ursescu theorem in the reflexive case without using the notion of ideally convex set. Theorem 3.13. Let \( W, X \) be Banach spaces, and let \( F : W \rightrightarrows X \) be a multimap with closed convex graph. If \( W \) is reflexive, then for every \( \left( {\bar{w},\bar{x}}\right) \) in (the graph of) \( F \) such that \( X = {\mathbb{R}}_{ + }\left( {F\left( W\right) - \bar{x}}\right) \), i.e., \( \bar{x} \in \operatorname{core}F\left( W\right) \), the multimap \( F \) is open at \( \left( {\bar{w},\bar{x}}\right) \) . In fact, \( F \) is open at \( \left( {\bar{w},\bar{x}}\right) \) with a linear rate in the sense that there exist some \( c > 0 \) , \( \bar{r} > 0 \) such that \[ \forall r \in \left( {0,\bar{r}}\right) ,\;B\left( {\bar{x}, r}\right) \subset F\left( {B\left( {\bar{w},{cr}}\right) }\right) . \] Proof. Let us define a function \( f : X \rightarrow {\mathbb{R}}_{\infty } \) by \[ f\left( x\right) \mathrel{\text{:=}} d\left( {\bar{w},{F}^{-1}\left( x\right) }\right) \mathrel{\text{:=}} \inf \{ \parallel w - \bar{w}\parallel : w \in W, x \in F\left( w\right) \} , \] with the convention that \( \inf \varnothing = + \infty \) . Since \( f\left( x\right) = \inf \left\{ {\parallel w - \bar{w}\parallel + {\iota }_{F}\left( {w, x}\right) : w \in W}\right\} \) and since \( \left( {w, x}\right) \mapsto \parallel w - \bar{w}\parallel + {\iota }_{F}\left( {w, x}\right) \) is convex, \( f \) is convex. Let us prove that \( f \) is lower semicontinuous on \( X \) by showing that for every \( r \in \mathbb{R} \), its sublevel set \( {S}_{f}\left( r\right) \mathrel{\text{:=}} {f}^{-1}(\left( {-\infty, r\rbrack }\right) \) is closed. Let \( \left( {x}_{n}\right) \) be a sequence of \( {S}_{f}\left( r\right) \) converging to some \( x \in X \) . Since \( X \) is reflexive and for all \( n \in \mathbb{N} \) the set \( {F}^{-1}\left( {x}_{n}\right) \) is closed, convex, hence weakly closed, there exists some \( {w}_{n} \in {F}^{-1}\left( {x}_{n}\right) \) such that \( \begin{Vmatrix}{{w}_{n} - \bar{w}}\end{Vmatrix} = f\left( {x}_{n}\right) \) . The sequence \( \left( {w}_{n}\right) \), being contained in the sequentially weakly compact ball \( B\left\lbrack {\bar{w}, r}\right\rbrack \) , has a subsequence that weakly converges to some \( w \in B\left\lbrack {\bar{w}, r}\right\rbrack \) . Since \( F \) is weakly closed in \( W \times X \), we have \( \left( {w, x}\right) \in F \), hence \( f\left( x\right) \leq \parallel w - \bar{w}\parallel \leq \mathop{\liminf }\limits_{n}\begin{Vmatrix}{{w}_{n} - \bar{w}}\end{Vmatrix} \leq r \) . Thus \( {S}_{f}\left( r\right) \) is closed and \( f \) is lower semicontinuous, hence is continuous on the core of its domain \( F\left( X\right) \) by Proposition 3.4. In fact, \( f \) is locally Lipschitzian around \( \bar{x} \) , so that there exist \( c > 0,\bar{r} > 0 \) such that \( f\left( x\right) = \left| {f\left( x\right) - f\left( \bar{x}\right) }\right| \leq {cd}\left( {x,\bar{x}}\right) \) for all \( x \in \) \( B\left( {\bar{x},\bar{r}}\right) \) . Thus for \( r \in \left( {0,\bar{r}}\right) \) and \( x \in B\left( {\bar{x}, r}\right) \), one can find \( w \in {F}^{-1}\left( x\right) \) with \( \parallel w - \bar{w}\parallel < \) \( {cr} \) : the last assertion is proved. The openness property of Theorem 3.13 can be strengthened to openness at a linear rate around \( \left( {\bar{w},\bar{x}}\right) \) . Corollary 3.14. Let \( F : W \rightrightarrows X \) be a multimap with convex graph between two normed spaces. Suppose that for some \( \rho, r > 0 \) and some \( \left( {\bar{w},\bar{x}}\right) \in F \) one has \( B\left( {\bar{x}, r}\right) \subset F\left( {B\left( {\bar{w},\rho }\right) }\right) \) . Then for every \( s \in (0, r/3\rbrack \) there exists some \( c > 0 \) such that \[ \forall \left( {w, x}\right) \in B\left( {\bar{w},\rho }\right) \times B\left( {\bar{x}, s}\right) ,\;d\left( {w,{F}^{-1}\left( x\right) }\right) \leq {cd}\left( {x, F\left( w\right) }\right) . \] Proof. Given \( s \in (0, r/3\rbrack \), let \( c \mathrel{\text{:=}} 4{\left( r - s\right) }^{-1}\rho \) and let \( \left( {w, z}\right) \in B\left( {\bar{w},\rho }\right) \times B\left( {\bar{x}, s}\right) \) with \( z \in F\left( w\right) \) . Then for every \( x \in B\left( {\bar{x}, s}\right), y \in F\left( w\right) \smallsetminus B\left( {\bar{x}, r}\right) \) one has \[ d\left( {x, y}\right) \geq d\left( {y,\bar{x}}\right) - d\left( {\bar{x}, x}\right) > r - s \geq {2s} > d\left( {x, z}\right) , \] hence \( d\left( {x, F\left( w\right) }\right) \leq d\left( {x, z}\right) < {2s} \leq d\left( {x, F\left( w\right) \smallsetminus B\left( {\bar{x}, r}\right) }\right) \) and \( d\left( {x, F\left( w\right) \cap B\left( {\bar{x}, r}\right) }\right) = \) \( d\le

Corollary 3.10. Every convex function that is continuous on an open convex subset \( U \) of a normed space is locally Lipschitzian on \( U \) .

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Proposition 3.4 The image under \( T \) of the closed unit ball of \( C\left( Y\right) \) is a relatively compact subset of \( C\left( X\right) \) . We say that \( T \) is a compact operator from \( C\left( Y\right) \) to \( C\left( X\right) \) (see Chapter 6). Proof. It is clear that \( T\left( {\bar{B}\left( {C\left( Y\right) }\right) }\right) \) is bounded by \[ M = \mu \left( Y\right) \mathop{\max }\limits_{{\left( {x, y}\right) \in X \times Y}}\left| {K\left( {x, y}\right) }\right| . \] On the other hand, \( K \) is uniformly continuous on \( X \times Y \) ; in particular, for all \( \varepsilon \), there exists \( \eta > 0 \) such that \[ \left| {K\left( {{x}_{1}, y}\right) - K\left( {{x}_{2}, y}\right) }\right| < \varepsilon \;\text{ for all }y \in Y\text{ and }{x}_{1},{x}_{2} \in X\text{ with }d\left( {{x}_{1},{x}_{2}}\right) < \eta . \] Thus, for all \( f \in \bar{B}\left( {C\left( Y\right) }\right) \), we have \( \left| {{Tf}\left( {x}_{1}\right) - {Tf}\left( {x}_{2}\right) }\right| \leq \mu \left( Y\right) \varepsilon \) . Therefore the subset \( T\left( {\bar{B}\left( {C\left( Y\right) }\right) }\right) \) of \( C\left( X\right) \) is equicontinuous, and we can apply Ascoli's Theorem. ## Exercises 1. For each \( n \in \mathbb{N} \), let \( {f}_{n} \) be the function from \( \left\lbrack {0,1}\right\rbrack \) to \( \mathbb{R} \) defined by \( {f}_{n}\left( x\right) = {x}^{n} \) . At what points in the interval \( \left\lbrack {0,1}\right\rbrack \) is the family \( {\left\{ {f}_{n}\right\} }_{n \in \mathbb{N}} \) equicontinuous? 2. a. Let \( X \) be a metric space and \( \left( {f}_{n}\right) \) a sequence in \( C\left( X\right) \) . Prove that, if \( {\left\{ {f}_{n}\right\} }_{n \in \mathbb{N}} \) is equicontinuous at a point \( x \) of \( X \), for any sequence \( \left( {x}_{n}\right) \) of \( X \) that converges to \( x \) the sequence \( \left( {{f}_{n}\left( x\right) - {f}_{n}\left( {x}_{n}\right) }\right) \) converges to 0 . b. Set \( {f}_{n}\left( x\right) = \sin {nx} \) . Prove that \( {\left\{ {f}_{n}\right\} }_{n \in \mathbb{N}} \) is not equicontinuous at any point \( x \) of \( \mathbb{R} \) . Hint. Consider the sequence \( \left( {x}_{n}\right) \) defined by \( {x}_{n} = x + \pi /\left( {2n}\right) \) . 3. Let \( X \) be a compact metric space. Prove that, if \( H \) is an equicontinuous subset of \( C\left( X\right) \), the closure \( \bar{H} \) of \( H \) in \( C\left( X\right) \) is equicontinuous. 4. Let \( X \) be a compact metric space, and let \( H \) be an equicontinuous family of elements of \( C\left( X\right) \) . a. Prove that the set of points \( x \) of \( X \) such that the set \( \{ f\left( x\right) : f \in H\} \) is bounded is open and closed. b. Assume that \( X \) is connected. Prove that, if there exists a point \( x \in X \) for which \( \{ f\left( x\right) : f \in H\} \) is bounded, \( H \) is a relatively compact subset of \( C\left( X\right) \) . 5. a. For \( \alpha \in \left( {0,1}\right) \), let \( {C}^{\alpha }\left( \left\lbrack {0,1}\right\rbrack \right) \) be the set of functions \( f \) from \( \left\lbrack {0,1}\right\rbrack \) to \( \mathbb{R} \) such that \[ {\left| f\right| }_{\alpha } = \mathop{\sup }\limits_{\substack{{0 \leq x, y \leq 1} \\ {x \neq y} }}\frac{\left| f\left( x\right) - f\left( y\right) \right| }{{\left| x - y\right| }^{\alpha }} \] is finite (such an \( f \) is called a Hölder function of exponent \( \alpha \) ). As usual, we denote by \( \parallel \cdot \parallel \) the uniform norm. i. Prove that \( {C}^{\alpha }\left( \left\lbrack {0,1}\right\rbrack \right) \), with the norm \( \parallel \cdot {\parallel }_{\alpha } = {\left| \cdot \right| }_{\alpha } + \parallel \cdot \parallel \), is a Banach space. ii. Prove that \( \bar{B}\left( {{C}^{\alpha }\left( \left\lbrack {0,1}\right\rbrack \right) }\right) \), the closed unit ball in \( {C}^{\alpha }\left( \left\lbrack {0,1}\right\rbrack \right) \), is a compact subset of \( C\left( \left\lbrack {0,1}\right\rbrack \right) \) . iii. Suppose \( 1 > \beta > \alpha > 0 \) . A. Take \( f \in {C}^{\beta }\left( \left\lbrack {0,1}\right\rbrack \right) \) . Prove that, for all \( \eta > 0 \) , \[ {\left| f\right| }_{\alpha } \leq \max \left( {{\left| f\right| }_{\beta }{\eta }^{\beta - \alpha },2\parallel f\parallel {\eta }^{-\alpha }}\right) . \] Deduce that, if \( \left( {f}_{n}\right) \) is a bounded sequence in \( {C}^{\beta } \) that converges uniformly to \( f \in {C}^{\beta } \), then \( {\begin{Vmatrix}{f}_{n} - f\end{Vmatrix}}_{\alpha } \rightarrow 0 \) . B. Deduce that \( \bar{B}\left( {{C}^{\beta }\left( \left\lbrack {0,1}\right\rbrack \right) }\right) \) is compact in \( {C}^{\alpha }\left( \left\lbrack {0,1}\right\rbrack \right) \) . b. Let \( m \) be a nonnegative integer. We give \( {C}^{m}\left( \left\lbrack {0,1}\right\rbrack \right) \) the norm defined by \[ \parallel f{\parallel }_{m} = \mathop{\sum }\limits_{{k = 0}}^{m}\mathop{\sup }\limits_{{x \in \left\lbrack {0,1}\right\rbrack }}\left| {{f}^{\left( k\right) }\left( x\right) }\right| . \] i. Prove that with this norm \( {C}^{m}\left( \left\lbrack {0,1}\right\rbrack \right) \) is a Banach space. ii. Prove that if \( m \) and \( n \) are nonnegative integers such that \( m > n \) , then \( \bar{B}\left( {{C}^{m}\left( \left\lbrack {0,1}\right\rbrack \right) }\right) \) is a relatively compact subset of \( {C}^{n}\left( \left\lbrack {0,1}\right\rbrack \right) \) . (You might start with \( m = 1 \) and \( n = 0 \) .) Is the ball \( \bar{B}\left( {{C}^{m}\left( \left\lbrack {0,1}\right\rbrack \right) }\right) \) closed in \( {C}^{n}\left( \left\lbrack {0,1}\right\rbrack \right) \) ? c. Take \( m \in \mathbb{N} \) and \( \alpha \in \left( {0,1}\right) \) . Denote by \( {C}^{m + \alpha }\left( \left\lbrack {0,1}\right\rbrack \right) \) the vector space consisting of functions of \( {C}^{m}\left( \left\lbrack {0,1}\right\rbrack \right) \) whose \( m \) -th derivative is an element of \( {C}^{\alpha }\left( \left\lbrack {0,1}\right\rbrack \right) \), and define on this vector space a norm \( \parallel \cdot {\parallel }_{m + \alpha } \) by setting \( \parallel f{\parallel }_{m + \alpha } = \parallel f{\parallel }_{m} + {\left| {f}^{\left( m\right) }\right| }_{\alpha } \) . i. Prove that \( {C}^{m + \alpha }\left( \left\lbrack {0,1}\right\rbrack \right) \), with the norm \( \parallel \cdot {\parallel }_{m + \alpha } \), is a Banach space. ii. Take \( p, q \in \mathbb{R} \) such that \( q > p \geq 0 \) . Prove that \( \bar{B}\left( {{C}^{q}\left( \left\lbrack {0,1}\right\rbrack \right) }\right) \) is a relatively compact subset of \( {C}^{p}\left( \left\lbrack {0,1}\right\rbrack \right) \) . 6. Ascoli’s Theorem in \( \mathbb{R} \) a. Let \( {f}_{n} \) be the function defined for all \( x \in \mathbb{R} \) by \[ {f}_{n}\left( x\right) = \left\{ \begin{array}{ll} \min \left( {1, n/x}\right) & \text{ if }x \neq 0 \\ 1 & \text{ if }x = 0 \end{array}\right. \] Prove that the subset \( {\left\{ {f}_{n}\right\} }_{n \in \mathbb{N}} \) of \( {C}_{0}\left( \mathbb{R}\right) \) is bounded and equicontinuous (see Exercise 8 on page 40 for the definition of \( {C}_{0}\left( \mathbb{R}\right) \) ), but the sequence \( \left( {f}_{n}\right) \) has no uniformly convergent subsequence. Hint. The sequence \( \left( {f}_{n}\right) \) converges pointwise but not uniformly to the constant function 1 . b. Let \( H \) be a subset of \( {C}_{0}\left( \mathbb{R}\right) \) . Prove that \( H \) is relatively compact in \( {C}_{0}\left( \mathbb{R}\right) \) if and only if it is bounded and equicontinuous at every point of \( \mathbb{R} \) and satisfies that condition that for any \( \varepsilon > 0 \) there exists \( A > 0 \) such that \[ \left| {h\left( x\right) }\right| < \varepsilon \text{ for all }h \in H\text{ and }x \in \mathbb{R}\text{ with }\left| x\right| \geq A. \] Hint. Use Ascoli’s Theorem in the space \( C\left( \mathbb{U}\right) \) (refer again to Exercise 8 on page 40). 7. A particular case of Peano’s Theorem. Let \( f \) be a continuous function from \( \left\lbrack {0,1}\right\rbrack \times \mathbb{R} \) to \( \mathbb{R} \) for which there exists a constant \( M > 0 \) such that \[ \left| {f\left( {x, t}\right) }\right| \leq M\left( {1 + \left| x\right| }\right) \;\text{ for all }t \in \left\lbrack {0,1}\right\rbrack \text{ and }x \in \mathbb{R}. \] a. Let \( n \) be a positive integer. We define points \( {x}_{j}^{n} \), for \( 0 \leq j \leq n \), by setting \( {x}_{0}^{n} = 0 \) and \[ {x}_{j + 1}^{n} = {x}_{j}^{n} + \frac{1}{n}f\left( {\frac{j}{n},{x}_{j}^{n}}\right) \;\text{ for }0 \leq j \leq n - 1. \] i. Prove that \( \left| {x}_{j}^{n}\right| \leq {\left( 1 + M/n\right) }^{j} - 1 \leq {e}^{M} - 1 \) for \( 0 \leq j \leq n \) . ii. Let \( {\varphi }_{n} \) be the continuous function on \( \left\lbrack {0,1}\right\rbrack \) that is affine on each interval \( \left\lbrack {j/n,\left( {j + 1}\right) /n}\right\rbrack \) and satisfies \( {\varphi }_{n}\left( {j/n}\right) = {x}_{j}^{n} \) for \( 0 \leq j \leq \) \( n \) . That is, for \( 0 \leq j \leq n - 1 \) and \( t \in \left\lbrack {j/n,\left( {j + 1}\right) /n}\right\rbrack \) we have \[ {\varphi }_{n}\left( t\right) = {x}_{j}^{n} + \left( {t - \frac{j}{n}}\right) f\left( {\frac{j}{n},{x}_{j}^{n}}\right) . \] Prove that for \( s, t \in \left\lbrack {0,1}\right\rbrack \) we have \( \left| {{\varphi }_{n}\left( t\right) - {\varphi }_{n}\left( s\right) }\right| \leq M{e}^{M}\left| {t - s}\right| \) . iii. For \( s \in \left\lbrack {0,1}\right\rbrack \), set \[ {\psi }_{n}\left( s\right) = \mathop{\sum }\limits_{{j = 0}}^{{n - 1}}{1}_{\lbrack j/n,\left( {j + 1}\right) /n)}\left( s\right) f\left( {\frac{j}{n},{\varphi }_{n}\left( \frac{j}{n}\right) }\right) . \] Prove that \( {\varphi }_{n}\left( t\right) = {\int }_{0}^{t}{\psi }_{n}\left( s\right) {ds} \) for all \( t \in \left\lbrack {0,1}\right\rbrack \) . b. i. Show that there exist

Proposition 3.4 The image under \( T \) of the closed unit ball of \( C\left( Y\right) \) is a relatively compact subset of \( C\left( X\right) \) .

It is clear that \( T\left( {\bar{B}\left( {C\left( Y\right) }\right) }\right) \) is bounded by \[ M = \mu \left( Y\right) \mathop{\max }\limits_{{\left( {x, y}\right) \in X \times Y}}\left| {K\left( {x, y}\right) }\right| . \] On the other hand, \( K \) is uniformly continuous on \( X \times Y \) ; in particular, for all \( \varepsilon \), there exists \( \eta > 0 \) such that \[ \left| {K\left( {{x}_{1}, y}\right) - K\left( {{x}_{2}, y}\right) }\right| < \varepsilon \;\text{ for all }y \in Y\text{ and }{x}_{1},{x}_{2} \in X\text{ with }d\left( {{x}_{1},{x}_{2}}\right) < \eta . \] Thus, for all \( f \in \bar{B}\left( {C\left( Y\right) }\right) \), we have \( \left| {{Tf}\left( {x}_{1}\right) - {Tf}\left( {x}_{2}\right) }\right| \leq \mu \left( Y\right) \varepsilon \) . Therefore the subset \( T\left( {\bar{B}\left( {C\left( Y\right) }\right) }\right) \) of \( C\left( X\right) \) is equicontinuous, and we can apply Ascoli's Theorem.

Lemma 7.1.3. Let \( p \geq 3 \), let \( K = {\mathbb{F}}_{p} \), let \( E \) be a degenerate curve over \( {\mathbb{F}}_{p} \) as above, and let \( {c}_{6} = {c}_{6}\left( E\right) \) be the invariant defined in Section 7.1.2. Then \( E \) has a cusp (respectively a double point with tangents defined over \( {\mathbb{F}}_{p} \) , respectively a double point with tangents not defined over \( K \) ) if and only if \( \left( \frac{-{c}_{6}}{p}\right) = 0 \) (respectively \( \left( \frac{-{c}_{6}}{p}\right) = 1 \), respectively \( \left( \frac{-{c}_{6}}{p}\right) = - 1 \) ). Proof. Since \( E \) is degenerate it has a singular point, and changing coordinates we may assume that it is at the origin, so that we can choose the equation of our curve to be \( {y}^{2} = {x}^{2}\left( {x + a}\right) \) for some \( a \in {\mathbb{F}}_{p} \) . One computes that \( {c}_{6}\left( E\right) = - {64}{a}^{3} \) . On the other hand, the general equation of a line through the origin is \( y = {tx} \), so such a line is tangent if and only if \( {t}^{2} = a \) . Thus, we have a cusp when \( a = 0 \) (so that \( \left( \frac{{c}_{6}}{p}\right) = 0 \) ), and the tangents are defined over \( {\mathbb{F}}_{p} \) if and only if \( a \) is a square; in other words, \( \left( \frac{-{c}_{6}}{p}\right) = 1 \) since \( - {c}_{6} = {\left( 8a\right) }^{2}a. \) ## 7.1.4 The Group Law In the sequel we let \( E \) be an elliptic curve defined over a field \( K \) by a generalized Weierstrass equation given in affine form as \( {y}^{2} + {a}_{1}{xy} + {a}_{3}y = \) \( {x}^{3} + {a}_{2}{x}^{2} + {a}_{4}x + {a}_{6} \) . One of the most important properties of elliptic curves is that there is a natural abelian group law on \( E \) that is defined by rational equations. More precisely, we have the following: Theorem 7.1.4. Let \( {y}^{2} + {a}_{1}{xy} + {a}_{3}y = {x}^{3} + {a}_{2}{x}^{2} + {a}_{4}x + {a}_{6} \) be the generalized Weierstrass equation of an elliptic curve \( E \) (hence nonsingular). We define an addition law on \( E \) by asking that the point at infinity \( \mathcal{O} \) be the identity element, and that \( {P}_{1} + {P}_{2} + {P}_{3} = \mathcal{O} \) if and only if the points \( {P}_{i} \) are (projectively) collinear. (1) This addition law defines an abelian group structure on \( E \) . (2) If \( {P}_{1} = \left( {{x}_{1},{y}_{1}}\right) \) and \( {P}_{2} = \left( {{x}_{2},{y}_{2}}\right) \) are two points on \( E \) different from \( \mathcal{O} \) , their sum is equal to \( \mathcal{O} \) if \( {x}_{1} = {x}_{2} \) and \( {y}_{2} = - {y}_{1} - {a}_{1}x - {a}_{3} \), and otherwise is given as follows. Set \[ m = \left\{ \begin{array}{ll} \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} & \text{ if }{P}_{1} \neq {P}_{2}, \\ \frac{3{x}_{1}^{2} + 2{a}_{2}{x}_{1} + {a}_{4} - {a}_{1}{y}_{1}}{2{y}_{1} + {a}_{1}{x}_{1} + {a}_{3}} & \text{ if }{P}_{1} = {P}_{2}. \end{array}\right. \] Then \( {P}_{1} + {P}_{2} = {P}_{3} = \left( {{x}_{3},{y}_{3}}\right) \) with \( {x}_{3} = m\left( {m + {a}_{1}}\right) - {x}_{1} - {x}_{2} - {a}_{2} \) and \( {y}_{3} = m\left( {{x}_{1} - {x}_{3}}\right) - {y}_{1} - {a}_{1}{x}_{3} - {a}_{3}. \) Proof. The proof of (1) is very classical and not too difficult, although associativity is painful if one tries to prove it directly from the formulas given in (2), but it is immediate in terms of divisors; see Exercise 6. The formulas of (2) follow from an immediate computation; see Exercise 5. Remarks. (1) If we write the equation as \( f\left( {x, y}\right) = 0 \), the case \( {P}_{1} = {P}_{2} \) gives \( m = - {f}_{x}^{\prime }/{f}_{y}^{\prime } \) as it should. (2) In the case of a simple Weierstrass equation \( {y}^{2} = {x}^{3} + {ax} + b \), the formulas reduce to \( m = \left( {{y}_{2} - {y}_{1}}\right) /\left( {{x}_{2} - {x}_{1}}\right) \) if \( {P}_{1} \neq {P}_{2}, m = \left( {3{x}_{1}^{2} + a}\right) /\left( {2{y}_{1}}\right) \) if \( {P}_{1} = {P}_{2} \), and \( {x}_{3} = {m}^{2} - {x}_{1} - {x}_{2},{y}_{3} = m\left( {{x}_{1} - {x}_{3}}\right) - {y}_{1} \) . (3) The opposite of \( \left( {x, y}\right) \) is \( \left( {x, - y - {a}_{1}x - {a}_{3}}\right) \) (hence \( \left( {x, - y}\right) \) in the simple case), and in particular the points of order dividing 2 are \( \mathcal{O} \) together with those such that \( {2y} + {a}_{1}x + {a}_{3} = 0 \) (hence \( y = 0 \) in the simple case). When the equation defines a singular curve, we still have a group law, but now on the set of nonsingular points: Proposition 7.1.5. Let \( {y}^{2} = {x}^{3} + {a}_{2}{x}^{2} + {a}_{4}x + {a}_{6} \) be a singular curve \( E \) with singular point \( {P}_{0} = \left( {\alpha ,0}\right) \) . The set \( G = E\left( K\right) \smallsetminus \left\{ {P}_{0}\right\} \) has a natural group structure, and furthermore \( G \simeq \left( {K, + }\right) \) when \( {P}_{0} \) is a cusp, \( G \simeq \left( {{K}^{ * }, \times }\right) \) when \( {P}_{0} \) is a double point with distinct tangents defined over \( K \), and finally \( G \) is isomorphic to the multiplicative group of elements of relative norm 1 in the quadratic extension of \( K \) generated by the slopes of the tangents at \( {P}_{0} \) when \( {P}_{0} \) is a double point with distinct tangents not defined over \( K \) . Proof. As in the preceding section we may assume that our equation has the form \( {y}^{2} = {x}^{2}\left( {x - \beta }\right) \) and is parametrized by \( \left( {\beta + 1/{t}^{2},\beta /t + 1/{t}^{3}}\right) \) . Thus if the points \( {P}_{1} \) and \( {P}_{2} \) (which we may assume to be different from the point at infinity) correspond to the respective parameters \( {t}_{1} \) and \( {t}_{2} \), a short computation using the formulas given above, which are still valid in the nonsingular case, shows that \( {P}_{3} = {P}_{1} + {P}_{2} \) is a nonsingular point that corresponds to the parameter \[ {t}_{3} = \frac{{t}_{1} + {t}_{2}}{1 - \beta {t}_{1}{t}_{2}} \] Since one also checks that \[ \beta + \frac{1}{{t}_{3}^{2}} = \frac{{t}_{1}^{2}{t}_{2}^{2}\left( {\beta + 1/{t}_{1}^{2}}\right) \left( {\beta + 1/{t}_{2}^{2}}\right) }{{\left( {t}_{1} + {t}_{2}\right) }^{2}}, \] it follows that \( {t}_{3} \) cannot correspond to the singular point (recall that \( {t}_{i} \neq 0 \) since we assume that the points \( {P}_{i} \) are not at infinity). The results in the elliptic curve case thus imply that we have an abelian group law on the nonsingular points. However, as stated, we can say more about the group structure \( G \) : if \( \beta = 0 \), we simply have \( {t}_{3} = {t}_{1} + {t}_{2} \), so evidently \( G \simeq \left( {K, + }\right) \) (the excluded value of \( t \) being \( \infty \) ). If \( - \beta = {\gamma }^{2} \) is a square in \( {K}^{ * } \) (the case in which we have two distinct tangents defined over \( K \) ), we set \( {u}_{i} = \left( {1 + \gamma {t}_{i}}\right) /\left( {1 - \gamma {t}_{i}}\right) \) , in other words \( {t}_{i} = \left( {{u}_{i} - 1}\right) /\left( {\gamma \left( {{u}_{i} + 1}\right) }\right) \) . An easy computation shows that \( {u}_{3} = \) \( {u}_{1}{u}_{2} \), so \( G \simeq \left( {{K}^{ * }, \cdot }\right) \) (the excluded values of \( u \) being 0 and \( \infty \), corresponding to \( 1/{t}^{2} = - \beta \) ). If \( - \beta \) is not a square in \( K \), we let \( L = K\left( \gamma \right) \) with \( {\gamma }^{2} = - \beta \) . The group law on \( G \) still corresponds to the multiplicative group law on \( {L}^{ * } \) , but now the acceptable values of \( u \) are such that \( u\bar{u} = 1 \), where \( \bar{u} \) is the Galois conjugate of \( u \) over \( K \) . Conversely, if \( u\bar{u} = 1 \) then by Hilbert’s Theorem 90 (which is here trivial by setting \( \alpha = u + u\bar{u} = u + 1 \) ), there exists \( \alpha \in {L}^{ * } \) such that \( u = \alpha /\bar{\alpha } \) . Writing \( \alpha = s + {\gamma t} \) with \( s \) and \( t \) in \( K \), we note that if \( s \neq 0 \), dividing by \( s \) if necessary we may assume that \( s = 1 \), hence that \( u = \left( {1 + {\gamma t}}\right) /\left( {1 - {\gamma t}}\right) \) as claimed. But the case \( s = 0 \) corresponds to \( t = \infty \) , giving the value \( u = - 1 \) . To summarize, in this last case \( G \) is isomorphic to the multiplicative group of elements of \( K\left( \gamma \right) \) of norm 1 . Because of this proposition, the case that \( {P}_{0} \) is a cusp is also called a case of additive reduction, while the case that \( {P}_{0} \) is a double point with distinct tangents defined over \( K \) (respectively not defined over \( K \) ) is called a case of split multiplicative reduction (respectively nonsplit multiplicative reduction). Example. Assume that \( K = {\mathbb{F}}_{q} \) . (1) In the case of additive reduction we have \( G \simeq {\mathbb{F}}_{q} \), a cyclic group of order \( q \), so that \( \left| {E\left( {\mathbb{F}}_{q}\right) }\right| = q + 1 \) . (2) In the case of split multiplicative reduction we have \( G \simeq {\mathbb{F}}_{q}^{ * } \), a cyclic group of order \( q - 1 \), so that \( \left| {E\left( {\mathbb{F}}_{q}\right) }\right| = q \) . (3) In the case of nonsplit multiplicative reduction, up to isomorphism \( {\mathbb{F}}_{{q}^{2}} \) is the unique quadratic extension of \( K \), and since \( {\mathcal{N}}_{{\mathbb{F}}_{{q}^{2}}/{\mathbb{F}}_{q}}\left( x\right) = {x}^{q + 1} \), the group \( G \) isomorphic to the unique (cyclic) subgroup of \( {\mathbb{F}}_{{q}^{2}}^{ * } \) of order \( q + 1 \) , so that \( \left| {E\left( {\mathbb{F}}_{q}\right) }\right| = q + 2 \) . ## 7.1.5 Isogenies As usual in mathematics, when a class of objects is defined, we must also define and study the natural maps between these objects (this is exactly the definition of a category, the maps being called morphisms). An elliptic curve \( E \) has three natural structures. The first one is the definition itself, which says that \( E \) is an algebraic curve (of genus 1 with a rational point), whether

Lemma 7.1.3. Let \( p \geq 3 \), let \( K = {\mathbb{F}}_{p} \), let \( E \) be a degenerate curve over \( {\mathbb{F}}_{p} \) as above, and let \( {c}_{6} = {c}_{6}\left( E\right) \) be the invariant defined in Section 7.1.2. Then \( E \) has a cusp (respectively a double point with tangents defined over \( {\mathbb{F}}_{p} \), respectively a double point with tangents not defined over \( K \) ) if and only if \( \left( \frac{-{c}_{6}}{p}\right) = 0 \) (respectively \( \left( \frac{-{c}_{6}}{p}\right) = 1 \), respectively \( \left( \frac{-{c}_{6}}{p}\right) = - 1 \) ).

Since \( E \) is degenerate it has a singular point, and changing coordinates we may assume that it is at the origin, so that we can choose the equation of our curve to be \( {y}^{2} = {x}^{2}\left( {x + a}\right) \) for some \( a \in {\mathbb{F}}_{p} \). One computes that \( {c}_{6}\left( E\right) = - {64}{a}^{3} \). On the other hand, the general equation of a line through the origin is \( y = {tx} \), so such a line is tangent if and only if \( {t}^{2} = a \). Thus, we have a cusp when \( a = 0 \) (so that \( \left( \frac{{c}_{6}}{p}\right) = 0 \) ), and the tangents are defined over \( {\mathbb{F}}_{p} \) if and only if \( a \) is a square; in other words, \( \left( \frac{-{c}_{6}}{p}\right) = 1 \) since \( - {c}_{6} = {\left( 8a\right) }^{2}a. \)

Proposition 11.4. Let \( T = {\left( {S}^{1}\right) }^{k} \) and let \( t = \left( {{e}^{{2\pi }{\theta }_{1}},\ldots ,{e}^{{2\pi }{\theta }_{k}}}\right) \) be an element of \( T \) . Then \( t \) generates a dense subgroup of \( T \) if and only if the numbers \[ 1,{\theta }_{1},\ldots ,{\theta }_{k} \] are linearly independent over the field \( \mathbb{Q} \) of rational numbers. The \( k = 1 \) case of this result is Exercise 9 in Chapter 1. In particular, if \( x \) is any transcendental real number and we define \( {\theta }_{j} = {x}^{j}, j = 1,\ldots, k \), then \( t \) will generate a dense subgroup of \( T \) . See Figure 11.2 for an example of an element that generates a dense subgroup of \( {S}^{1} \times {S}^{1} \) . Lemma 11.5. If \( T \) is a torus and \( t \) is an element of \( T \), then the subgroup generated by \( t \) is not dense in \( T \) if and only if there exists a nonconstant homomorphism \( \Phi \) : \( T \rightarrow {S}^{1} \) such that \( \Phi \left( t\right) = 1 \) . Proof. Suppose first that there exists a nonconstant homomorphism \( \Phi : T \rightarrow {S}^{1} \) with \( T\left( t\right) = 1 \) . Then \( \ker \left( \Phi \right) \) is a closed subgroup of \( T \) that contains \( t \) and thus the group generated by \( t \) . But since \( \Phi \) is nonconstant, \( \ker \left( \Phi \right) \neq T \), which means that the closure of the group generated by \( t \) is not all of \( T \) . In the other direction, let \( S \) be the closure of the group generated by \( t \) and suppose that \( S \) is not all of \( T \) . We will proceed by describing the preimage of \( S \) under the exponential map, using an extension of Lemma 11.3. Thus, let \( \Lambda \) be the set of all \( H \in \mathfrak{t} \) such that \( {e}^{2\pi H} \in S \) . Since \( S \) is a closed subgroup of \( T \), the set \( \Lambda \) will be a closed subgroup of the additive group \( \mathfrak{t} \) . Now let \( {\Lambda }_{0} \) be the identity component of \( \Lambda \), which must be a subspace of \( \mathfrak{t} \) . (Indeed, by Corollaries 3.47 and \( {3.52},{\Lambda }_{0} \) must be equal to the Lie algebra of \( \Lambda \) .) Since \( S \) is not all of \( T \), the subspace \( {\Lambda }_{0} \) cannot be all of \( \mathrm{t} \) . The entire group \( \Lambda \) now decomposes as \( {\Lambda }_{0} \times {\Lambda }_{1} \), where \[ {\Lambda }_{1} \mathrel{\text{:=}} \Lambda \cap {\left( {\Lambda }_{0}\right) }^{ \bot } \]  Fig. 11.2 A portion of the dense subgroup generated by \( t \) in \( {S}^{1} \times {S}^{1} \) is a closed subgroup of \( {\left( {\Lambda }_{0}\right) }^{ \bot } \) . Furthermore, the identity component of \( {\Lambda }_{1} \) must be trivial, which means that the Lie algebra of \( {\Lambda }_{1} \) must be \( \{ 0\} \) . Thus, by Theorem 3.42, \( {\Lambda }_{1} \) is discrete. Let us now define a homomorphism \( \phi : \mathfrak{t} \rightarrow {S}^{1} \) by setting \[ \phi \left( H\right) = {e}^{{2\pi i\xi }\left( H\right) } \] for some linear functional \( \xi \) on \( \mathfrak{t} \) . By Lemma 11.3, \( {\Lambda }_{1} \) is the integer span of linearly independent vectors \( {v}_{1},\ldots ,{v}_{l} \) . Since \( {\Lambda }_{0} \neq \mathfrak{t} \), we can arrange things so that \( \xi \) is zero on \( {\Lambda }_{0} \), the values of \( \xi \) on \( {v}_{1},\ldots ,{v}_{l} \) are integers, but \( \xi \) is not identically zero. Then \( \ker \left( \phi \right) \) will contain \( \Lambda \), and, in particular, the kernel of the map \( H \mapsto {e}^{2\pi H} \) . Thus, there is a nonconstant, continuous homomorphism \( \Phi : T \rightarrow {S}^{1} \) satisfying \[ \Phi \left( {e}^{2\pi H}\right) = \phi \left( H\right) \] for all \( H \in \mathfrak{t} \) . If we choose \( H \) so that \( {e}^{2\pi H} = t \in S \), then \( H \in \Lambda \), which means that \[ \Phi \left( t\right) = \Phi \left( {e}^{2\pi H}\right) = 1 \] but \( \Phi \) is not constant. Proof of Proposition 11.4. In light of Lemma 11.5, we may reformulate the proposition as follows: The numbers \( 1,{\theta }_{1},\ldots ,{\theta }_{k} \) are linearly dependent over \( \mathbb{Q} \) if and only if there exists a nonconstant homomorphism \( \Phi : T \rightarrow {S}^{1} \) with \( \left( {{e}^{{2\pi i}{\theta }_{1}},\ldots ,{e}^{{2\pi i}{\theta }_{k}}}\right) \in \ker \left( \Phi \right) \) . Suppose first that there is a dependence relation among \( 1,{\theta }_{1},\ldots ,{\theta }_{k} \) over \( \mathbb{Q} \) . Then after clearing the denominators from this relation, we find that there exist integers \( {m}_{1},\ldots ,{m}_{k} \), not all zero, such that \[ {m}_{1}{\theta }_{1} + \cdots + {m}_{k}{\theta }_{k} \in \mathbb{Z}. \] Thus, we may define a nonconstant \( \Phi : T \rightarrow {S}^{1} \) by \[ \Phi \left( {{u}_{1},\ldots ,{u}_{k}}\right) = {u}_{1}^{{m}_{1}}\cdots {u}_{k}^{{m}_{k}} \] (11.1) and the kernel of \( \Phi \) will contain \( \left( {{e}^{{2\pi i}{\theta }_{1}},\ldots ,{e}^{{2\pi i}{\theta }_{k}}}\right) \) . In the other direction, Exercise 2 tells us that every continuous homomorphism \( \Phi : T \rightarrow {S}^{1} \) is of the form (11.1) for some set of integers \( {m}_{1},\ldots ,{m}_{k} \) . Furthermore, if \( \Phi \) is nonconstant, these integers cannot all be zero. Thus, if \[ 1 = \Phi \left( {{e}^{{2\pi i}{\theta }_{1}},\ldots ,{e}^{{2\pi i}{\theta }_{k}}}\right) = {e}^{{2\pi i}\left( {{m}_{1}{\theta }_{1} + \cdots + {m}_{k}{\theta }_{k}}\right) }, \] we must have \( {m}_{1}{\theta }_{1} + \cdots + {m}_{k}{\theta }_{k} = n \) for some integer \( n \), which implies that \( 1,{\theta }_{1},\ldots ,{\theta }_{k} \) are linearly dependent over \( \mathbb{Q} \) . ## 11.2 Maximal Tori and the Weyl Group In this section, we introduce a the concept of a maximal torus, which plays the same role in the compact group approach to representation theory as the Cartan subalgebra plays in the Lie algebra approach. Definition 11.6. A subgroup \( T \) of \( K \) is a torus if \( T \) is isomorphic to \( {\left( {S}^{1}\right) }^{k} \) for some \( k \) . A subgroup \( T \) of \( K \) is a maximal torus if it is a torus and is not properly contained in any other torus in \( K \) . If \( K = \mathrm{{SU}}\left( n\right) \), we may consider \[ T = \left\{ {\left. \left( \begin{matrix} {e}^{i{\theta }_{1}} & & & \\ & \ddots & & \\ & & {e}^{i{\theta }_{n - 1}} & \\ & & & {e}^{-i\left( {{\theta }_{1} + \cdots + {\theta }_{n - 1}}\right) } \end{matrix}\right) \right| \;{\theta }_{j} \in \mathbb{R}}\right\} , \] which is a torus of dimension \( n - 1 \) . If \( T \) is contained in another torus \( S \subset \mathrm{{SU}}\left( n\right) \) , then every element \( s \) of \( S \) would commute with every element \( t \) of \( T \) . If we choose \( t \) to have distinct eigenvalues, then by Proposition A.2, \( s \) would have to be diagonal in the standard basis, meaning that \( s \in T \) . Thus, \( T \) is actually a maximal torus. Proposition 11.7. If \( T \) is a maximal torus, the Lie algebra \( \mathfrak{t} \) of \( T \) is a maximal commutative subalgebra of \( \mathfrak{k} \) . Conversely, if \( \mathfrak{t} \) is a maximal commutative subalgebra of \( \mathfrak{k} \), the connected Lie subgroup \( T \) of \( \mathfrak{k} \) with Lie algebra \( \mathfrak{t} \) is a maximal torus. Proof. If \( T \) is a maximal torus, it is commutative, which means that its Lie algebra \( \mathrm{t} \) is also commutative (Proposition 3.22). Suppose \( t \) is contained in a commutative subalgebra \( \mathfrak{s} \) . Then it is also contained in a maximal commutative subalgebra \( {\mathfrak{s}}^{\prime } \) containing \( \mathfrak{s} \) . The connected Lie subgroup \( {S}^{\prime } \) with Lie algebra \( {\mathfrak{s}}^{\prime } \) must be commutative (since \( {S}^{\prime } \) is generated by exponentials of elements of \( {\mathfrak{s}}^{\prime } \) ) and closed (Proposition 5.24) and hence compact. Thus, by Theorem 11.2, \( {S}^{\prime } \) is a torus. Since \( T \) is a maximal torus, we must have \( {S}^{\prime } = T \) and thus \( {\mathfrak{s}}^{\prime } = \mathfrak{s} = \mathfrak{t} \), showing that \( \mathfrak{t} \) is maximal commutative. In the other direction, if \( \mathfrak{t} \) is maximal commutative, the connected Lie subgroup \( T \) with Lie algebra \( \mathfrak{t} \) is closed (Proposition 5.24), hence compact. But \( T \) is also commutative and connected, hence a torus, by Theorem 11.2. If \( T \) is contained in a torus \( S \), then \( t \) is contained in the commutative Lie algebra \( \mathfrak{s} \) of \( S \) . Since \( t \) is maximal commutative, we have \( \mathfrak{s} = \mathfrak{t} \) and since \( S \) is connected, \( S = T \), showing that \( T \) is a maximal torus. Definition 11.8. If \( T \) is a maximal torus in \( K \), then the normalizer of \( T \), denoted \( N\left( T\right) \), is the group of elements \( x \in K \) such that \( {xT}{x}^{-1} = T \) . The quotient group \[ W \mathrel{\text{:=}} N\left( T\right) /T \] ## is the Weyl group of \( T \) . Note that \( T \) is, almost by definition, a normal subgroup of \( N\left( T\right) \) . If \( w \) is an element of \( W \) represented by \( x \in N\left( T\right) \), then \( w \) acts on \( T \) by the formula \[ w \cdot t = {xt}{x}^{-1},\;t \in T. \] If \( x \in N\left( T\right) \), the conjugation action of \( x \) maps \( T \) onto \( T \) . It follows that \( {\operatorname{Ad}}_{x} \) maps the Lie algebra \( \mathfrak{t} \) of \( T \) into itself. We define an action of \( W \) on \( \mathfrak{t} \) by \[ w \cdot H = {\operatorname{Ad}}_{x}\left( H\right) ,\;H \in \mathfrak{t}. \] (11.2) Since our inner product is invariant under the adjoint action of \( K \), the action of \( W \) on \( t \) is by orthogonal linear transformations. We will see in Sect. 11.7 that the centraliz

Proposition 11.4. Let \( T = {\left( {S}^{1}\right) }^{k} \) and let \( t = \left( {{e}^{{2\pi }{\theta }_{1}},\ldots ,{e}^{{2\pi }{\theta }_{k}}}\right) \) be an element of \( T \) . Then \( t \) generates a dense subgroup of \( T \) if and only if the numbers \[ 1,{\theta }_{1},\ldots ,{\theta }_{k} \] are linearly independent over the field \( \mathbb{Q} \) of rational numbers.

In light of Lemma 11.5, we may reformulate the proposition as follows: The numbers \( 1,{\theta }_{1},\ldots ,{\theta }_{k} \) are linearly dependent over \( \mathbb{Q} \) if and only if there exists a nonconstant homomorphism \( \Phi : T \rightarrow {S}^{1} \) with \( \left( {{e}^{{2\pi i}{\theta }_{1}},\ldots ,{e}^{{2\pi i}{\theta }_{k}}}\right) \in \ker \left( \Phi \right) \) . Suppose first that there is a dependence relation among \( 1,{\theta }_{1},\ldots ,{\theta }_{k} \) over \( \mathbb{Q} \) . Then after clearing the denominators from this relation, we find that there exist integers \( {m}_{1},\ldots ,{m}_{k} \), not all zero, such that \[ {m}_{1}{\theta }_{1} + \cdots + {m}_{k}{\theta }_{k} \in \mathbb{Z}. \] Thus, we may define a nonconstant \( \Phi : T \rightarrow {S}^{1} \) by \[ \Phi \left( {{u}_{1},\ldots ,{u}_{k}}\right) = {u}_{1}^{{m}_{1}}\cdots {u}_{k}^{{m}_{k}} \] and the kernel of \( \Phi \) will contain \( \left( {{e}^{{2\pi i}{\theta }_{1}},\ldots ,{e}^{{2\pi i}{\theta }_{k}}}\right) \) . In the other direction, Exercise 2 tells us that every continuous homomorphism \( \Phi : T \rightarrow {S}^{1} \) is of the form (11.1) for some set of integers \( {m}_{1},\ldots ,{m}_{k} \) . Furthermore, if \( \Phi \) is nonconstant, these integers cannot all be zero. Thus, if \[ 1 = \Phi \left( {{e}^{{2\pi i}{\theta }_{1}},\ldots ,{e}^{{2\pi i}{\theta }_{k}}}\right) = {e}^{{2\pi i}\left( {{m}_{1}{\theta }_{1} + \cdots + {m}_{k}{\theta }_{k}}\right) }, \] we must have \( {m}_{1}{\theta }_{1} + \cdots + {m}_{k}{\theta }_{k} = n \) for some integer \( n \), which implies that \( 1,{\theta }_{1},\ldots ,{\theta }_{k} \) are linearly dependent over \( \mathbb{Q} \) .

Proposition 9.6.22 (Stirling’s formula). As \( n \rightarrow \infty \) we have \[ n! \sim {n}^{n}{e}^{-n}\sqrt{2\pi n} \] or equivalently, \[ \log \left( {n!}\right) = \left( {n + \frac{1}{2}}\right) \log \left( n\right) - n + \frac{1}{2}\log \left( {2\pi }\right) + o\left( 1\right) . \] Proof. Once again there are several classical proofs. Certainly the most classical is as follows: if we set \( {u}_{n} = \log \left( {n!/\left( {{n}^{n}{e}^{-n}\sqrt{n}}\right) }\right) \) then \[ {u}_{n + 1} - {u}_{n} = 1 - \left( {n + \frac{1}{2}}\right) \log \left( {1 + \frac{1}{n}}\right) \sim - \frac{1}{{12}{n}^{2}}; \] hence this is the general term of an absolutely convergent series, so as \( n \rightarrow \infty \) , \( {u}_{n} \) tends to some limit \( \log \left( A\right) \), say (we could also apply the Euler-MacLaurin summation formula). To obtain \( A \) we can use Wallis’s formulas. We let \( {C}_{n} = \) \( {\int }_{0}^{\pi /2}{\cos }^{n}\left( t\right) {dt} \) . By integrating by parts, it is immediate that for \( n \geq 2 \) we have \( {C}_{n} = \left( {n - 1}\right) \left( {{C}_{n - 2} - {C}_{n}}\right) \), hence \( {C}_{n} = \left( {\left( {n - 1}\right) /n}\right) {C}_{n - 2} \) . Since \( {C}_{0} = \pi /2 \) and \( {C}_{1} = 1 \), we deduce that \[ {C}_{2k} = \frac{\left( {2k}\right) !}{{2}^{2k}{\left( k!\right) }^{2}}\frac{\pi }{2}\;\text{ and }\;{C}_{{2k} + 1} = \frac{{2}^{2k}{\left( k!\right) }^{2}}{\left( {{2k} + 1}\right) !}. \] On the other hand, the sequence \( {C}_{n} \) is clearly decreasing, so that in particular \( {C}_{{2k} + 1} \leq {C}_{2k} \leq {C}_{{2k} - 1} \) . If we replace \( {C}_{n} \) by its asymptotic value \( {n}^{n}{e}^{-n}{n}^{1/2}A \) (where \( A \) is the unknown nonzero constant above) a short computation shows that \( {A}^{2} = {2\pi } \), proving Stirling’s formula. Of course the \( o\left( 1\right) \) in the expression for \( \log \left( {n!}\right) \) can be given a complete asymptotic expansion by the Euler-MacLaurin summation formula; see Section 9.2.5. A more sophisticated, but easier to generalize, way of finding the value of the constant \( A \) has been explained in Section 9.2.5: by derivation of the formulas of Proposition 9.2.13, which come from the Euler-MacLaurin summation formula, we find that \[ \log \left( {n!}\right) = \left( {n + 1/2}\right) \log \left( n\right) - n - {\zeta }^{\prime }\left( 0\right) + O\left( {1/n}\right) . \] The value of \( {\zeta }^{\prime }\left( 0\right) \) is immediate to compute from the functional equation for the zeta function, which itself is a simple application of the Poisson summation formula; see Section 10.2.4. Once again we urge the reader to study Exercise 44 for generalizations of this idea. Corollary 9.6.23. For any \( m \geq 1 \) we have \[ \log \Gamma \left( s\right) = \left( {s - \frac{1}{2}}\right) \log \left( s\right) - s + \frac{\log \left( {2\pi }\right) }{2} + \mathop{\sum }\limits_{{k = 1}}^{m}\frac{{B}_{2k}}{{2k}\left( {{2k} - 1}\right) {s}^{{2k} - 1}} \] \[ - \frac{1}{{2m} + 1}{\int }_{0}^{\infty }\frac{{B}_{{2m} + 1}\left( {\{ t\} }\right) }{{\left( t + s\right) }^{{2m} + 1}}{dt} \] Proof. Clear from Stirling's formula and Proposition 9.6.16. Proposition 9.6.24. We have the following expansions: (1) \[ \pi \operatorname{cotan}\left( {\pi x}\right) = \frac{1}{x} + {2x}\mathop{\sum }\limits_{{n \geq 1}}\frac{1}{{x}^{2} - {n}^{2}}. \] (2) \[ {\left( \frac{\pi }{\sin \left( {\pi x}\right) }\right) }^{2} = \mathop{\sum }\limits_{{n \in \mathbb{Z}}}\frac{1}{{\left( x - n\right) }^{2}}. \] (3) \[ \sin \left( {\pi x}\right) = {\pi x}\mathop{\prod }\limits_{{n \geq 1}}\left( {1 - \frac{{x}^{2}}{{n}^{2}}}\right) . \] Proof. Let \( a \notin \mathbb{Z} \) be a parameter, and define \( f\left( x\right) \) to be the \( {2\pi } \) -periodic function such that \( f\left( x\right) = \cos \left( {ax}\right) \) for \( - \pi \leq x \leq \pi \) . This function is clearly continuous and piecewise differentiable. It is thus everywhere equal to the sum of its Fourier series. A short computation gives \[ f\left( x\right) = \frac{\sin \left( {\pi a}\right) }{\pi }\left( {\frac{1}{a} + {2a}\mathop{\sum }\limits_{{n \geq 1}}\frac{{\left( -1\right) }^{n}\cos \left( {nx}\right) }{{a}^{2} - {n}^{2}}}\right) , \] and taking \( x = \pi \) gives \[ \pi \operatorname{cotan}\left( {\pi a}\right) = \frac{1}{a} + {2a}\mathop{\sum }\limits_{{n \geq 1}}\frac{1}{{a}^{2} - {n}^{2}}, \] proving (1). Note incidentally that this formula immediately implies the formulas for \( \zeta \left( {2k}\right) \) ; see Exercise 74. (2) follows by differentiation, after writing \( {2x}/\left( {{x}^{2} - {n}^{2}}\right) = 1/\left( {x - n}\right) + 1/\left( {x + n}\right) \) . For (3), consider the function \( g\left( a\right) = \sin \left( {\pi a}\right) /\left( {{\pi a}\mathop{\prod }\limits_{{n \geq 1}}\left( {1 - {a}^{2}/{n}^{2}}\right) }\right) \) . Clearly the product converges absolutely, so the function is defined for all \( a \notin \mathbb{Z} \) . In addition, as \( a \) tends to 0 it is clear that \( g\left( a\right) \) tends to 1, and since by writing \( \left( {1 - {a}^{2}/{n}^{2}}\right) = \left( {1 - a/n}\right) \left( {1 + a/n}\right) \) it is clear that \( g\left( a\right) \) is a periodic function of period 1, it follows that \( g\left( a\right) \) tends to 1 as \( a \) tends to any integer. Moreover, it is also clear that \( g\left( a\right) \) is differentiable (in fact infinitely). If we compute the logarithmic derivative of \( g\left( a\right) \) we find using (1) that \[ \frac{{g}^{\prime }\left( a\right) }{g\left( a\right) } = \pi \operatorname{cotan}\left( {\pi a}\right) - \left( {\frac{1}{a} + {2a}\mathop{\sum }\limits_{{n \geq 1}}\frac{1}{{a}^{2} - {n}^{2}}}\right) = 0. \] It follows that \( g\left( a\right) \) is a constant, and since \( g\left( 0\right) = 1 \), that \( g\left( a\right) = 1 \) for all \( a \) , proving (3). Proposition 9.6.25. For all \( s \) such that \( \Re \left( s\right) > 0 \) we have \[ \log \left( s\right) = {\int }_{0}^{\infty }\frac{{e}^{-t} - {e}^{-{st}}}{t}{dt} \] where the left-hand side is the principal determination of the logarithm. More generally, if \( \Re \left( {s}_{1}\right) > 0 \) and \( \Re \left( {s}_{2}\right) > 0 \) we have \[ \log \left( {{s}_{1}/{s}_{2}}\right) = {\int }_{0}^{\infty }\frac{{e}^{-{s}_{2}t} - {e}^{-{s}_{1}t}}{t}{dt} \] Proof. Let \( I\left( s\right) \) be the first integral above. It is clearly absolutely convergent for \( \Re \left( s\right) > 0 \), and its (for the moment formal) derivative with respect to \( s \) is \( {\int }_{0}^{\infty }{e}^{-{st}}{dt} \), which is normally convergent in the domain \( \Re \left( s\right) \geq \varepsilon > 0 \) for any fixed \( \varepsilon \) . It follows that the derivation under the integral sign is justified; hence \( {I}^{\prime }\left( s\right) = 1/s \), so that \( I\left( s\right) = \log \left( s\right) \) with the principal determination of the logarithm, since clearly \( I\left( 1\right) = 0 \) . The second formula follows from \( \log \left( {{s}_{1}/{s}_{2}}\right) = \log \left( {s}_{1}\right) - \log \left( {s}_{2}\right) \) when \( \Re \left( {s}_{i}\right) > 0 \), with the principal determinations. ## 9.6.4 Properties of the Gamma Function With this out the way, we can now begin our detailed study of the gamma function. Recall that we have set \( {u}_{n}\left( s\right) = {n}^{s - 1}n!/\left( {s\left( {s + 1}\right) \cdots \left( {s + n - 1}\right) }\right) \) and that we have defined \( \Gamma \left( s\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{u}_{n}\left( s\right) \) in Definition 9.6.18. Since \( {\mathbb{R}}_{ > 0} \) is not a discrete subset of \( \mathbb{C} \smallsetminus {\mathbb{Z}}_{ \leq 0} \), an important remark is that all the identities that we prove on the real gamma function (usually as byproducts of corresponding results for the Hurwitz zeta function) will be automatically valid for the complex gamma function by analytic continuation. In particular, by Proposition 9.6.14 we have \( \Gamma \left( {s + 1}\right) = {s\Gamma }\left( s\right) \) for all \( s \in \mathbb{C} \smallsetminus {\mathbb{Z}}_{ \leq 0} \) . Proposition 9.6.26 (Hadamard product). We have \[ \Gamma \left( {s + 1}\right) = {e}^{-{\gamma s}}\mathop{\prod }\limits_{{n \geq 1}}\frac{{e}^{s/n}}{1 + s/n}, \] where \( \gamma = {0.57721}\ldots \) is Euler’s constant. Note that this is the Hadamard product expansion of the entire function \( 1/\Gamma \left( {s + 1}\right) \) ; see Theorem 10.7.6. Proof. If we divide the numerator and the denominator of \( {u}_{n}\left( s\right) \) by \( n! = \) \( 1 \cdot 2\cdots n \) we obtain \[ {u}_{n}\left( {s + 1}\right) = \frac{{n}^{s}}{\mathop{\prod }\limits_{{1 \leq k \leq n}}\left( {1 + s/k}\right) } = {n}^{s}{e}^{-s{H}_{n}}\mathop{\prod }\limits_{{1 \leq k \leq n}}\frac{{e}^{s/k}}{1 + s/k}, \] where \( {H}_{n} = \mathop{\sum }\limits_{{1 \leq k \leq n}}1/k \) is the harmonic sum. Since \[ {n}^{s}{e}^{-s{H}_{n}} = {e}^{-s\left( {{H}_{n} - \log \left( n\right) }\right) }, \] by definition of \( \gamma \) this tends to \( {e}^{-{\gamma s}} \), proving the proposition. Note that this implies that the infinite product is convergent, which is clear directly by noting that the logarithm of its general term is \( O\left( {1/{n}^{2}}\right) \) . Proposition 9.6.27 (Complex Stirling formula). For any \( s \in \mathbb{C} \) set \( \rho \left( s\right) = \max \left( {\Re \left( s\right) ,\left| {\Im \left( s\right) }\right| }\right) \) . Then as \( \rho \left( s\right) \rightarrow \infty \) we have \[ \log \Gamma \left( s\right) = \left( {s - \frac{1}{2}}\right) \log \left( s\right) - s + \frac{1}{2}\log \left( {2\pi }\right) + O\left( {1/\rho \left( s\right) }\right) . \] Proof. First note that the region \( {R}_{N} = \{ s \in \mathbb{C}/\rho \left( s\right) \geq N\} \) is a subset of \( \mathbb{C} \smallsetminus {\mathbb{R}}_{ \leq 0} \), hence we can choose

Proposition 9.6.22 (Stirling’s formula). As \( n \rightarrow \infty \) we have \[ n! \sim {n}^{n}{e}^{-n}\sqrt{2\pi n} \] or equivalently, \[ \log \left( {n!}\right) = \left( {n + \frac{1}{2}}\right) \log \left( n\right) - n + \frac{1}{2}\log \left( {2\pi }\right) + o\left( 1\right) . \]

Proof. Once again there are several classical proofs. Certainly the most classical is as follows: if we set \( {u}_{n} = \log \left( {n!/\left( {{n}^{n}{e}^{-n}\sqrt{n}}\right) }\right) \) then \[ {u}_{n + 1} - {u}_{n} = 1 - \left( {n + \frac{1}{2}}\right) \log \left( {1 + \frac{1}{n}}\right) \sim - \frac{1}{{12}{n}^{2}}; \] hence this is the general term of an absolutely convergent series, so as \( n \rightarrow \infty \) , \( {u}_{n} \) tends to some limit \( \log \left( A\right) \), say (we could also apply the Euler-MacLaurin summation formula). To obtain \( A \) we can use Wallis’s formulas. We let \( {C}_{n} = \) \( {\int }_{0}^{\pi /2}{\cos }^{n}\left( t\right) {dt} \) . By integrating by parts, it is immediate that for \( n \geq 2 \) we have \( {C}_{n} = \left( {n - 1}\right) \left( {{C}_{n - 2} - {C}_{n}}\right) \), hence \( {C}_{n} = \left( {\left( {n - 1}\right) /n}\right) {C}_{n - 2} \) . Since \( {C}_{0} = \pi /2 \) and \( {C}_{1} = 1 \), we deduce that \[ {C}_{2k} = \frac{\left( {2k}\right) !}{{2}^{2k}{\left( k!\right) }^{2}}\frac{\pi }{2}\;\text{ and }\;{C}_{{2k} + 1} = \frac{{2}^{2k}{\left( k!\right) }^{2}}{\left( {{2k} + 1}\right) !}. \] On the other hand, the sequence \( {C}_{n} \) is clearly decreasing, so that in particular \( {C}_{{2k} + 1} \leq {C}_{2k} \leq {C}_{{2k} - 1} \) . If we replace \( {C}_{n} \) by its asymptotic value \( {n}^{n}{e}^{-n}{n}^{1/2}A \) (where \( A \) is the unknown nonzero constant above) a short computation shows that \( {A}^{2} = {2\pi } \), proving Stirling’s formula. Of course the \( o\left( 1\right) \) in the expression for \( \log \left( {n!}\right) \) can be given a complete asymptotic expansion by the Euler-MacLaurin summation formula; see Section 9.2.5.

Theorem 5.23 (Jordan Curve Theorem \( {}^{2} \) ). If \( \gamma \) is a simple closed path in \( \mathbb{C} \), then (a) \( \mathbb{C} \) - range \( \gamma \) has exactly two connected components, one of which is bounded. (b) Range \( \gamma \) is the boundary of each of these components, and (c) \( I\left( {\gamma, c}\right) = 0 \) for all \( c \) in the unbounded component of the complement of the range of \( \gamma .I\left( {\gamma, c}\right) = \pm 1 \) for all \( c \) in the bounded component of the complement of the range of \( \gamma \) . The choice of sign depends only on the choice of direction for traversal on \( \gamma \) . Definition 5.24. For a simple closed path \( \gamma \) in \( \mathbb{C} \) we define the interior of \( \gamma, i\left( \gamma \right) \) , to be the bounded component of \( \mathbb{C} \) - range \( \gamma \) and the exterior of \( \gamma, e\left( \gamma \right) \), to be the unbounded component of \( \mathbb{C} \) -range \( \gamma \) . If \( I\left( {\gamma, c}\right) = + 1 \) (respectively -1) for \( c \) in \( i\left( \gamma \right) \) then we say that \( \gamma \) is a Jordan curve with positive (respectively negative) orientation. We shall not prove the above theorem. It is a deep result. In all of our applications, it will be obvious that our Jordan curves have the above properties. Remark 5.25. Another important (and nontrivial to prove) property of Jordan curves is the fact that the interior of a Jordan curve is always a simply connected domain in \( \mathbb{C} \) . If we view the Jordan curve as lying on the Riemann sphere \( \widehat{\mathbb{C}} \), then each component of the complement of its range is simply connected. This property allows us to prove the following result. Theorem 5.26 (Cauchy’s Theorem (Extended Version)). Let \( {\gamma }_{0},\ldots ,{\gamma }_{n} \) be \( n + 1 \) positively oriented Jordan curves. Assume that \[ \text{range}{\gamma }_{j} \subset e\left( {\gamma }_{k}\right) \cap i\left( {\gamma }_{0}\right) \] --- \( {}^{2} \) For a proof see the appendix to Ch. IX of J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, vol. X, Academic Press, 1960 or Chap. 10 of J. R. Munkres, Topology (Second Edition), Dover, 2000. ---  Fig. 5.2 Jordan curves and the domain they define for all \( 1 \leq j \neq k \leq n \), see Fig. 5.2. If \( f \) is a holomorphic function on a neighborhood \( N \) of the closure of the domain \[ D = i\left( {\gamma }_{0}\right) \cap e\left( {\gamma }_{1}\right) \cap \cdots \cap e\left( {\gamma }_{n}\right) , \] then \[ {\int }_{{\gamma }_{0}}f\left( z\right) \mathrm{d}z = \mathop{\sum }\limits_{{k = 1}}^{n}{\int }_{{\gamma }_{k}}f\left( z\right) \mathrm{d}z. \] Proof. Adjoin nonintersecting curves \( {\delta }_{j} \) in \( D \) from \( {\gamma }_{0} \) to \( {\gamma }_{j} \) for \( j = 1,\ldots, n \), as in Fig. 5.2. Then the cycle \[ \delta = \left( {{\gamma }_{0},{\delta }_{1} * {\gamma }_{{1}_{ - }} * {\delta }_{{1}_{ - }},\ldots ,{\delta }_{n} * {\gamma }_{{n}_{ - }} * {\delta }_{{n}_{ - }}}\right) \] is homologous to zero in \( N \) . Thus Theorem 5.21 implies that \[ \left( {{\int }_{{\gamma }_{0}} + \mathop{\sum }\limits_{{k = 1}}^{n}\left( {{\int }_{{\delta }_{k}} + {\int }_{{\gamma }_{{k}_{ - }}} + {\int }_{{\delta }_{{k}_{ - }}}}\right) }\right) f\left( z\right) \mathrm{d}z = 0, \] and the result follows by noting that the integral over each \( {\delta }_{k} \) is canceled by the corresponding integral over \( {\delta }_{{k}_{ - }} \) . An immediate consequence is Theorem 5.27 (Cauchy's Integral Formula (Extended Version)). With the hypotheses as in the extended version of Cauchy's Theorem 5.26, we have \[ {2\pi }\imath f\left( c\right) = {\int }_{{\gamma }_{0}}\frac{f\left( z\right) }{z - c}\mathrm{\;d}z - \mathop{\sum }\limits_{{k = 1}}^{n}{\int }_{{\gamma }_{k}}\frac{f\left( z\right) }{z - c}\mathrm{\;d}z \] for all \( c \in D \) . Proof. We can apply Theorem 5.2 to the function \( f \), using the neighborhood \( N \) of Theorem 5.26 and the cycle \( \delta \) constructed in its proof, since \( \delta \) is homologous to zero in \( N \) and \( I\left( {\delta, c}\right) = + 1 \) . As before, the integral over each \( {\delta }_{k} \) is canceled by the corresponding integral over \( {\delta }_{{k}_{ - }} \) . ## 5.4 The Mean Value Property The next concept applies in a broader context than that of holomorphic functions, as we will see in Chap. 9. Definition 5.28. Let \( f \) be a function defined on a domain \( D \) in \( \mathbb{C} \) . We say that \( f \) has the mean value property (MVP) if for each \( c \in D \) there exists \( {r}_{0} > 0 \) with \( U\left( {c,{r}_{0}}\right) \subseteq D \) and \[ f\left( c\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }f\left( {c + r{\mathrm{e}}^{t\theta }}\right) \mathrm{d}\theta \text{ for all }0 \leq r < {r}_{0}. \] (5.7) Remark 5.29. A holomorphic function \( f \) on a domain \( D \) has the MVP (with \( {r}_{0} = \) the distance of \( c \in D \) to \( \partial D \) ). Hence so do its real and imaginary parts. Theorem 5.30 (Maximum Modulus Principle). Suppose \( f \) is a continuous complex-valued function defined on a domain \( D \) in \( \mathbb{C} \) that has the MVP. If \( \left| f\right| \) has a relative maximum at a point \( c \in D \), then \( f \) is constant in a neighborhood of \( c \) . Proof. The result is clear if \( f\left( c\right) = 0 \) . If \( f\left( c\right) \neq 0 \), replacing \( f \) by \( {\mathrm{e}}^{-{t\theta }}f \) for some \( \theta \in \mathbb{R} \), we may assume that \( f\left( c\right) > 0 \) . Write \( f = u + {\iota v} \) and choose \( {r}_{0} > 0 \) such that (1) \( \operatorname{cl}U\left( {c,{r}_{0}}\right) \subset D \) . (2) Equation (5.7) holds. (3) \( \left| {f\left( z\right) }\right| \leq f\left( c\right) \) for \( z \in \operatorname{cl}U\left( {c,{r}_{0}}\right) \) . If we define \[ M\left( r\right) = \sup \{ \left| {f\left( z\right) }\right| ;\left| {z - c}\right| = r\} \text{ for }0 \leq r \leq {r}_{0}, \] then it follows from (3) that \[ M\left( r\right) \leq f\left( c\right) \text{ for }0 \leq r \leq {r}_{0} \] Since the MVP implies that \[ f\left( c\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }f\left( {c + r{\mathrm{e}}^{i\theta }}\right) \mathrm{d}\theta \text{ for }0 \leq r < {r}_{0}, \] we also have \( f\left( c\right) \leq M\left( r\right) \), and we conclude that \( f\left( c\right) = M\left( r\right) \) for \( 0 \leq r \leq {r}_{0} \) . Now observe that \[ \frac{1}{2\pi }{\int }_{0}^{2\pi }M\left( r\right) \mathrm{d}\theta = M\left( r\right) = f\left( c\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }u\left( {c + r{\mathrm{e}}^{\iota \theta }}\right) \mathrm{d}\theta , \] (5.8) where the last equality holds because \( f\left( c\right) \) is real. Also note that \[ M\left( r\right) - u\left( {c + r{\mathrm{e}}^{\iota \theta }}\right) \geq 0, \] (5.9) from the definition of \( M\left( r\right) \) . But from (5.8) we obtain that \[ {\int }_{0}^{2\pi }\left\lbrack {M\left( r\right) - u\left( {c + r{\mathrm{e}}^{\iota \theta }}\right) }\right\rbrack \mathrm{d}\theta = 0, \] and hence we must have equality in (5.9) for all \( \theta \) . Finally, \[ M\left( r\right) \geq {\left( {u}^{2}\left( c + r{\mathrm{e}}^{\iota \theta }\right) + {v}^{2}\left( c + r{\mathrm{e}}^{\iota \theta }\right) \right) }^{\frac{1}{2}} = {\left( M{\left( r\right) }^{2} + {v}^{2}\left( c + r{\mathrm{e}}^{\iota \theta }\right) \right) }^{\frac{1}{2}} \] which implies that \( v\left( {c + r{\mathrm{e}}^{t\theta }}\right) = 0 \) for \( 0 \leq r \leq {r}_{0} \) and \( 0 \leq \theta \leq {2\pi } \) . Therefore \( f\left( z\right) = u\left( z\right) = M\left( \left| z\right| \right) = f\left( c\right) \) for all \( \left| z\right| \leq {r}_{0} \) . It is now easy to deduce that if \( f \) is a nonconstant holomorphic function on a bounded domain that extends to a continuous function on the closure of the domain, then \( \left| f\right| \) assumes its maximum on the boundary of that domain. More is true, as seen in Corollary 5.31. Suppose \( D \) is a bounded domain and \( f \in {\mathbf{C}}^{0}\left( {\mathrm{{cl}}D}\right) \) satisfies the MVP in D. If \[ M = \sup \{ \left| {f\left( z\right) }\right| ;z \in \partial D\} \] then (a) \( \left| {f\left( z\right) }\right| \leq M \) for all \( z \in D \) . (b) If \( \left| {f\left( c\right) }\right| = M \) for some \( c \in D \), then \( f \) is constant in \( D \) . Proof. If \[ {M}^{\prime } = \sup \{ \left| {f\left( z\right) }\right| ;z \in \operatorname{cl}D\} , \] then \[ M \leq {M}^{\prime } < + \infty \] We know that there exists a \( c \) in \( \operatorname{cl}D \) such that \( \left| {f\left( c\right) }\right| = {M}^{\prime } \) . If \( c \in D \), then \( f \) is constant in a neighborhood of \( c \) by the MMP. Let \[ {D}^{\prime } = \left\{ {z \in D;\left| {f\left( z\right) }\right| = {M}^{\prime }}\right\} . \] The set \( {D}^{\prime } \) is closed and open in \( D \) ; hence, if nonempty, it is all of \( D \) . In this latter case \( f \) is constant on \( D \), since it is locally constant and \( D \) is connected, and thus also constant on \( \operatorname{cl}D \) . Then \( M = {M}^{\prime } \), and (a) and (b) are trivially true. On the other hand, if \( {D}^{\prime } = \varnothing \), then \( M = {M}^{\prime } \) ; (a) follows and (b) is trivially true. In particular, since a function that is holomorphic in a domain satisfies the MVP, we have Corollary 5.32 (The Maximum Modulus Principle for Analytic Functions). If \( f \) is a nonconstant holomorphic function on a domain \( D \), then \( \left| f\right| \) has no relative maximum in \( D \) . Further, if \( D \) is bounded and \( f \) is continuous on the boundary of \( D \), then \( \left| f\right| \) assumes its maximum on the boundary of \( D \)

Theorem 5.23 (Jordan Curve Theorem \( {}^{2} \)). If \( \gamma \) is a simple closed path in \( \mathbb{C} \), then (a) \( \mathbb{C} \) - range \( \gamma \) has exactly two connected components, one of which is bounded. (b) Range \( \gamma \) is the boundary of each of these components, and (c) \( I\left( {\gamma, c}\right) = 0 \) for all \( c \) in the unbounded component of the complement of the range of \( \gamma .I\left( {\gamma, c}\right) = \pm 1 \) for all \( c \) in the bounded component of the complement of the range of \( \gamma \) . The choice of sign depends only on the choice of direction for traversal on \( \gamma \) .

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Lemma 1.6. Let \( R = k \) be a field, and let \( V \) be a \( k \) -vector space. Let \( B \) be a maximal linearly independent subset of \( V \) ; then \( B \) is a basis of \( V \) . Again, this should be contrasted with the situation over rings: \( \{ 2\} \) is a maximal linearly independent subset of \( \mathbb{Z} \), but it is not a basis. Proof. Let \( v \in V, v \notin B \) . Then \( B \cup \{ v\} \) is not linearly independent, by the maximality of \( B \) ; therefore, there exist \( {c}_{0},\ldots ,{c}_{t} \in k \) and (distinct) \( {b}_{1},\ldots ,{b}_{t} \in B \) such that \[ {c}_{0}v + {c}_{1}{b}_{1} + \cdots + {c}_{t}{b}_{t} = 0, \] \( {}^{3} \) This terminology is also often used for free modules over any ring. with not all \( {c}_{0},\ldots ,{c}_{t} \) equal to 0 . Now, \( {c}_{0} \neq 0 \) : otherwise we would get a linear dependence relation among elements of \( B \) . Since \( k \) is a field, \( {c}_{0} \) is a unit; but then \[ v = \left( {-{c}_{0}^{-1}{c}_{1}}\right) {b}_{1} + \cdots + \left( {-{c}_{0}^{-1}{c}_{t}}\right) {b}_{t}, \] proving that \( v \) is in the span of \( B \) . It follows that \( B \) generates \( V \), as needed. Summarizing, Proposition 1.7. Let \( R = k \) be a field, and let \( V \) be a \( k \) -vector space. Let \( S \) be a linearly independent set of vectors of \( V \) . Then there exists a basis \( B \) of \( V \) containing \( S \) . In particular, \( V \) is free as a \( k \) -module. Proof. Put Lemma 1.2, Lemma 1.5, and Lemma 1.6 together. We could also contemplate this situation from the 'mirror' point of view of generating sets: Lemma 1.8. Let \( R = k \) be a field, and let \( V \) be a \( k \) -vector space. Let \( B \) be a minimal generating set for \( V \) ; then \( B \) is a basis of \( V \) . Every set generating \( V \) contains a basis of \( V \) . ## Proof. Exercise 1.6. Lemma 1.8 also fails on more general rings (Exercise 1.5). To reiterate, over fields (but not over general rings) a subset \( B \) of a vector space is a basis \( \Leftrightarrow \) it is a maximal linearly independent subset \( \Leftrightarrow \) it is a minimal generating set. 1.4. Recovering \( B \) from \( {F}^{R}\left( B\right) \) . We are ready for the ’reconstruction’ of a set \( B \) (up to a bijection!) from the corresponding free module \( {F}^{R}\left( B\right) \) . This is the result justifying the notion of dimension of a vector space, or, more generally, the rank of a free module. Again, we prove a somewhat stronger statement. Proposition 1.9. Let \( R \) be an integral domain, and let \( M \) be a free \( R \) -module. Let \( B \) be a maximal linearly independent subset of \( M \), and let \( S \) be a linearly independent subset. Then 4 \( \left| S\right| \leq \left| B\right| \) . In particular, any two maximal linearly independent subsets of a free module over an integral domain have the same cardinality. Proof. By taking fields of fractions, the general case over an integral domain is easily reduced to the case of vector spaces over a field; see Exercise 1.7 We may then assume that \( R = k \) is a field and \( M = V \) is a \( k \) -vector space. We have to prove that there is an injective map \( j : S \hookrightarrow B \), and this can be done by an inductive process, replacing elements of \( B \) by elements of \( S \) ’one-by-one’. For this, let \( \leq \) be a well-ordering on \( S \), let \( v \in S \), and assume we have defined \( j \) for all \( w \in S \) with \( w < v \) . Let \( {B}^{\prime } \) be the set obtained from \( B \) by replacing all --- \( {}^{4} \) Here, \( \left| A\right| \) denotes the cardinality of the set \( A \), a notion with which the reader is hopefully familiar. The reader will not lose much by only considering the case in which \( B, S \) are finite sets; but the fact is true for 'infinite-dimensional spaces' as well, as the argument shows. --- \( j\left( w\right) \) by \( w \), for \( w < v \), and assume (inductively) that \( {B}^{\prime } \) is still a maximal linearly independent subset of \( V \) . Then I claim that \( j\left( v\right) \in B \) may be defined so that - \( j\left( v\right) \neq j\left( w\right) \) for all \( w < v \) ; - the set \( {B}^{\prime \prime } \) obtained from \( {B}^{\prime } \) by replacing \( j\left( v\right) \) by \( v \) is still a maximal linearly independent subset. (Transfinite) induction (Claim V13.2) then shows that \( j \) is defined and injective on \( S \), as needed. To verify my claim, since \( {B}^{\prime } \) is a maximal linearly independent set, \( {B}^{\prime } \cup \{ v\} \) is linearly dependent (as an indexed set 5), so that there exists a linear dependence relation (*) \[ {c}_{0}v + {c}_{1}{b}_{1} + \cdots + {c}_{t}{b}_{t} = 0 \] with not all \( {c}_{i} \) equal to zero and the \( {b}_{i} \) distinct in \( {B}^{\prime } \) . Necessarily \( {c}_{0} \neq 0 \) (because \( {B}^{\prime } \) is linearly independent); also, necessarily not all the \( {b}_{i} \) with \( {c}_{i} \neq 0 \) are elements of \( S \) (because \( S \) is linearly independent). Without loss of generality we may then assume that \( {c}_{1} \neq 0 \) and \( {b}_{1} \in {B}^{\prime } \smallsetminus S \) . This guarantees that \( {b}_{1} \neq j\left( w\right) \) for all \( w < v \) ; I set \( j\left( v\right) = {b}_{1} \) . All that is left now is the verification that the set \( {B}^{\prime \prime } \) obtained by replacing \( {b}_{1} \) by \( v \) in \( {B}^{\prime } \) is a maximal linearly independent subset. But by using (*) to write \[ v = - {c}_{0}^{-1}{c}_{1}{b}_{1} - \cdots - {c}_{0}^{-1}{c}_{t}{b}_{t} \] this is an easy consequence of the fact that \( {B}^{\prime } \) is a maximal linearly independent subset. Further details are left to the reader. Example 1.10. An uncountable subset of \( \mathbb{C}\left\lbrack x\right\rbrack \) is necessarily linearly dependent. Indeed, \( \mathbb{C}\left\lbrack x\right\rbrack \) has a countable basis over \( \mathbb{C} \) : for example, \( \left\{ {1, x,{x}^{2},{x}^{3},\ldots }\right\} \) . Corollary 1.11. Let \( R \) be an integral domain, and let \( A, B \) be sets. Then \[ {F}^{R}\left( A\right) \cong {F}^{R}\left( B\right) \Leftrightarrow \text{ there is a bijection }A \cong B. \] Proof. Exercise 1.8. Remark 1.12. We have learned in Lemma 1.2 that we can 'complete' every linearly independent subset \( S \) to a maximal one. The argument used in the proof of Proposition 1.9 shows that we can in fact do this by borrowing elements of a given maximal linearly independent subset. Remark 1.13. As a particular case of Corollary 1.11, we see that if \( R \) is an integral domain, then \( {R}^{m} \cong {R}^{n} \) if and only if \( m = n \) . This says that integral domains satisfy the 'IBN (Invariant Basis Number) property'. Strange as it may seem, this obvious-looking fact does not hold over arbitrary rings: for example, the ring of endomorphisms of an infinite-dimensional vector space does not satisfy the IBN property. On the other hand, integral domains are a bit of an overshoot: all commutative rings satisfy the IBN property (Exercise 1.11). \( {}^{5} \) This allows for the possibility that \( v \in {B}^{\prime } \) ’already’. In this case, the reader can check that the process I am about to describe gives \( j\left( v\right) = v \) . One way to think about this is that the category of finitely generated free modules over (say) an integral domain is ’classified’ by \( {\mathbb{Z}}^{ \geq 0} \) : up to isomorphisms, there is exactly one finitely generated free module for any nonnegative integer. The task of describing this category then amounts to describing the homomorphisms between objects corresponding to two given nonnegative integers; this will be done in \( §{2.1} \) As a byproduct of the result of Proposition 1.9, we can now give the following important definition. Definition 1.14. Let \( R \) be an integral domain. The rank of a free \( R \) -module \( M \) , denoted \( {\operatorname{rk}}_{R}M \), is the cardinality of a maximal linearly independent subset of \( M \) . The rank of a vector space is called the dimension, denoted \( {\dim }_{k}V \) . This definition will in fact be adopted for more general finitely generated modules when the time comes, in [5.3] Finite-dimensional vector spaces over a fixed field form a category. Since vector spaces are free modules (Proposition 1.7), Corollary 1.11 implies that two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension. The subscripts \( R, k \) are often omitted, if the context permits. But note that, for example, viewing the complex numbers as a real vector space, we have \( {\dim }_{\mathbb{R}}\mathbb{C} = 2 \) , while \( {\dim }_{\mathbb{C}}\mathbb{C} = 1 \) . So some care is warranted. Proposition 1.9 tells us that every linearly independent subset \( S \) of a free \( R \) - module \( M \) must have cardinality lower than or equal to \( {\operatorname{rk}}_{R}M \) . Similarly, every generating set must have cardinality higher than or equal to the rank. Indeed, Proposition 1.15. Let \( R \) be an integral domain, and let \( M \) be a free \( R \) -module; assume that \( M \) is generated by \( S : M = \langle S\rangle \) . Then \( S \) contains a maximal linearly independent subset of \( M \) . Proof. By Exercise 1.7 we may assume that \( R \) is a field and \( M = V \) is a vector space. Use Zorn’s lemma to obtain a linearly independent subset \( B \subseteq S \) which is maximal among subsets of \( S \) . Arguing as in the proof of Lemma 1.6 shows that \( S \) is in the span of \( B \), and it follows that \( B \) generates \( V \) . Thus \( B \) is a basis, and hence a maximal linearly independent subset of \( V \), as needed. Remark 1.16. I have used again the trick of switching from an integral domain to its field of fractions. The second part of the argument would not work over an arbitrary integral dom

Lemma 1.6. Let \( R = k \) be a field, and let \( V \) be a \( k \) -vector space. Let \( B \) be a maximal linearly independent subset of \( V \) ; then \( B \) is a basis of \( V \) .

Proof. Let \( v \in V, v \notin B \) . Then \( B \cup \{ v\} \) is not linearly independent, by the maximality of \( B \) ; therefore, there exist \( {c}_{0},\ldots ,{c}_{t} \in k \) and (distinct) \( {b}_{1},\ldots ,{b}_{t} \in B \) such that \[ {c}_{0}v + {c}_{1}{b}_{1} + \cdots + {c}_{t}{b}_{t} = 0, \] with not all \( {c}_{0},\ldots ,{c}_{t} \) equal to 0 . Now, \( {c}_{0} \neq 0 \) : otherwise we would get a linear dependence relation among elements of \( B \) . Since \( k \) is a field, \( {c}_{0} \) is a unit; but then \[ v = \left( {-{c}_{0}^{-1}{c}_{1}}\right) {b}_{1} + \cdots + \left( {-{c}_{0}^{-1}{c}_{t}}\right) {b}_{t}, \] proving that \( v \) is in the span of \( B \) . It follows that \( B \) generates \( V \), as needed.

Proposition 1.5 A scalar product satisfies the Cauchy-Schwarz \( {}^{1} \) inequality \[ b{\left( x, y\right) }^{2} \leq q\left( x\right) q\left( y\right) ,\;\forall x, y \in E. \] The equality holds true if and only if \( x \) and \( y \) are colinear. Proof. The polynomial \[ t \mapsto q\left( {{tx} + y}\right) = q\left( x\right) {t}^{2} + {2b}\left( {x, y}\right) t + q\left( y\right) \] takes nonnegative values for \( t \in \mathbb{R} \) . Hence its discriminant \( 4\left( {b{\left( x, y\right) }^{2} - q\left( x\right) q\left( y\right) }\right) \) is nonpositive. When the latter vanishes, the polynomial has a real root \( {t}_{0} \), which implies that \( {t}_{0}x + y = 0 \) . The Cauchy-Schwarz inequality implies immediately \[ q\left( {x + y}\right) \leq {\left( \sqrt{q\left( x\right) } + \sqrt{q\left( y\right) }\right) }^{2}, \] which means that the square root \( \parallel \cdot \parallel \mathrel{\text{:=}} {q}^{1/2} \) satisfies the triangle inequality \[ \parallel x + y\parallel \leq \parallel x\parallel + \parallel y\parallel \] --- \( {}^{1} \) In Cauchy-Schwarz, the name Schwarz (1843-1921) is spelled without a \( t \) . --- Because \( \parallel \cdot \parallel \) is positively homogeneous, it is thus a norm over \( E \) : every Euclidean space is a normed space. The converse is obviously false. The space \( {\mathbb{R}}^{n} \) is endowed with a canonical scalar product \[ \langle x, y\rangle \mathrel{\text{:=}} {x}_{1}{y}_{1} + \cdots + {x}_{n}{y}_{n} \] The corresponding norm is \[ \parallel x\parallel = {\left( {x}_{1}^{2} + \cdots + {x}_{n}^{2}\right) }^{1/2}. \] It is denoted \( \parallel \cdot {\parallel }_{2} \) in Chapter 7. ## 1.3.6 Hermitian Spaces When the scalar field is that of complex numbers \( \mathbb{C} \), the complex conjugation yields an additional structure. Definition 1.3 Let \( E \) be a complex space, and \( \phi : E \times E \rightarrow \mathbb{C} \) be a scalar-valued map. We say that \( \phi \) is a sesquilinear form if it satisfies the following Linearity: For every \( x \in E, y \mapsto \phi \left( {x, y}\right) \) is linear, Anti-linearity: For every \( y \in E, x \rightarrow \phi \left( {x, y}\right) \) is antilinear, meaning \[ \phi \left( {{\lambda x} + {x}^{\prime }, y}\right) = \bar{\lambda }\phi \left( {x, y}\right) + \phi \left( {{x}^{\prime }, y}\right) . \] Given a sesquilinear form \( \phi \), the formula \( \psi \left( {x, y}\right) \mathrel{\text{:=}} \overline{\phi \left( {y, x}\right) } \) defines another sesquilinear form, in general different from \( \phi \) . The equality case is especially interesting: Definition 1.4 An Hermitian form is a sesquilinear form satisfying in addition \[ \phi \left( {y, x}\right) = \overline{\phi \left( {x, y}\right) },\;\forall x, y \in E. \] For an Hermitian form, the function \( q\left( x\right) \mathrel{\text{:=}} \phi \left( {x, x}\right) \) is real-valued and satisfies \[ q\left( {\lambda x}\right) = {\left| \lambda \right| }^{2}q\left( x\right) \] The form \( \phi \) can be retrieved from \( q \) via the formula \[ \phi \left( {x, y}\right) = \frac{1}{4}\left( {q\left( {x + y}\right) - q\left( {x - y}\right) - \mathrm{i}q\left( {x + \mathrm{i}y}\right) + \mathrm{i}q\left( {x - \mathrm{i}y}\right) }\right) . \] (1.1) Definition 1.5 An Hermitian form is said to be positive definite if \( q\left( x\right) > 0 \) for every \( x \neq 0 \) . There are also semipositive-definite Hermitian forms, satisfying \( q\left( x\right) \geq 0 \) for every \( x \in E \) . A semipositive-definite form satisfies the Cauchy-Schwarz inequality \[ {\left| \phi \left( x, y\right) \right| }^{2} \leq q\left( x\right) q\left( y\right) ,\;\forall x, y \in E. \] In the positive-definite case, the equality holds if and only if \( x \) and \( y \) are colinear. Definition 1.6 An Hermitian space is a pair \( \left( {E,\phi }\right) \) where \( E \) is a complex space and \( \phi \) is a positive-definite Hermitian form. As in the Euclidean case, an Hermitian form is called a scalar product. An Hermitian space is a normed space, where the norm is given by \[ \parallel x\parallel \mathrel{\text{:=}} \sqrt{q\left( x\right) }. \] The space \( {\mathbb{C}}^{n} \) is endowed with a canonical scalar product \[ \langle x, y\rangle \mathrel{\text{:=}} {\bar{x}}_{1}{y}_{1} + \cdots + {\bar{x}}_{n}{y}_{n} \] The corresponding norm is \[ \parallel x\parallel = {\left( {\left| {x}_{1}\right| }^{2} + \cdots + {\left| {x}_{n}\right| }^{2}\right) }^{1/2}. \] ## Chapter 2 What Are Matrices ## 2.1 Introduction In real life, a matrix is a rectangular array with prescribed numbers \( n \) of rows and \( m \) of columns \( \left( {n \times m\text{matrix}}\right) \) . To make this array as clear as possible, one encloses it between delimiters; we choose parentheses in this book. The position at the intersection of the \( i \) th row and \( j \) th column is labeled by the pair \( \left( {i, j}\right) \) . If the name of the matrix is \( M \) (respectively, \( A, X \), etc.), the entry at the \( \left( {i, j}\right) \) th position is usually denoted \( {m}_{ij} \) (respectively, \( {a}_{ij},{x}_{ij} \) ). An entry can be anything provided it gives the reader information. Here is a the real-life example. \[ M = \left( \begin{matrix} {11} & {27} & {83} \\ \text{ blue } & \text{ green } & \text{ yellow } \\ \text{ undefined Republican Democrat } & & \end{matrix}\right) \] Perhaps this matrix gives the age, the preferred color, and the political tendency of three people. In the present book, however, we restrict to matrices whose entries are mathematical objects. In practice, they are elements of a ring \( A \) . In most cases, this ring is Abelian; if it is a field, then it is denoted \( k \) or \( K \), unless it is one of the classical number fields \( \mathbb{Q},\mathbb{R},\mathbb{C},{\mathbb{F}}_{p} \) . When writing a matrix blockwise, it becomes a smaller matrix whose elements are themselves matrices, and thus belong to some spaces that are not even rings; having possibly different sizes, these submatrices may even belong to distinct sets. In some circumstances (extraction of matrices or minors, e.g.) the rows and the columns can be numbered in a different way, using nonconsecutive numbers \( i \) and \( j \) . In general one needs only two finite sets \( I \) and \( J \), one for indexing the rows and the other for indexing the columns. For instance, the following extraction from a \( 4 \times 5 \) matrix \( M \) corresponds to the choice \( I = \left( {1,3}\right), J = \left( {2,5,3}\right) \) . \[ {M}_{I}^{J} = \left( \begin{array}{lll} {m}_{12} & {m}_{15} & {m}_{13} \\ {m}_{32} & {m}_{35} & {m}_{33} \end{array}\right) . \] Notice that the indices need not be taken in increasing order. ## 2.1.1 Addition of Matrices The set of matrices of size \( n \times m \) with entries in \( A \) is denoted by \( {\mathbf{M}}_{n \times m}\left( A\right) \) . It is an additive group, where \( M + {M}^{\prime } \) denotes the matrix \( {M}^{\prime \prime } \) whose entries are given by \( {m}_{ij}^{\prime \prime } = {m}_{ij} + {m}_{ij}^{\prime } \) . ## 2.1.2 Multiplication by a Scalar One defines the multiplication by a scalar \( a \in A : {M}^{\prime } \mathrel{\text{:=}} {aM} \) by \( {m}_{ij}^{\prime } = a{m}_{ij} \) . One has the formulæ \( a\left( {bM}\right) = \left( {ab}\right) M, a\left( {M + {M}^{\prime }}\right) = \left( {aM}\right) + \left( {a{M}^{\prime }}\right) \), and \( \left( {a + b}\right) M = \) \( \left( {aM}\right) + \left( {bM}\right) \) . Likewise we define \( {M}^{\prime \prime } = {Ma} \) by \( {m}_{ij}^{\prime \prime } \mathrel{\text{:=}} {m}_{ij}a \) and we have similar properties, together with \( \left( {aM}\right) b = a\left( {Mb}\right) \) . With these operations, \( {\mathbf{M}}_{n \times m}\left( A\right) \) is a left and right \( A \) -module. If \( A \) is Abelian, then \( {aM} = {Ma} \) . When the set of scalars is a field \( K,{\mathbf{M}}_{n \times m}\left( K\right) \) is a \( K \) -vector space. The zero matrix is denoted by 0, or \( {0}_{nm} \) when one needs to avoid ambiguity: \[ {0}_{n \times m} = \left( \begin{matrix} 0 & \cdots & 0 \\ \vdots & & \vdots \\ 0 & \cdots & 0 \end{matrix}\right) \] When \( m = n \), one writes simply \( {\mathbf{M}}_{n}\left( K\right) \) instead of \( {\mathbf{M}}_{n \times n}\left( K\right) \), and \( {0}_{n} \) instead of \( {0}_{nn} \) . The matrices of sizes \( n \times n \) are called square matrices of size \( n \) . When \( A \) has a unit 1, one writes \( {I}_{n} \) for the identity matrix, a square matrix of order \( n \) defined by \[ {m}_{ij} = {\delta }_{i}^{j} = \left\{ \begin{array}{ll} 0, & \text{ if }i \neq j \\ 1, & \text{ if }i = j \end{array}\right. \] In other words, \[ {I}_{n} = \left( \begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{matrix}\right) \] ## 2.1.3 Special Matrices The identity matrix is a special case of a permutation matrix, which is a square matrix having exactly one nonzero entry in each row and each column, that entry being a 1 . In other words, a permutation matrix \( M \) reads \[ {m}_{ij} = {\delta }_{i}^{\sigma \left( j\right) } \] for some permutation \( \sigma \in {S}_{n} \) . A square matrix for which \( i < j \) implies \( {m}_{ij} = 0 \) is called a lower-triangular matrix. It is upper-triangular if \( i > j \) implies \( {m}_{ij} = 0 \) . It is strictly upper- (respectively, lower)-triangular if \( i \geq j \) (respectively, \( i \leq j \) ) implies \( {m}_{ij} = 0 \) . It is diagonal if \( {m}_{ij} \) vanishes for every pair \( \left( {i, j}\right) \) such that \( i \neq j \) . When \( {d}_{1},\ldots ,{d}_{n} \in A \) are given, one denotes by \( \operatorname{diag}\left( {{d}_{1},\ldots ,{d}_{n}}\right) \) the diagonal matrix \( M \) whose diagonal term \( {m}_{ii} \) equals \( {d}_{i} \) for every index \( i

Proposition 1.5 A scalar product satisfies the Cauchy-Schwarz inequality \[ b{\left( x, y\right) }^{2} \leq q\left( x\right) q\left( y\right) ,\;\forall x, y \in E. \] The equality holds true if and only if \( x \) and \( y \) are colinear.

The polynomial \[ t \mapsto q\left( {{tx} + y}\right) = q\left( x\right) {t}^{2} + {2b}\left( {x, y}\right) t + q\left( y\right) \] takes nonnegative values for \( t \in \mathbb{R} \). Hence its discriminant \( 4\left( {b{\left( x, y\right) }^{2} - q\left( x\right) q\left( y\right) }\right) \) is nonpositive. When the latter vanishes, the polynomial has a real root \( {t}_{0} \), which implies that \( {t}_{0}x + y = 0 \).

Proposition 7.5. Let \( X \) be a homogeneous space for \( G \) . If \( X \) is strictly unimodular, then there exists a left G-invariant volume form on \( X \), unique up to a constant multiple. Proof. We want to define the invariant form on \( G/H \) by translating a given volume form \( {\omega }_{e} \) on \( {T}_{e}\left( {G/H}\right) \) . On \( G/H \), the left translation \( {L}_{h} \) is induced by conjugation \( {\mathbf{c}}_{h} \) on \( G \) . By Proposition 7.4 and the hypothesis, we have \[ \text{det}T{L}_{h}\left( {e}_{G/H}\right) = \det T{\mathbf{c}}_{h}\left( {e}_{G/H}\right) = 1\text{.} \] Hence \( {L}_{h}{\omega }_{e}\left( {G/H}\right) = {\omega }_{e\left( {G/H}\right) } \), that is \( {\omega }_{e\left( {G/H}\right) } \) is invariant under translations by elements of \( H \) . Then for any \( g \in G \) we define \[ {\omega }_{gH} = {L}_{g}{\omega }_{e}\left( {G/H}\right) \] The value on the right is independent of the coset representative \( g \), and it is then clear that translation yields the desired \( G \) -invariant volume form on \( G/H \) . The uniqueness up to a constant factor follows because the invariant forms are determined linearly from their values at the origin, and the forms at the origin constitute a 1-dimensional space. This concludes the proof. Remark. If both \( G \) and \( H \) are unimodular, then so is \( G/H \) . If \( H \) is compact, then \( H \) is unimodular. If \( G \) is unimodular in addition, so is \( G/H \) . The same goes for strict unimodularity. When one applies the above considerations to Haar measures and integration, what matters is modularity, not the strict modularity. Cf. Chapter XVI, Theorem 4.3, and Chapter XVI, \( §5 \) for a derivation of some of these results in the context of Haar measure. Of course, the existence of an invariant volume form on Lie groups was known at the end of nineteenth century. At the time, into the twentieth century, it was a problem whether an invariant measure could be found on any locally compact group, and this problem was solved by Haar, whence the name Haar measure. In the next section, we shall accordingly define Haar forms, to fit into the psychology which has developed since Haar's result, even though invariant forms were known long before this result. I found dealing with the Haar forms rather than Haar measure to provide additional flexibility. Then one has to make a distinction between modularity and strict modularity, but it isn't at all serious for local results. In all examples I know, the number of components is finite, and local results can be reduced to the case when the groups are connected, sometimes by passing to finite covering. ## XV, §8. HOMOGENEOUSLY FIBERED SUBMERSIONS In [He 72], Helgason obtained a formula for the Laplacian in a Riemannin submersion admitting horizontal metric sections. The result was reproduced in his book [He 84], Chapter II, Theorem 3.7, and concerns the case when there is a homogeneity condition on the fibers of the submersion. The present section developed from the attempt by \( \mathrm{{Wu}} \) and myself to understand Helgason's situation better, from the point of view of local Riemannian geometry. The results of \( \$ 6 \) were developed with this goal in mind, and will thus have their first application here, together with an important fact due to \( \mathrm{{Wu}} \) . We start without a Riemannian structure. For the first two basic properties, we don’t need finite dimensionality. So let \( X, Z \) be connected possibly infinite dimensional manifolds, and let \[ \pi : X \rightarrow Z \] be a submersion. We shall say that the submersion is homogeneously fibered if it satisfies the following condition. HF Condition. There is a possibly infinite dimensional Lie group \( H \) acting as a group of differential automorphisms on \( X \), preserving the fibers, such that at each point \( x \), we have a differential isomorphism \[ H/{H}_{x} \rightarrow {Y}_{\pi \left( x\right) } = {\pi }^{-1}\left( {\pi \left( x\right) }\right) \] of \( H/{H}_{x} \) -principal homogeneous space given by \( h \mapsto {hx} \) . Note that a submersion always admits local differential sections, but in general these do not need to be metric. Furthermore, the submersion need not admit a global section. The next proposition applies to local sections when the need arises, but we shall not use it in this book. Proposition 8.1. Suppose there is a section \( \sigma : Z \rightarrow X \) of a homogeneously fibered submersion. Define \[ \gamma : H \times Z \rightarrow X\;\text{ by }\;\gamma \left( {h, z}\right) = {h\sigma }\left( z\right) . \] Then \( \gamma \) is a submersion. Proof. The tangent map \( {T\gamma }\left( {h, z}\right) \) is a surjective homomorphism of tangent spaces at each point. In fact, if we let \( {\gamma }_{h}\left( {\sigma \left( z\right) }\right) = h\left( {\sigma z}\right) = \gamma \left( {h, z}\right) \) , then \( T{\gamma }_{h}\left( {\sigma \left( z\right) }\right) \) gives a linear isomorphism of the tangent spaces to the fiber. On the other hand, \( {T\sigma } \) gives a linear isomorphism of the tangent space \( {T}_{z}Z \) to a subspace of \( {T}_{\sigma \left( z\right) }X \), and we have the direct sum decomposition at the point \( x = \sigma \left( z\right) \) , \[ {T}_{\sigma \left( z\right) }X = {T}_{x}\left( {Hx}\right) \oplus {\sigma }_{ * }{T}_{z}Z \] This concludes the proof. Suppose in addition that \( \pi : X \rightarrow Z \) is a Riemannian submersion, and \( H \) acts isometrically. We then say that \( \pi : X \rightarrow Z \) is a metrically homogeneously fibered submersion. We suppose this is the case from now on. An immediate question which arises about the isotropy groups \( {H}_{x} \) is the extent to which they can vary (up to conjugation). I owe the next key result to Wu. Theorem 8.2 (Wu). Let \( \pi : X \rightarrow Z \) be a metrically homogeneously fibered submersion. For any two points \( x, y \in X \), the isotropy groups \( {H}_{x} \) , \( {H}_{y} \) are conjugate in \( H \) . In fact, let \( x, y \) be points of \( X \) which can be joined by the horizontal lift of a curve in \( Z \) . Then \( {H}_{x} = {H}_{y} \), and the flow of the horizontal lift induces an H-homogeneous space isomorphism between the fibers at \( x \) and at \( y \) . Proof. We recall that the horizontal lift was defined in Chapter XIV, §3. Suppose first that \( x, y \) can be joined by a horizontal lift \( A \) . Let \( h \in {H}_{x} \) . Since \( H \) acts isometrically on \( X, h \circ A \) is the unique horizontal lift from \( {hx} = x \) to \( {hy} \) . But \( h \circ A \) has the same initial conditions as \( A \), and so coincides with \( A \) by the uniqueness of solutions of differential equations. Hence \( {hy} = y \), and \( h \in {H}_{y} \) . The reverse inclusion \( {H}_{y} \subset {H}_{x} \) follows by symmetry, so \( {H}_{x} = {H}_{y} \) . Next, for arbitrary points \( x, y \in X \), consider any curve in \( Z \) between \( \pi \left( x\right) \) and \( \pi \left( y\right) \) . Then the horizontal lift of this curve in \( X \) joins \( x \) to a point \( {y}^{\prime } \) in the same fiber as \( y \), and the isotropy groups of \( y \) and \( {y}^{\prime } \) are conjugate. Finally, let \( F \) be the flow of horizontal lifts, that is \( {F}_{t}\left( x\right) = {A}_{x}\left( t\right) \), where \( {A}_{x} \) is the horizontal lift of a curve \( {\alpha }_{\pi \left( x\right) } \) with initial condition \( \pi \left( x\right) \) on \( Z \) . Then \( t \mapsto {F}_{t}\left( {hx}\right) \) and \( t \mapsto h{A}_{x}\left( t\right) \) are horizontal lifts with the same initial conditions, and so are equal. This concludes the proof. We now assume finite dimensionality, so we have volume forms. In addition, we assume that the fibers are strictly unimodular, i.e. \( H/{H}_{x} \) is strictly unimodular for all \( x \), in which case we say that the homogeneous fibration is strictly unimodular. We then select a fixed Haar form on one of the coset spaces \( H/{H}_{o} \) with one of the isotropy groups. Then conjugation transforms this Haar form to a Haar form \( {\operatorname{Haar}}_{H/{H}_{x}} \) for all \( x \in X \) . Let \( {Y}_{z} \) be a fiber of the submersion, with \( z = \pi \left( x\right) \), so we obtain a homogeneous space isomorphism \( H/{H}_{x}\overset{ \approx }{ \rightarrow }{Y}_{z} = {Y}_{\pi \left( x\right) } \) . Selecting two different points in the fiber above \( z \) give rise to different isomorphisms, but the unimodularity condition implies that the Haar form on \( H/{H}_{x} \) corresponds to a Haar form on \( {Y}_{z} \) independent of the choice of point \( x \) in the fiber. We denote this Haar form by \( {\operatorname{Haar}}_{{Y}_{z}} \) . Let \( {\Omega }_{X},{\Omega }_{Z} \) be the Riemannian volume forms on \( X \) (resp. \( Z \) ). Then there is a function \( \delta \) on \( Z \) such that for each \( z \), and \( y \in {Y}_{z} \) we have \[ {\Omega }_{{Y}_{z}}\left( y\right) = \delta \left( z\right) {\operatorname{Haar}}_{{Y}_{z}}\left( y\right) \] It is immediate that \( \delta \) is \( {C}^{\infty } \) (say from a local coordinate representation). We call \( \delta \) the Riemannian Haar density. The Haar form \( \Psi \) on \( X \) is defined to be the \( p \) -form \( \left( {p = \text{fiber dimension}}\right) \) whose restriction to each fiber is the Haar form as above, and which is 0 on decomposable elements containing a horizontal field. Equivalently, let \( \left\{ {{\xi }_{1},\ldots ,{\xi }_{p}}\right\} \) be a frame of vertical fields on some open subset of \( X \), and \( \left\{ {{\mu }_{1},\ldots ,{\mu }_{q}}\right\} \) a frame of horizontal fields. Then there exists a function \( \varphi \) such that with the dual frame \( \left\{ {{\xi }_{1}^{\prime },\ldots ,{\xi }_{p}^{\prime },{\mu }_{1}^{\prime },\ldots ,{\mu }_{q}^{\prime }}\right\} \) we have \[ \Psi = \varphi {\xi }_{1}^{\prime } \land \

Proposition 7.5. Let \( X \) be a homogeneous space for \( G \) . If \( X \) is strictly unimodular, then there exists a left G-invariant volume form on \( X \), unique up to a constant multiple.

We want to define the invariant form on \( G/H \) by translating a given volume form \( {\omega }_{e} \) on \( {T}_{e}\left( {G/H}\right) \) . On \( G/H \), the left translation \( {L}_{h} \) is induced by conjugation \( {\mathbf{c}}_{h} \) on \( G \) . By Proposition 7.4 and the hypothesis, we have \[ \text{det}T{L}_{h}\left( {e}_{G/H}\right) = \det T{\mathbf{c}}_{h}\left( {e}_{G/H}\right) = 1\text{.} \] Hence \( {L}_{h}{\omega }_{e}\left( {G/H}\right) = {\omega }_{e\left( {G/H}\right) } \), that is \( {\omega }_{e\left( {G/H}\right) } \) is invariant under translations by elements of \( H \) . Then for any \( g \in G \) we define \[ {\omega }_{gH} = {L}_{g}{\omega }_{e}\left( {G/H}\right) \] The value on the right is independent of the coset representative \( g \), and it is then clear that translation yields the desired \( G \) -invariant volume form on \( G/H \) . The uniqueness up to a constant factor follows because the invariant forms are determined linearly from their values at the origin, and the forms at the origin constitute a 1-dimensional space. This concludes the proof.

Theorem 4.2.5 If \( \mathcal{M} \) is a countable model of \( {PA} \), then there is \( \mathcal{M} \prec \mathcal{N} \) such that \( \mathcal{N} \) is a proper end extension of \( \mathcal{M} \) . Proof Consider the language \( {\mathcal{L}}^{ * } \) where we have constant symbols for all elements of \( M \) and a new constant symbol \( c \) . Let \( T = {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) \cup \{ c > m \) : \( m \in M\} \), and for \( a \in M \smallsetminus \mathbb{N} \) let \( {p}_{a} \) be the type \( \{ v < a, v \neq m : m \in M\} \) . Any \( \mathcal{N} \vDash T \) is a proper elementary extension of \( \mathcal{M} \) . If \( \mathcal{N} \) omits each \( {p}_{a} \) , then \( \mathcal{N} \) is an end extension of \( \mathcal{M} \) . By Theorem 4.2.4, it suffices to show that each \( {p}_{a} \) is nonisolated. Suppose that \( \phi \left( v\right) \) is an \( {\mathcal{L}}^{ * } \) formula isolating \( {p}_{a} \) . Let \( \phi \left( v\right) = \theta \left( {v, c}\right) \), where \( \theta \) is an \( {\mathcal{L}}_{M} \) -formula. Then \[ T \cup \theta \left( {v, c}\right) \vDash v < a. \] Because \( T \cup \{ \theta \left( {v, c}\right) \} \) is satisfiable, \[ \mathcal{M} \vDash \forall x\exists y > x\exists v < {a\theta }\left( {v, y}\right) . \] The Pigeonhole Principle is provable in Peano arithmetic. Thus \[ \mathcal{M} \vDash \left\lbrack {\forall x\exists y > x\exists v < {a\theta }\left( {v, y}\right) }\right\rbrack \rightarrow \exists v < a\forall x\exists y > {x\theta }\left( {v, y}\right) . \] \( \left( {* * }\right) \) Thus, there is \( m < a \) such that \[ \mathcal{M} \vDash \forall x\exists y > {x\theta }\left( {m, y}\right) . \] We claim that \( T \cup \{ \theta \left( {m, c}\right) \} \) is satisfiable. If not, there is \( n \in M \) such that \[ {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) + c > n \vDash \neg \theta \left( {m, c}\right) \] contradicting \( \left( {* * }\right) \) . Thus, \( \phi \left( v\right) \) does not isolate \( {p}_{a} \), a contradiction. ## Prime and Atomic Models We use the Omitting Types Theorem to study small models of a complete theory. For the remainder of this section, we will assume that \( \mathcal{L} \) is a countable language and \( T \) is a complete \( \mathcal{L} \) -theory with infinite models. Definition 4.2.6 We say that \( \mathcal{M} \vDash T \) is a prime model of \( T \) if whenever \( \mathcal{N} \vDash T \) there is an elementary embedding of \( \mathcal{M} \) into \( \mathcal{N} \) . For example, let \( T = {\mathrm{{ACF}}}_{0} \) . If \( K \vDash {\mathrm{{ACF}}}_{0} \), and \( F \) is the algebraic closure of \( \mathbb{Q} \), then there is an embedding of \( F \) into \( K \) . Because \( {\mathrm{{ACF}}}_{0} \) is model complete this embedding is elementary. Thus, \( F \) is a prime model of \( {\mathrm{{ACF}}}_{0} \) . Similarly, RCF has a prime model, the real closure of \( \mathbb{Q} \) . For a third example, consider \( \mathcal{L} = \{ + , \cdot , < ,0,1\} \) and let \( T \) be \( \operatorname{Th}\left( \mathbb{N}\right) \) , true arithmetic. If \( \mathcal{M} \vDash T \), then we can view \( \mathbb{N} \) as an initial segment of \( \mathcal{M} \) . We claim that this embedding is elementary. We use the Tarski-Vaught test (Proposition 2.3.5). Let \( \phi \left( {v,{w}_{1},\ldots ,{w}_{m}}\right) \) be an \( \mathcal{L} \) -formula and let \( {n}_{1},\ldots ,{n}_{m} \in \mathbb{N} \) such that \( \mathcal{M} \vDash \exists {v\phi }\left( {v,\bar{n}}\right) \) . Let \( \psi \) be the \( \mathcal{L} \) -sentence \[ \exists {v\phi }\left( {v,\underset{{n}_{1} - \text{ times }}{\underbrace{1 + \ldots + 1}},\ldots ,\underset{{n}_{m} - \text{ times }}{\underbrace{1 + \ldots + 1}}}\right) . \] Then, \( \mathcal{M} \vDash \psi \) and \( \mathbb{N} \vDash \psi \) because \( \mathcal{M} \equiv \mathbb{N} \) . But then, for some \( s \in \mathbb{N} \) , \[ \mathbb{N} \vDash \phi \left( {s,\underset{{n}_{1} - \text{ times }}{\underbrace{1 + \ldots + 1}},\ldots ,\underset{{n}_{m} - \text{ times }}{\underbrace{1 + \ldots + 1}}}\right) \] and \[ \mathbb{N} \vDash \phi \left( {\underset{\mathrm{s} - \text{ times }}{\underbrace{1 + \ldots + 1}},\underset{{n}_{1} - \text{ times }}{\underbrace{1 + \ldots + 1}},\ldots ,\underset{{n}_{m} - \text{ times }}{\underbrace{1 + \ldots + 1}}}\right) . \] Because the latter statement is an \( \mathcal{L} \) -sentence, \[ \mathcal{M} \vDash \phi \left( {\underset{\mathrm{s} - \text{ times }}{\underbrace{1 + \ldots + 1}},\underset{{n}_{1} - \text{ times }}{\underbrace{1 + \ldots + 1}},\ldots ,\underset{{n}_{m} - \text{ times }}{\underbrace{1 + \ldots + 1}}}\right) \] and \( \mathcal{M} \vDash \phi \left( {s,{n}_{1},\ldots ,{n}_{m}}\right) \) . By the Tarski-Vaught test, \( \mathbb{N} \prec \mathcal{M} \) . Thus, \( \mathbb{N} \) is a prime model of \( T \) . Suppose \( \mathcal{M} \) is a prime model of \( T \) . Suppose that \( j : \mathcal{M} \rightarrow \mathcal{N} \) is an elementary embedding. If \( \bar{a} \in {M}^{n} \) realizes \( p \in {S}_{n}\left( T\right) \), then so does \( j\left( \bar{a}\right) \) . If \( p \in {S}_{n}\left( T\right) \) is nonisolated, there is \( \mathcal{N} \) such that \( \mathcal{N} \) omits \( p \) . If \( \mathcal{M} \) realizes \( p \) , then we can not elementarily embed \( \mathcal{M} \) into \( \mathcal{N} \) ; thus, \( \mathcal{M} \) must also omit \( p \) . In particular, if \( \bar{a} \in {M}^{n} \), then \( {\operatorname{tp}}^{\mathcal{M}}\left( \bar{a}\right) \) must be isolated. This leads us to the following definition. Definition 4.2.7 We say that \( \mathcal{M} \vDash T \) is atomic if \( {\operatorname{tp}}^{\mathcal{M}}\left( \bar{a}\right) \) is isolated for all \( \bar{a} \in {M}^{n} \) . We have just argued that prime models are atomic. For countable models, the converse is also true. Theorem 4.2.8 Let \( \mathcal{L} \) be a countable language and let \( T \) be a complete \( \mathcal{L} \) -theory with infinite models. Then, \( \mathcal{M} \vDash T \) is prime if and only if it is countable and atomic. ## Proof \( \left( \Rightarrow \right) \) We have argued that prime models are atomic. Because \( \mathcal{L} \) is countable, \( T \) has a countable model. Thus, the prime model must be countable. \( \left( \Leftarrow \right) \) Let \( \mathcal{M} \) be countable and atomic. Let \( \mathcal{N} \vDash T \) . We must construct an elementary embedding of \( \mathcal{M} \) into \( \mathcal{N} \) . Let \( {m}_{0},{m}_{1},\ldots ,{m}_{n},\ldots \) be an enumeration of \( M \) . For each \( i \), let \( {\theta }_{i}\left( {{v}_{0},\ldots ,{v}_{i}}\right) \) isolate the type of \( \left( {{m}_{0},\ldots ,{m}_{i}}\right) \) . We will build \( {f}_{0} \subseteq {f}_{1} \subseteq \ldots \) a sequence of partial elementary maps from \( \mathcal{M} \) into \( \mathcal{N} \) where the domain of \( {f}_{i} \) is \( \left\{ {{m}_{0},\ldots ,{m}_{i - 1}}\right\} \) . Then, \( f = \mathop{\bigcup }\limits_{{i = 0}}^{\infty }{f}_{i} \) is an elementary embedding of \( \mathcal{M} \) into \( \mathcal{N} \) . Let \( {f}_{0} = \varnothing \) . Because \( \mathcal{M} \equiv \mathcal{N},{f}_{0} \) is partial elementary. Given \( {f}_{s} \), let \( {n}_{i} = f\left( {m}_{i}\right) \) for \( i < s \) . Because \( {\theta }_{s}\left( {{m}_{0},\ldots ,{m}_{s}}\right) \) and \( {f}_{s} \) is partial elementary, \[ \mathcal{N} \vDash \exists v{\theta }_{s}\left( {{n}_{0},\ldots ,{n}_{s - 1}, v}\right) \] Let \( {n}_{s} \in N \) such that \( \mathcal{N} \vDash {\theta }_{s}\left( {{n}_{0},\ldots ,{n}_{s}}\right) \) . Because \( {\theta }_{s} \) isolates \( {\operatorname{tp}}^{\mathcal{M}}\left( {{m}_{0},\ldots ,{m}_{s}}\right) \) \[ {\operatorname{tp}}^{\mathcal{M}}\left( {{m}_{0},\ldots ,{m}_{s}}\right) = {\operatorname{tp}}^{\mathcal{N}}\left( {{n}_{0},\ldots ,{n}_{s}}\right) \] Thus, \( {f}_{s + 1} = {f}_{s} \cup \left\{ \left( {{m}_{s},{n}_{s}}\right) \right\} \) is a partial elementary map. Theorem 4.2.8 will lead to a criterion for the existence of prime models. We need one preparatory lemma. Lemma 4.2.9 Suppose that \( \left( {\bar{a},\bar{b}}\right) \in {M}^{m + n} \) realizes an isolated type in \( {S}_{m + n}\left( T\right) \) . Then \( \bar{a} \) realizes an isolated type in \( {S}_{m}\left( T\right) \) . Indeed if \( A \subseteq M \) and \( \left( {\bar{a},\bar{b}}\right) \in {M}^{m + n} \) realizes an isolated type in \( {S}_{m + n}^{\mathcal{M}}\left( A\right) \), then \( {\operatorname{tp}}^{\mathcal{M}}\left( {\bar{a}/A}\right) \) is isolated. Proof Let \( \phi \left( {\bar{v},\bar{w}}\right) \) isolate \( {\operatorname{tp}}^{\mathcal{M}}\left( {\bar{a},\bar{b}/A}\right) \) . We claim that \( \exists {w\phi }\left( {\bar{v},\bar{w}}\right) \) isolates \( {\operatorname{tp}}^{\mathcal{M}}\left( {\bar{a}/A}\right) \) . Let \( \psi \left( \bar{v}\right) \) be any \( {\mathcal{L}}_{A} \) -formula such that \( \mathcal{M} \vDash \psi \left( \bar{a}\right) \) . We must show that \[ {\operatorname{Th}}_{A}\left( \mathcal{M}\right) \vDash \exists \overline{w}\left( {\phi \left( {\overline{v},\overline{w}}\right) \rightarrow \psi \left( \overline{v}\right) }\right) . \] Suppose not. Then, there is \( \bar{c} \in {M}^{m} \) such that \[ \mathcal{M} \vDash \exists \bar{w}\left( {\phi \left( {\bar{c},\bar{w}}\right) \land \neg \psi \left( \bar{c}\right) }\right) . \] Let \( \bar{d} \in {M}^{n} \) such that \( \mathcal{M} \vDash \phi \left( {\bar{c},\bar{d}}\right) \land \neg \psi \left( \bar{c}\right) \) . Because \( \phi \left( {\bar{v},\bar{w}}\right) \) isolates \( {\operatorname{tp}}^{\mathcal{M}}\left( {\overline{a},\overline{b}/A}\right) , \) \[ {\operatorname{Th}}_{A}\left( \mathcal{M}\right) \vDash \phi \left( {\bar{v},\bar{w}}\right) \rightarrow \psi \left( \bar{v}\right) \] This is a contradiction because \[ \psi \left( \bar{v}\right) \in {\operatorname{tp}}^{\mathcal{M}}

Theorem 4.2.5 If \( \mathcal{M} \) is a countable model of \( {PA} \), then there is \( \mathcal{M} \prec \mathcal{N} \) such that \( \mathcal{N} \) is a proper end extension of \( \mathcal{M} \).

Consider the language \( {\mathcal{L}}^{ * } \) where we have constant symbols for all elements of \( M \) and a new constant symbol \( c \). Let \( T = {\operatorname{Diag}}_{\mathrm{{el}}}\left( \mathcal{M}\right) \cup \{ c > m : m \in M\} \), and for \( a \in M \smallsetminus \mathbb{N} \) let \( {p}_{a} \) be the type \( \{ v < a, v \neq m : m \in M\} \). Any \( \mathcal{N} \vDash T \) is a proper elementary extension of \( \mathcal{M} \). If \( \mathcal{N} \) omits each \( {p}_{a} \), then \( \mathcal{N} \) is an end extension of \( \mathcal{M} \). By Theorem 4.2.4, it suffices to show that each \( {p}_{a} \) is nonisolated. Suppose that \( \phi \left( v\right) \) is an \( {\mathcal{L}}^{ * } \) formula isolating \( {p}_{a} \). Let \( \phi \left( v\right) = \theta \left( {v, c}\right) \), where \( \theta \) is an \( {\mathcal{L}}_{M} \) -formula. Then \[ T \cup \theta \left( {v, c}\right) \vDash v < a. \] Because \( T \cup \{ \theta \left( {v, c}\right) \} \) is satisfiable, \[ \mathcal{M} \vDash \forall x\exists y > x\exists v < {a\theta }\left( {v, y}\right) . \] The Pigeonhole Principle is provable in Peano arithmetic. Thus \[ \mathcal{M} \vDash \left\lbrack {\forall x\exists y > x\exists v < {a\theta }\left( {v, y}\right) }\right\rbrack \rightarrow \exists v < a\forall x\exists y > {x\theta }\left( {v, y}\right) . \] Thus, there is \( m < a \) such that \[ \mathcal{M}

Proposition 5.1.16. If \( q \) is nondegenerate and represents 0, then there exists a hyperbolic form \( h \) and a nondegenerate form \( {q}^{\prime } \) such that \( q \sim h \oplus {q}^{\prime } \) . Furthermore, \( q \) represents all elements of \( K \) . Proof. This is Lemma 5.1.5 and Corollary 5.1.6. The fact that \( {q}^{\prime } \) is nondegenerate follows from Proposition 5.1.2. Corollary 5.1.17. Let \( q \) be a nondegenerate quadratic form in \( n \) variables and let \( c \in {K}^{ * } \) . The following three conditions are equivalent: (1) The form \( q \) represents \( c \) . (2) There exists a quadratic form \( {q}_{1} \) in \( n - 1 \) variables such that \( q \sim c{x}_{0}^{2} \oplus {q}_{1} \) . (3) The quadratic form \( q \ominus c{x}_{0}^{2} \) represents 0 in \( K \) . Proof. (2) implies (1) is trivial by taking \( {x}_{0} = 1 \) and the other variables equal to 0 . Conversely, if \( q \) represents \( c \), there exists \( x \in {K}^{n} \) such that \( q\left( x\right) = \) \( x \cdot x = c \) . Since \( q \) is nondegenerate, if \( H = {x}^{ \bot } \) we have \( {K}^{n} = H{ \oplus }^{ \bot }{Kx} \) , hence \( q \sim {q}_{1} \oplus c{x}_{0}^{2} \), where \( {q}_{1} \) is the quadratic form corresponding to a choice of basis of \( H \), proving (2). (1) implies (3) is trivial by choosing \( {x}_{0} = 1 \) and \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) a representation of \( c \) . Finally, if \( - c{\alpha }_{0}^{2} + q\left( {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right) = 0 \), then either \( {\alpha }_{0} \neq 0 \), in which case \( c = q\left( {{\alpha }_{1}/{\alpha }_{0},\ldots ,{\alpha }_{n}/{\alpha }_{0}}\right) \), or \( {\alpha }_{0} = 0 \), and we conclude by Proposition 5.1.16. Corollary 5.1.18. Let \( {q}_{1} \) and \( {q}_{2} \) be two nonzero nondegenerate quadratic forms and let \( q = {q}_{1} \ominus {q}_{2} \) as defined above. The following properties are equivalent: (1) The form \( q \) represents 0 . (2) There exists \( c \in {K}^{ * } \) represented by both \( {q}_{1} \) and \( {q}_{2} \) . (3) There exists \( c \in {K}^{ * } \) such that both \( {q}_{1} \ominus c{x}_{0}^{2} \) and \( {q}_{2} \ominus c{x}_{0}^{2} \) represent 0 . Proof. The equivalence of (2) and (3) follows from the above corollary, and (2) implies (1) is trivial. Let us prove that (1) implies (2). If \( q = {q}_{1} \ominus {q}_{2} \) represents 0 there exist \( x \) and \( y \) in the corresponding quadratic modules such that \( {q}_{1}\left( x\right) = {q}_{2}\left( y\right) \), with \( x \) and \( y \) not both zero. Consider the element \( c = {q}_{1}\left( x\right) = {q}_{2}\left( y\right) \) . If \( c \neq 0 \), then (2) is proved. Otherwise, since \( x \) and \( y \) are not both zero, at least one of the forms, say \( {q}_{1} \), represents 0 ; hence by Proposition 5.1.16 it represents all elements of \( K \), and in particular all nonzero values taken by \( {q}_{2} \) . Theorem 5.1.19. Let \( q \) be a quadratic form in \( n \) variables. There exists an equivalent form that is a diagonal quadratic form; in other words, there exist \( {a}_{i} \in K \) such that \( q \sim \mathop{\sum }\limits_{{1 \leq i \leq n}}{a}_{i}{x}_{i}^{2} \) . Proof. This is a translation of Proposition 5.1.7. As already mentioned, this can be proved computationally using Gauss's reduction of quadratic forms into sums of squares, which gives an explicit algorithm for finding the \( {a}_{i} \) and the linear equivalence from \( q \) to the diagonal form. Note that things are much easier to read in diagonal form: the rank of \( q \) is equal to the number of nonzero \( {a}_{i} \), and \( q \) is nondegenerate if and only if the rank is equal to \( n \), in which case \( d\left( q\right) = \mathop{\prod }\limits_{{1 \leq i \leq n}}{a}_{i} \) up to squares as usual. However, note that the above theorem is valid only over a field. For instance, the quadratic form \( {x}_{1}^{2} + {x}_{1}{x}_{2} + {x}_{2}^{2} \) is not equivalent to a diagonal form over \( \mathbb{Z} \), but only over \( \mathbb{Q} \) (Exercise 2). Theorem 5.1.20 (Witt). Let \( q = {q}_{1} \oplus {q}_{2} \) and \( {q}^{\prime } = {q}_{1}^{\prime } \oplus {q}_{2}^{\prime } \) be two nondegenerate quadratic forms. If \( q \sim {q}^{\prime } \) and \( {q}_{1} \sim {q}_{1}^{\prime } \), then \( {q}_{2} \sim {q}_{2}^{\prime } \) . Proof. This is the translation of Corollary 5.1.13, the corollary to Witt's theorem. It is this theorem to which one usually refers when talking about Witt's theorem. Corollary 5.1.21. If \( q \) is nondegenerate there exist hyperbolic forms \( {h}_{i} \) for \( 1 \leq i \leq m \) and a form \( {q}^{\prime } \) that does not represent 0 such that \[ q \sim {h}_{1} \oplus \cdots \oplus {h}_{m} \oplus {q}^{\prime } \] and this decomposition is unique up to equivalence. Proof. Existence follows from Proposition 5.1.16. Let us prove uniqueness. With evident notation let \[ q \sim \mathop{\sum }\limits_{{1 \leq i \leq m}}{h}_{i} \oplus {q}_{1} \sim \mathop{\sum }\limits_{{1 \leq i \leq {m}^{\prime }}}{h}_{i}^{\prime } \oplus {q}_{2}, \] and assume for instance that \( {m}^{\prime } \leq m \) . Since all hyperbolic forms are equivalent the above theorem implies that \( \mathop{\sum }\limits_{{1 \leq i \leq m - {m}^{\prime }}}{h}_{i} \oplus {q}_{1} \sim {q}_{2} \), which is a contradiction if \( m \neq {m}^{\prime } \), since \( {q}_{2} \) does not represent 0, while a hyperbolic form does. ## 5.2 Quadratic Forms over Finite and Local Fields After having studied general properties of quadratic forms, we now specialize the base field \( K \) . ## 5.2.1 Quadratic Forms over Finite Fields We begin with the simplest possible fields, the finite fields. Recall that what makes them simple is not the fact that they are finite, but the fact that their structure and hierarchy are very simple; see Chapter 2. For instance, finite groups are extremely complicated and only partly understood objects. Thus let \( q = {p}^{k} \) be a prime power, with \( p \neq 2 \) . Proposition 5.2.1. A quadratic form over \( {\mathbb{F}}_{q} \) of rank \( n \geq 2 \) represents all elements of \( {\mathbb{F}}_{q}^{ * } \), and a quadratic form of rank \( n \geq 3 \) represents all elements of \( {\mathbb{F}}_{q} \) . Proof. Corollary 5.1.17 tells us that both statements are equivalent. To prove the first we could apply the Chevalley-Warning Theorem 2.5.2, but we give a direct proof that is a direct application of the so-called pigeonhole principle ("principe des tiroirs" in French). Indeed, by Theorem 5.1.19 we may assume that our quadratic form is \( \mathop{\sum }\limits_{{1 \leq i \leq m}}{a}_{i}{x}_{i}^{2} \) with \( {a}_{1}{a}_{2} \neq 0 \) . Let \( a \in {\mathbb{F}}_{q} \) . We choose \( {x}_{i} = 0 \) for \( i \geq 3 \) . Since \( q \) is odd the map \( x \mapsto {x}^{2} \) is a group homomorphism of \( {\mathbb{F}}_{q}^{ * } \) onto itself, and its kernel has two elements since \( {\mathbb{F}}_{q} \) is a field. It follows that its image has \( \left( {q - 1}\right) /2 \) elements, so adding the element 0 there are \( \left( {q + 1}\right) /2 \) squares in \( {\mathbb{F}}_{q} \) . Since \( {a}_{1}{a}_{2} \neq 0 \) it follows that the subsets \( \left\{ {{a}_{1}{x}_{1}^{2}}\right\} \) and \( \left\{ {a - {a}_{2}{x}_{2}^{2}}\right\} \) of \( {\mathbb{F}}_{q} \) also have \( \left( {q + 1}\right) /2 \) elements hence have a nonempty intersection, proving the proposition. Proposition 5.2.2. Let \( c \in {\mathbb{F}}_{q}^{ * } \) that is not a square in \( {\mathbb{F}}_{q}^{ * } \) . A nondegenerate quadratic form over \( {\mathbb{F}}_{q} \) is equivalent to \( {x}_{1}^{2} + \cdots + {x}_{n - 1}^{2} + a{x}_{n}^{2} \) with \( a = 1 \) if its discriminant is a square, and with \( a = c \) otherwise. Proof. Since \( q \) is odd, the map \( x \mapsto {x}^{2} \) is a group homomorphism from \( {\mathbb{F}}_{q}^{ * } \) to itself with kernel \( \{ \pm 1\} \) ; hence \( {\mathbb{F}}_{q}^{ * }/{\mathbb{F}}_{q}^{*2} \) has order 2, generated by the class modulo squares of \( c \) . Thus if \( n = 1 \) the result is true. Assume \( n \geq 2 \) and the result true by induction up to \( n - 1 \) . Thanks to the preceding proposition the form represents any nonzero element, hence 1 , so thanks to Corollary 5.1.17 there exists a quadratic form \( g \) in \( n - 1 \) variables such that our form is equivalent to \( {x}_{0}^{2} + g \), and the result follows by induction. Corollary 5.2.3. Two nondegenerate quadratic forms over \( {\mathbb{F}}_{q} \) are equivalent if and only if they have the same rank and the same discriminant in \( {\mathbb{F}}_{q}^{ * }/{\mathbb{F}}_{q}^{*2} \) . Proof. Clear from the above proposition. ## 5.2.2 Definition of the Local Hilbert Symbol The crucial case in the study of quadratic forms over \( p \) -adic fields is the case of three variables, so it is necessary to study in detail this case first. For this, we introduce the Hilbert symbol, which will be sufficient for the local study of quadratic forms. In this section, we let \( \mathcal{K} \) be a completion of \( \mathbb{Q} \), in other words either \( {\mathbb{Q}}_{p} \) or \( \mathbb{R} \) . Definition 5.2.4. If \( a \) and \( b \) are in \( {\mathcal{K}}^{ * } \), we set \( \left( {a, b}\right) = 1 \) if the equation \( a{x}^{2} + b{y}^{2} = {z}^{2} \) has a nontrivial solution (in other words with \( \left( {x, y, z}\right) \neq \) \( \left( {0,0,0}\right) ) \), and \( \left( {a, b}\right) = - 1 \) otherwise. The number \( \left( {a, b}\right) \) is called the (local) Hilbert symbol of a and b. When using several completions as we will do in the global situation, we will write \( {\left( a, b\right) }_{p} \) or \( {\left( a, b\right) }_{\infty } \) to specify the prime, or simply \( {\left( a, b\right) }_{v} \) to indicate the place \( v \) . It is clear that \( \left( {a, b}\right) \) does not change when \( a \) or \( b \) is multiplied by a nonzero

Proposition 5.1.16. If \( q \) is nondegenerate and represents 0, then there exists a hyperbolic form \( h \) and a nondegenerate form \( {q}^{\prime } \) such that \( q \sim h \oplus {q}^{\prime } \) . Furthermore, \( q \) represents all elements of \( K \) .

This is Lemma 5.1.5 and Corollary 5.1.6. The fact that \( {q}^{\prime } \) is nondegenerate follows from Proposition 5.1.2.

Exercise 2.4.4 Using Dirichlet's hyperbola method, show that \[ \mathop{\sum }\limits_{{n \leq x}}\frac{f\left( n\right) }{\sqrt{n}} = {2L}\left( {1,\chi }\right) \sqrt{x} + O\left( 1\right) \] where \( f\left( n\right) = \mathop{\sum }\limits_{{d \mid n}}\chi \left( d\right) \) and \( \chi \neq {\chi }_{0} \) . Exercise 2.4.5 If \( \chi \neq {\chi }_{0} \) is a real character, deduce from the previous exercise that \( L\left( {1,\chi }\right) \neq 0 \) . Exercise 2.4.6 Prove that \[ \mathop{\sum }\limits_{{n > x}}\frac{\chi \left( n\right) }{n} = O\left( \frac{1}{x}\right) \] whenever \( \chi \) is a nontrivial character \( \left( {\;\operatorname{mod}\;q}\right) \) . Exercise 2.4.7 Let \[ {a}_{n} = \mathop{\sum }\limits_{{d \mid n}}\chi \left( d\right) \] where \( \chi \) is a nonprincipal character \( \left( {\;\operatorname{mod}\;q}\right) \) . Show that \[ \mathop{\sum }\limits_{{n \leq x}}{a}_{n} = {xL}\left( {1,\chi }\right) + O\left( \sqrt{x}\right) \] Exercise 2.4.8 Deduce from the previous exercise that \( L\left( {1,\chi }\right) \neq 0 \) for \( \chi \) real. Thus, we have proved the following Theorem: Theorem 2.4.9 (Dirichlet) For any natural number \( q \), and a coprime residue class \( a\left( {\;\operatorname{mod}\;q}\right) \), there are infinitely many primes \( p \equiv a\left( {\;\operatorname{mod}\;q}\right) \) . ## 2.5 Supplementary Problems Exercise 2.5.1 Let \( {d}_{k}\left( n\right) \) be the number of ways of writing \( n \) as a product of \( k \) numbers. Show that \[ \mathop{\sum }\limits_{{n \leq x}}{d}_{k}\left( n\right) = \frac{x{\left( \log x\right) }^{k - 1}}{\left( {k - 1}\right) !} + O\left( {x{\left( \log x\right) }^{k - 2}}\right) \] for every natural number \( k \geq 2 \) . Exercise 2.5.2 Show that \[ \mathop{\sum }\limits_{{n \leq x}}\log \frac{x}{n} = x + O\left( {\log x}\right) \] Exercise 2.5.3 Let \( A\left( x\right) = \mathop{\sum }\limits_{{n \leq x}}{a}_{n} \) . Show that for \( x \) a positive integer, \[ \mathop{\sum }\limits_{{n \leq x}}{a}_{n}\log \frac{x}{n} = {\int }_{1}^{x}\frac{A\left( t\right) {dt}}{t} \] Exercise 2.5.4 Let \( \{ x\} \) denote the fractional part of \( x \) . Show that \[ \mathop{\sum }\limits_{{n \leq x}}\left\{ \frac{x}{n}\right\} = \left( {1 - \gamma }\right) x + O\left( {x}^{1/2}\right) \] where \( \gamma \) is Euler’s constant. Exercise 2.5.5 Prove that \[ \mathop{\sum }\limits_{{n \leq x}}{\log }^{k}\frac{x}{n} = O\left( x\right) \] for any \( k > 0 \) . Exercise 2.5.6 Show that for \( x \geq 3 \) , \[ \mathop{\sum }\limits_{{3 \leq n \leq x}}\frac{1}{n\log n} = \log \log x + B + O\left( \frac{1}{x\log x}\right) . \] Exercise 2.5.7 Let \( \chi \) be a nonprincipal character \( \left( {\;\operatorname{mod}\;q}\right) \) . Show that \[ \mathop{\sum }\limits_{{n \geq x}}\frac{\chi \left( n\right) }{\sqrt{n}} = O\left( \frac{1}{\sqrt{x}}\right) \] Exercise 2.5.8 For any integer \( k \geq 0 \), show that \[ \mathop{\sum }\limits_{{n \leq x}}\frac{{\log }^{k}n}{n} = \frac{{\log }^{k + 1}x}{k + 1} + O\left( 1\right) \] Exercise 2.5.9 Let \( d\left( n\right) \) be the number of divisors of \( n \) . Show that for some constant \( c \) , \[ \mathop{\sum }\limits_{{n \leq x}}\frac{d\left( n\right) }{n} = \frac{1}{2}{\log }^{2}x + {2\gamma }\log x + c + O\left( \frac{1}{\sqrt{x}}\right) \] for \( x \geq 1 \) . Exercise 2.5.10 Let \( \alpha \geq 0 \) and suppose \( {a}_{n} = O\left( {n}^{\alpha }\right) \) and \[ A\left( x\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{n \leq x}}{a}_{n} = O\left( {x}^{\delta }\right) \] for some fixed \( \delta < 1 \) . Define \[ {b}_{n} = \mathop{\sum }\limits_{{d \mid n}}{a}_{d} \] Prove that \[ \mathop{\sum }\limits_{{n \leq x}}{b}_{n} = {cx} + O\left( {x}^{\left( {1 - \delta }\right) \left( {1 + \alpha }\right) /\left( {2 - \delta }\right) }\right) , \] for some constant \( c \) . Exercise 2.5.11 Let \( \chi \) be a nontrivial character \( \left( {\;\operatorname{mod}\;q}\right) \) and set \[ f\left( n\right) = \mathop{\sum }\limits_{{d \mid n}}\chi \left( d\right) \] Show that \[ \mathop{\sum }\limits_{{n \leq x}}f\left( n\right) = {xL}\left( {1,\chi }\right) + O\left( {q\sqrt{x}}\right) \] where the constant implied is independent of \( q \) . Exercise 2.5.12 Suppose that \( {a}_{n} \geq 0 \) and that for some \( \delta > 0 \), we have \[ \mathop{\sum }\limits_{{n \leq x}}{a}_{n} \ll \frac{x}{{\left( \log x\right) }^{\delta }} \] Let \( {b}_{n} \) be defined by the formal Dirichlet series \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{b}_{n}}{{n}^{s}} = {\left( \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{{n}^{s}}\right) }^{2} \] Show that \[ \mathop{\sum }\limits_{{n \leq x}}{b}_{n} \ll x{\left( \log x\right) }^{1 - {2\delta }} \] Exercise 2.5.13 Let \( \left\{ {a}_{n}\right\} \) be a sequence of nonnegative numbers. Show that there exists \( {\sigma }_{0} \in \mathbb{R} \) (possibly infinite) such that \[ f\left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{{n}^{s}} \] converges for \( \operatorname{Re}\left( s\right) > {\sigma }_{0} \) and diverges for \( \operatorname{Re}\left( s\right) < {\sigma }_{0} \) . Moreover, show that the series converges uniformly in \( \operatorname{Re}\left( s\right) \geq {\sigma }_{0} + \delta \) for any \( \delta > 0 \) and that \[ {f}^{\left( k\right) }\left( s\right) = {\left( -1\right) }^{k}\mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}{\left( \log n\right) }^{k}}{{n}^{s}} \] for \( \operatorname{Re}\left( s\right) > {\sigma }_{0} \) ( \( {\sigma }_{0} \) is called the abscissa of convergence of the Dirichlet series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}/{n}^{s} \) ). Exercise 2.5.14 (Landau’s theorem) Let \( {a}_{n} \geq 0 \) be a sequence of nonnegative numbers. Let \( {\sigma }_{0} \) be the abscissa of convergence of \[ f\left( s\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{a}_{n}}{{n}^{s}} \] Show that \( s = {\sigma }_{0} \) is a singular point of \( f\left( s\right) \) (that is, \( f\left( s\right) \) cannot be extended to define an analytic function at \( s = {s}_{0} \) ). Exercise 2.5.15 Let \( \chi \) be a nontrivial character \( \left( {\;\operatorname{mod}\;q}\right) \) and define \[ {\sigma }_{a,\chi } = \mathop{\sum }\limits_{{d \mid n}}\chi \left( d\right) {d}^{a} \] If \( {\chi }_{1},{\chi }_{2} \) are two characters \( \left( {\;\operatorname{mod}\;q}\right) \), prove that for \( a, b \in \mathbb{C} \) , \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\sigma }_{a,{\chi }_{1}}\left( n\right) {\sigma }_{b,{\chi }_{2}}\left( n\right) {n}^{-s} \] \[ = \frac{\zeta \left( s\right) L\left( {s - a,{\chi }_{1}}\right) L\left( {s - b,{\chi }_{2}}\right) L\left( {s - a - b,{\chi }_{1}{\chi }_{2}}\right) }{L\left( {{2s} - a - b,{\chi }_{1}{\chi }_{2}}\right) }. \] as formal Dirichlet series. Exercise 2.5.16 Let \( \chi \) be a nontrivial character \( \left( {\;\operatorname{mod}\;q}\right) \) . Set \( a = \bar{b},{\chi }_{1} = \) \( \chi \) and \( {\chi }_{2} = \bar{\chi } \) in the previous exercise to deduce that \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left| {\sigma }_{a,\chi }\left( n\right) \right| }^{2}{n}^{-s} = \frac{\zeta \left( s\right) L\left( {s - a,\chi }\right) L\left( {s - \bar{a},\bar{\chi }}\right) L\left( {s - a - \bar{a},{\chi }_{0}}\right) }{L\left( {{2s} - a - \bar{a},{\chi }_{0}}\right) } \] Exercise 2.5.17 Using Landau's theorem and the previous exercise, show that \( L\left( {1,\chi }\right) \neq 0 \) for any non-trivial real character \( \left( {\;\operatorname{mod}\;q}\right) \) . Exercise 2.5.18 Show that \( \zeta \left( s\right) \neq 0 \) for \( \operatorname{Re}\left( s\right) > 1 \) . Exercise 2.5.19 (Landau’s theorem for integrals) Let \( A\left( x\right) \) be right continuous for \( x \geq 1 \) and of bounded finite variation on each finite interval. Suppose that \[ f\left( s\right) = {\int }_{1}^{\infty }\frac{A\left( x\right) }{{x}^{s + 1}}{dx} \] with \( A\left( x\right) \geq 0 \) . Let \( {\sigma }_{0} \) be the infimum of all real \( s \) for which the integral converges. Show that \( f\left( s\right) \) has a singularity at \( s = {\sigma }_{0} \) . Exercise 2.5.20 Let \( \lambda \) denote Liouville’s function and set \[ S\left( x\right) = \mathop{\sum }\limits_{{n \leq x}}\lambda \left( n\right) \] Show that if \( S\left( x\right) \) is of constant sign for all \( x \) sufficiently large, then \( \zeta \left( s\right) \neq 0 \) for \( \operatorname{Re}\left( s\right) > \frac{1}{2} \) . (The hypothesis is an old conjecture of Pólya. It was shown by Haselgrove in 1958 that \( S\left( x\right) \) changes sign infinitely often.) Exercise 2.5.21 Prove that \[ {b}_{n}\left( x\right) = \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {B}_{n - k}{x}^{k} \] where \( {b}_{n}\left( x\right) \) is the nth Bernoulli polynomial and \( {B}_{n} \) denotes the nth Bernoulli number. Exercise 2.5.22 Prove that \[ {b}_{n}\left( {1 - x}\right) = {\left( -1\right) }^{n}{b}_{n}\left( x\right) \] where \( {b}_{n}\left( x\right) \) denotes the \( n \) th Bernoulli polynomial. Exercise 2.5.23 Let \[ {s}_{k}\left( n\right) = {1}^{k} + {2}^{k} + {3}^{k} + \cdots + {\left( n - 1\right) }^{k}. \] Prove that for \( k \geq 1 \) , \[ \left( {k + 1}\right) {s}_{k}\left( n\right) = \mathop{\sum }\limits_{{i = 0}}^{k}\left( \begin{matrix} k + 1 \\ i \end{matrix}\right) {B}_{i}{n}^{k + i - i}. \] ## 3 The Prime Number Theorem Let \( \pi \left( x\right) \) denote the number of primes \( p \leq x \) . The prime number theorem is the assertion that \[ \mathop{\lim }\limits_{{x \rightarrow \infty }}\frac{\pi \left( x\right) }{x/\log x} = 1 \] It was proved independently by Hadamard and de la Vallée Poussin in 1896. It is the goal of this chapter to prove this theorem following a method evolved by Wiener and Ikehara in the early twentieth century. As far as w

Exercise 2.4.4 Using Dirichlet's hyperbola method, show that \[ \mathop{\sum }\limits_{{n \leq x}}\frac{f\left( n\right) }{\sqrt{n}} = {2L}\left( {1,\chi }\right) \sqrt{x} + O\left( 1\right) \] where \( f\left( n\right) = \mathop{\sum }\limits_{{d \mid n}}\chi \left( d\right) \) and \( \chi \neq {\chi }_{0} \) .

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Theorem 11.16 (Kolmogorov). Let \( \left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) be a probability space, and let \( {\left( {X}_{n}\right) }_{n \in \mathbb{N}} \subseteq {\mathrm{L}}^{1}\left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) be a sequence of independent and identically distributed real random variables. Then \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\left( {{X}_{1} + \cdots + {X}_{n}}\right) = \mathrm{E}\left( {X}_{1}\right) \;\text{ P-almost surely. } \] Proof. Since the \( {X}_{j} \) are identically distributed, \( v \mathrel{\text{:=}} {\mathrm{P}}_{{X}_{j}} \) (the distribution of \( {X}_{j} \) ) is a Borel probability measure on \( \mathbb{R} \), independent of \( j \), and \[ \mathrm{E}\left( {X}_{1}\right) = {\int }_{\mathbb{R}}t\mathrm{\;d}v\left( t\right) \] is the common expectation. Define the product space \[ \mathrm{X} \mathrel{\text{:=}} \left( {X,\sum ,\mu }\right) \mathrel{\text{:=}} \left( {{\mathbb{R}}^{\mathbb{N}},{\bigotimes }_{\mathbb{N}}\operatorname{Bo}\left( \mathbb{R}\right) ,{\bigotimes }_{\mathbb{N}}v}\right) . \] As mentioned in Section 5.1.5, the left shift \( \tau \) is a measurable transformation of \( \left( {X,\sum }\right) \) and \( \mu \) is \( \tau \) -invariant. The measure-preserving system \( \left( {\mathrm{X};\tau }\right) \) is ergodic, and this can be shown in exactly the same way as it was done for the finite state space Bernoulli shift (Proposition 6.20). For \( n \in \mathbb{N} \) let \( {Y}_{n} : X \rightarrow \mathbb{R} \) be the \( {n}^{\text{th }} \) projection and write \( g \mathrel{\text{:=}} {Y}_{1} \) . Then \( {Y}_{j + 1} = g \circ {\tau }^{j} \) for every \( j \geq 0 \), hence Corollary 11.2 yields that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\left( {{Y}_{1} + \cdots + {Y}_{n}}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\mathop{\sum }\limits_{{j = 0}}^{{n - 1}}\left( {g \circ {\tau }^{j}}\right) = {\int }_{X}g\mathrm{\;d}\mu \] pointwise \( \mu \) -almost everywhere. Note that \( {g}_{ * }\mu = v \) and hence \[ {\int }_{X}g\mathrm{\;d}\mu = {\int }_{\mathbb{R}}t\mathrm{\;d}v\left( t\right) \] It remains to show that the \( {\left( {Y}_{n}\right) }_{n \in \mathbb{N}} \) are in a certain sense "the same" as the originally given \( {\left( {X}_{n}\right) }_{n \in \mathbb{N}} \) . This is done by devising an injective lattice homomorphism \[ \Phi : {\mathrm{L}}^{0}\left( {\mathrm{X};\overline{\mathbb{R}}}\right) \rightarrow {\mathrm{L}}^{0}\left( {\Omega ,\mathcal{F},\mathrm{P};\overline{\mathbb{R}}}\right) \] which carries \( {Y}_{n} \) to \( \Phi \left( {Y}_{n}\right) = {X}_{n} \) for every \( n \in \mathbb{N} \) . Define \[ \varphi : \Omega \rightarrow X,\;\varphi \left( \omega \right) \mathrel{\text{:=}} {\left( {X}_{n}\left( \omega \right) \right) }_{n \in \mathbb{N}}\;\left( {\omega \in \Omega }\right) . \] Since \( {\left( {X}_{n}\right) }_{n \in \mathbb{N}} \) is an independent sequence, the push-forward measure satisfies \( {\varphi }_{ * }\mathrm{P} = \mu \) . Let \( \Phi = {T}_{\varphi } : f \mapsto f \circ \varphi \) be the Koopman operator induced by \( \varphi \) mapping \( {\mathrm{L}}^{0}\left( {\mathrm{X};\overline{\mathbb{R}}}\right) \) to \( {\mathrm{L}}^{0}\left( {\Omega ,\mathcal{F},\mathrm{P};\overline{\mathbb{R}}}\right) \) . The operator \( \Phi \) is well defined since \( \varphi \) is measure-preserving. By construction, \( \Phi {Y}_{n} = {Y}_{n} \circ \varphi = {X}_{n} \) for each \( n \in \mathbb{N} \) . Moreover, \( \Phi \) is clearly a homomorphism of lattices (see Chapter 7) satisfying \[ \mathop{\sup }\limits_{{n \in \mathbb{N}}}\Phi \left( {f}_{n}\right) = \Phi \left( {\mathop{\sup }\limits_{{n \in \mathbb{N}}}{f}_{n}}\right) \;\text{ and }\;\mathop{\inf }\limits_{{n \in \mathbb{N}}}\Phi \left( {f}_{n}\right) = \Phi \left( {\mathop{\inf }\limits_{{n \in \mathbb{N}}}{f}_{n}}\right) \] for every sequence \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) in \( {\mathrm{L}}^{0}\left( {\mathrm{X};\overline{\mathbb{R}}}\right) \) . Since the almost everywhere convergence of a sequence can be described in purely lattice theoretic terms involving only countable suprema and infima (cf. also (11.1)), one has \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n} = f\;\mu \text{-a.e. } \Rightarrow \mathop{\lim }\limits_{{n \rightarrow \infty }}\Phi \left( {f}_{n}\right) = \Phi \left( f\right) \;\text{ P-almost surely. } \] This, for \( {f}_{n} \mathrel{\text{:=}} \frac{1}{n}\left( {{Y}_{1} + \cdots + {Y}_{n}}\right) \) and \( f \mathrel{\text{:=}} \mathrm{E}\left( {X}_{1}\right) \mathbf{1} \), concludes the proof. Remark 11.17. By virtue of the same product construction one can show that the mean ergodic theorem implies a general weak law of large numbers. ## Final Remark: Birkhoff Versus von Neumann From the point of view of statistical mechanics, Birkhoff's theorem seems to outrun von Neumann's. By virtue of the dominated convergence theorem and the denseness of \( {\mathrm{L}}^{\infty } \) in \( {\mathrm{L}}^{2} \), the latter is even a corollary of the former (cf. the presentation in Walters (1982, Cor. 1.14.1)). Reed and Simon take a moderate viewpoint when they write in (1972, p. 60) (annotation in square brackets by the authors). This [i.e., the pointwise ergodic] theorem is closer to what one wants to justify [in?] statistical mechanics than the von Neumann theorem, and it is fashionable to say that the von Neumann theorem is unsuitable for statistical mechanics. We feel that this is an exaggeration. If we had only the von Neumann theorem we could probably live with it quite well. Typically, initial conditions are not precisely measurable anyway, so that one could well associate initial states with measures \( f\mathrm{\;d}\mu \) where \( \int f\mathrm{\;d}\mu = 1 \), in which case the von Neumann theorem suffices. However, the Birkhoff theorem does hold and is clearly a result that we are happier to use in justifying the statement that phase-space averages and time averages are equal. However, von Neumann's theorem inspired the operator theoretic concept of mean ergodicity and an enormous amount of research in the field of asymptotics of discrete (and continuous) operator semigroups with tantamount applications to various other fields. Certainly it would be too much to say that Birkhoff's theorem is overrated, but von Neumann's theorem should not be underestimated either. ## Notes and Further Reading The maximal ergodic theorem and a related result, called "Hopf's lemma" (Exercise 7) are from Hopf (1954) generalizing results from Yosida and Kakutani (1939). A slightly weaker form was already obtained by Wiener (1939). Our proof is due to Garsia (1965). The role of maximal inequalities for almost everywhere convergence results is known at least since Kolmogorov (1925) and is demonstrated impressively in Stein (1993). Employing a Baire category argument one can show that an abstract maximal inequality is indeed necessary for pointwise convergence results of quite a general type Krengel (1985, Ch. 1, Thm. 7.2); a thorough study of this connection has been carried out in Stein (1961a). Finally, we recommend Garsia (1970) for more results on almost everywhere convergence. ## Exercises 1. Let \( K \) be a compact topological space, let \( T \) be a Markov operator on \( \mathrm{C}\left( K\right) \) (see Exercise 10.2), and let \( \mu \in {\mathrm{M}}^{1}\left( K\right) \) be such that \( {T}^{\prime }\mu \leq \mu \) . Show that \( T \) extends uniquely to a positive Dunford-Schwartz operator on \( {\mathrm{L}}^{1}\left( {K,\mu }\right) \) . 2. Let \( \mathrm{X} \) be a measure space, \( 1 \leq p < \infty \), and let \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of bounded linear operators on \( E = {\mathrm{L}}^{p}\left( \mathrm{X}\right) \) . Moreover, let \( T \) be a bounded linear operator on \( E \) . If the associated maximal operator \( {T}^{ * } \) satisfies a maximal inequality, then the set \[ C \mathrel{\text{:=}} \left\{ {f \in E : {T}_{n}f \rightarrow {Tf}\text{ almost everywhere }}\right\} \] is a closed subspace of \( E \) . 3. Prove Corollaries 11.13 and 11.14. 4. Let \( \left( {K;\varphi }\right) \) be a topological system with Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \) and associated Cesàro averages \( {\mathrm{A}}_{n} = {\mathrm{A}}_{n}\left\lbrack T\right\rbrack, n \in \mathbb{N} \) . Let \( \mu \) be a \( \varphi \) -invariant probability measure on \( K \) . A point \( x \in K \) is called generic for \( \mu \) if \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{\mathrm{A}}_{n}f}\right) \left( x\right) = {\int }_{K}f\mathrm{\;d}\mu \] (11.3) for all \( f \in \mathrm{C}\left( K\right) \) . a) Show that \( x \in K \) is generic for \( \mu \) if (11.3) holds for each \( f \) from a dense subset of \( \mathrm{C}\left( K\right) \) . b) Show that in the case that \( \mathrm{C}\left( K\right) \) is separable and \( \mu \) is ergodic, \( \mu \) -almost every \( x \in K \) is generic for \( \mu \) . (Hint: Apply Corollary 11.2 to every \( f \) from a countable dense set \( D \subseteq \mathrm{C}\left( K\right) \) .) 5. A number \( x \in \left\lbrack {0,1}\right\rbrack \) is called normal (in base 10) if every finite combination (of length \( k \) ) of the digits \( \{ 0,1,\ldots ,9\} \) appears in the decimal expansion of \( x \) with asymptotic frequency \( {10}^{-k} \) . Prove that almost all numbers in \( \left\lbrack {0,1}\right\rbrack \) are normal. 6 (Dominated Ergodic Theorem). Let \( \mathrm{X} = \left( {X,\sum ,\mu }\right) \) be a measure space and let \( f \in {\mathrm{L}}_{ + }^{0}\left( \mathrm{X}\right) \) . Show that \[ {\int }_{X}f\mathrm{\;d}\mu = {\int }_{0}^{\infty }\mu \left\lbrack {f > t}\right\rbrack \mathrm{d}t \] and derive from this that \( \parallel f{\parallel }_{p}^{p} = {\int }_{0}^{\infty }p{t}^{p - 1}\mu \left\lbrack

Theorem 11.16 (Kolmogorov). Let \( \left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) be a probability space, and let \( {\left( {X}_{n}\right) }_{n \in \mathbb{N}} \subseteq {\mathrm{L}}^{1}\left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) be a sequence of independent and identically distributed real random variables. Then \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\left( {{X}_{1} + \cdots + {X}_{n}}\right) = \mathrm{E}\left( {X}_{1}\right) \;\text{ P-almost surely. } \]

Since the \( {X}_{j} \) are identically distributed, \( v \mathrel{\text{:=}} {\mathrm{P}}_{{X}_{j}} \) (the distribution of \( {X}_{j} \) ) is a Borel probability measure on \( \mathbb{R} \), independent of \( j \), and \[ \mathrm{E}\left( {X}_{1}\right) = {\int }_{\mathbb{R}}t\mathrm{\;d}v\left( t\right) \] is the common expectation. Define the product space \[ \mathrm{X} \mathrel{\text{:=}} \left( {X,\sum ,\mu }\right) \mathrel{\text{:=}} \left( {{\mathbb{R}}^{\mathbb{N}},{\bigotimes }_{\mathbb{N}}\operatorname{Bo}\left( \mathbb{R}\right) ,{\bigotimes }_{\mathbb{N}}v}\right) . \] As mentioned in Section 5.1.5, the left shift \( \tau \) is a measurable transformation of \( \left( {X,\sum }\right) \) and \( \mu \) is \( \tau \) -invariant. The measure-preserving system \( \left( {\mathrm{X};\tau }\right) \) is ergodic, and this can be shown in exactly the same way as it was done for the finite state space Bernoulli shift (Proposition 6.20). For \( n \in \mathbb{N} \) let \( {Y}_{n} : X \rightarrow \mathbb{R} \) be the \( {n}^{\text{th }} \) projection and write \( g \mathrel{\text{:=}} {Y}_{1} \) . Then \( {Y}_{j + 1} = g \circ {\tau }^{j} \) for every \( j \geq 0 \), hence Corollary 11.2 yields that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\left( {{Y}_{1} + \cdots + {Y}_{n}}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\mathop{\sum }\limits_{{j = 0}}^{{n - 1}}\left( {g \circ {\tau }^{j}}\right) = {\int }_{X}g\mathrm{\;d}\mu \] pointwise \( \mu \) -almost everywhere. Note that \( {g}_{ * }\mu = v \) and hence \[ {\int }_{X}g\mathrm{\;d}\mu = {\int }_{\mathbb{R}}t\mathrm{\;d}v\left( t\right) \] It remains to show that the \( {\left( {Y}_{n}\right) }_{n \in \mathbb{N}} \) are in a certain sense "the same" as the originally given \( {\left( {X}_{n}\right) }_{n \in \mathbb{N}} \) . This is done by devising an injective lattice homomorphism \[ \Phi : {\mathrm{L}}^{0}\left( {\mathrm{X};\overline{\mathbb{R}}}\right) \rightarrow {\mathrm{L}}^{0}\left( {\Omega ,\mathcal{F},\mathrm{P};\overline{\mathbb{R}}}\right) \] which carries \( {Y}_{n} \) to \( \(\Phi ({Y}_{n}) = {X}_{n}\) for every n∈ℕ. Define φ:Ω→X,φ(ω):=(X_n (ω))_( n∈ℕ)(ω∈Ω). Since ( X_n )_( n∈ℕ ) is an independent sequence, the push-forward measure satisfies φ_* P=μ. Let Φ=T_φ:f↦f∘φ be the Koopman operator induced by φ mapping L^0 ( X;ℝ¯ ) to L^0 ( Ω,F,P;ℝ¯ ). The operator Φ is well defined since φ is measure-preserving. By construction, Φ Y_n=Y_n∘φ= X_n for each n∈ℕ. Moreover, Φ is clearly a homomorphism of lattices (see Chapter 7) satisfying sup_( n∈ℕ )Φ f_n=Φ sup_( n∈ℕ )f_n and inf_( n∈ℕ )Φ f_n=Φ inf_( n∈ℕ )f_n for every sequence ( f_n )_( n∈ℕ ) in L^0 ( X;ℝ¯ ). Since the almost everywhere convergence of a sequence can be described in purely lattice theoretic terms involving only countable suprema and infima (cf. also (11.1)), one has lim_( n→∞ )f_

Example 9.1.8. As an illustration of Theorem 9.1.6 we consider a Goppa code \[ {\left. \Gamma \left( L, g\left( z\right) \right) = {C}_{\Omega }\left( {D}_{L},{G}_{0} - {P}_{\infty }\right) \right| }_{{\mathbb{F}}_{q}} \] (notation as in Definition 2.3.10 and Proposition 2.3.11). Let \( {g}_{1}\left( z\right) \in {\mathbb{F}}_{{q}^{m}}\left\lbrack z\right\rbrack \) be the polynomial of maximal degree such that \( {g}_{1}{\left( z\right) }^{q} \) divides \( g\left( z\right) \) . We set \( {G}_{1} \mathrel{\text{:=}} {\left( {g}_{1}\left( z\right) \right) }_{0} - {P}_{\infty } \) where \( {\left( {g}_{1}\left( z\right) \right) }_{0} \) is the zero divisor of \( {g}_{1}\left( z\right) \), and obtain from (9.12) the estimate \[ \dim \Gamma \left( {L, g\left( z\right) }\right) \geq n - m\left( {\deg g\left( z\right) - \deg {g}_{1}\left( z\right) }\right) . \] (9.14) In many cases, equality holds in (9.14). This will be shown in Proposition 9.2.13. ## 9.2 Weights of Trace Codes In this section we investigate some specific trace codes. The main idea is to relate the weights of their codewords to the number of rational places in certain algebraic function fields. The Hasse-Weil-Serre Bound then yields estimates for the weights and the minimum distance of these codes. First we introduce the codes to be considered. Definition 9.2.1. Let \( F \) be an algebraic function field over the constant field \( {\mathbb{F}}_{{q}^{m}} \) and let \( V \subseteq F \) be a finite-dimensional \( {\mathbb{F}}_{{q}^{m}} \) -subspace of \( F \) . Let \( {P}_{1},\ldots ,{P}_{n} \in {\mathbb{P}}_{F} \) be \( n \) distinct places of degree one such that \( {v}_{{P}_{i}}\left( f\right) \geq 0 \) for all \( f \in V \) and \( i = 1,\ldots, n \) . Set \( D \mathrel{\text{:=}} {P}_{1} + \ldots + {P}_{n} \) . Then we define \[ C\left( {D, V}\right) \mathrel{\text{:=}} \left\{ {\left( {f\left( {P}_{1}\right) ,\ldots, f\left( {P}_{n}\right) }\right) \mid f \in V}\right\} \subseteq {\left( {\mathbb{F}}_{{q}^{m}}\right) }^{n} \] and \[ {\operatorname{Tr}}_{D}\left( V\right) \mathrel{\text{:=}} \operatorname{Tr}\left( {C\left( {D, V}\right) }\right) \subseteq {\mathbb{F}}_{q}^{n} \] i.e., \( {\operatorname{Tr}}_{D}\left( V\right) \) is the trace code of \( C\left( {D, V}\right) \) with respect to the extension \( {\mathbb{F}}_{{q}^{m}}/{\mathbb{F}}_{q} \) . Note that \( C\left( {D, V}\right) \) is a code over \( {\mathbb{F}}_{{q}^{m}} \), whereas \( {\operatorname{Tr}}_{D}\left( V\right) \) is a code over \( {\mathbb{F}}_{q} \) . Our main objective in this section is to study the codes \( {\operatorname{Tr}}_{D}\left( V\right) \) . Let us first give some examples of such codes. Example 9.2.2. The codes \( C\left( {D, V}\right) \) are a generalization of algebraic geometry codes. Choosing \( V \mathrel{\text{:=}} \mathcal{L}\left( G\right) \) where \( G \) is a divisor with \( \operatorname{supp}G \cap \operatorname{supp}D = \varnothing \) as usual, we obtain \( C\left( {D, V}\right) = {C}_{\mathcal{L}}\left( {D, G}\right) \) . Example 9.2.3. Every code \( C \subseteq {\mathbb{F}}_{q}^{n} \) over \( {\mathbb{F}}_{q} \) can be represented as \( C = \) \( {\operatorname{Tr}}_{D}\left( V\right) \) for a suitable choice of \( V \) and \( D \) . This can be seen as follows. Choose \( m \in \mathbb{N} \) sufficiently large such that \( {q}^{m} \geq n \) . Let \( F \mathrel{\text{:=}} {\mathbb{F}}_{{q}^{m}}\left( z\right) \) be the rational function field over \( {\mathbb{F}}_{{q}^{m}} \) . Choose \( n \) distinct elements \( {\alpha }_{1},\ldots ,{\alpha }_{n} \in {\mathbb{F}}_{{q}^{m}} \) and denote by \( {P}_{i} \in {\mathbb{P}}_{F} \) the zero of \( z - {\alpha }_{i} \) . Choose a basis \( \left\{ {{a}^{\left( 1\right) },\ldots ,{a}^{\left( k\right) }}\right\} \) of \( C \) over \( {\mathbb{F}}_{q} \) . Write \( {a}^{\left( j\right) } = \left( {{a}_{1}^{\left( j\right) },\ldots ,{a}_{n}^{\left( j\right) }}\right) \) . For \( j = 1,\ldots, k \) choose a polynomial \( {f}_{j} = {f}_{j}\left( z\right) \in {\mathbb{F}}_{{q}^{m}}\left\lbrack z\right\rbrack \) satisfying \( {f}_{j}\left( {\alpha }_{i}\right) = {a}_{i}^{\left( j\right) } \) for \( i = 1,\ldots, n \) . Let \( V \subseteq F \) be the \( {\mathbb{F}}_{{q}^{m}} \) -vector space generated by \( {f}_{1},\ldots ,{f}_{k} \) . Then it is easily verified that \( C = {\operatorname{Tr}}_{D}\left( V\right) \) . More interesting than the previous example is the fact that specific classes of codes over \( {\mathbb{F}}_{q} \) can be represented as trace codes in a natural manner. In the following we give such a representation for cyclic codes. A code \( C \) over \( {\mathbb{F}}_{q} \) of length \( n \) is said to be cyclic if its automorphism group \( \operatorname{Aut}\left( C\right) \) contains the cyclic shift; i.e., \[ \left( {{c}_{0},{c}_{1},\ldots ,{c}_{n - 1}}\right) \in C \Rightarrow \left( {{c}_{1},\ldots ,{c}_{n - 1},{c}_{0}}\right) \in C. \] As is common in coding theory, we identify \( {\mathbb{F}}_{q}^{n} \) with the vector space of polynomials of degree \( \leq n - 1 \) over \( {\mathbb{F}}_{q} \) via \[ c = \left( {{c}_{o},\ldots ,{c}_{n - 1}}\right) \leftrightarrow c\left( x\right) = {c}_{o} + {c}_{1}x + \ldots + {c}_{n - 1}{x}^{n - 1} \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack . \] (9.15) We shall always assume that \[ \gcd \left( {n, q}\right) = 1\text{.} \] (9.16) Let \( m \) be the least integer \( \geq 1 \) satisfying \( {q}^{m} \equiv 1{\;\operatorname{mod}\;n} \) . Then the polynomial \( {x}^{n} - 1 \) factors over the field \( {\mathbb{F}}_{{q}^{m}} \supseteq {\mathbb{F}}_{q} \) as \[ {x}^{n} - 1 = \mathop{\prod }\limits_{{\nu = 0}}^{{n - 1}}\left( {x - {\beta }^{\nu }}\right) \] (9.17) where \( \beta \in {\mathbb{F}}_{{q}^{m}} \) is a primitive \( n \) -th root of unity. All linear factors in (9.17) are distinct. Let us briefly recall some basic facts about cyclic codes, cf. [28]. Given a cyclic code \( C \neq \{ 0\} \) of length \( n \) over \( {\mathbb{F}}_{q} \), there exists a unique monic polynomial \( g\left( x\right) \in C \) of minimal degree; it is called the generator polynomial of \( C \) . The generator polynomial divides \( {x}^{n} - 1 \), so \[ g\left( x\right) = \mathop{\prod }\limits_{{\nu \in I}}\left( {x - {\beta }^{\nu }}\right) \] (9.18) where \( \beta \) is a primitive \( n \) -th root of unity as in (9.17) and \( I \) is a certain subset of \( \{ 0,\ldots, n - 1\} \) . The elements \( {\beta }^{\nu } \) with \( \nu \in I \) are called the zeros of \( C \) because one has the following description of \( C \) : for \( c\left( x\right) \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack \) with \( \deg c\left( x\right) \leq n - 1 \) , \[ c\left( x\right) \in C \Leftrightarrow c\left( {\beta }^{\nu }\right) = 0\;\text{ for all }\nu \in I. \] (9.19) The conditions on the right-hand side of (9.19) can be weakened. To this end we define the cyclotomic coset \( \mathcal{C}\left( i\right) \) of an integer \( i \in \mathbb{Z},0 \leq i \leq n - 1 \), by \[ \mathcal{C}\left( i\right) \mathrel{\text{:=}} \left\{ {j \in \mathbb{Z} \mid 0 \leq j \leq n - 1\text{ and }j \equiv {q}^{l}i{\;\operatorname{mod}\;n}\text{ for some }l \geq 0}\right\} . \] It is easily checked that either \( \mathcal{C}\left( i\right) = \mathcal{C}\left( {i}^{\prime }\right) \) or \( \mathcal{C}\left( i\right) \cap \mathcal{C}\left( {i}^{\prime }\right) = \varnothing \) . Hence the set \( \{ 0,1,\ldots, n - 1\} \) is partitioned into pairwise disjoint cyclotomic cosets; i.e., \( \{ 0,1,\ldots, n - 1\} = \mathop{\bigcup }\limits_{{\mu = 1}}^{s}{\mathcal{C}}_{\mu } \) (with \( s \leq n \) and \( {\mathcal{C}}_{\mu } = \mathcal{C}\left( {i}_{\mu }\right) \) for some \( {i}_{\mu } \) , \( \left. {0 \leq {i}_{\mu } \leq n - 1}\right) \) . For \( \nu \in \mathbb{Z} \) denote by \( \widetilde{\nu } \) the unique integer in \( \{ 0,1,\ldots, n - 1\} \) with \( \nu \equiv \) \( \widetilde{\nu }{\;\operatorname{mod}\;n} \) . Let \( \varnothing \neq M \subseteq \{ 0,1,\ldots, n - 1\} \) . A subset \( {M}_{0} \subseteq \mathbb{Z} \) is called a complete set of cyclotomic coset representatives of \( M \), if for each \( \nu \in M \) there exists a unique \( {\nu }_{0} \in {M}_{0} \) such that \( {\widetilde{\nu }}_{0} \in \mathcal{C}\left( \nu \right) \) . It is evident that one can always find a complete set of cyclotomic coset representatives of \( M \) which is contained in \( \{ 0,1,\ldots, n - 1\} \) . Now consider the set \( I \subseteq \{ 0,1,\ldots, n - 1\} \) given by (9.18); i.e., \( \left\{ {{\beta }^{\nu } \mid \nu \in I}\right\} \) is the set of zeros of the cyclic code \( C \) . Let \( {I}_{0} \) be a complete set of cyclotomic coset representatives of \( I \) . Since for \( c\left( x\right) \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack \) , \[ c\left( {\beta }^{\nu }\right) = 0 \Leftrightarrow c\left( {\beta }^{{q}^{l}\nu }\right) = 0, \] we can replace (9.19) by the following condition: \[ c\left( x\right) \in C \Leftrightarrow c\left( {\beta }^{\nu }\right) = 0\;\text{ for all }\nu \in {I}_{0}, \] (9.20) where \( c\left( x\right) \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack \) and \( \deg c\left( x\right) \leq n - 1 \) . The dual code \( {C}^{ \bot } \) of a cyclic code \( C \) is cyclic as well. Let \( g\left( x\right) \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack \) be the generator polynomial of \( C \) and \[ h\left( x\right) \mathrel{\text{:=}} \left( {{x}^{n} - 1}\right) /g\left( x\right) \in {\mathbb{F}}_{q}\left\lbrack x\right\rbrack . \] (9.21) The polynomial \( h\left( x\right) \) is called the check polynomial of \( C \) . The reciprocal polynomial \( {h}^{ \bot }\left( x\right) \) of \( h\left( x\right) \) , \[ {h}^{ \bot }\left( x\right) \mathrel{\text{:=}} h{\left( 0\right) }^{-1} \cdot {x}^{\deg h\left( x\right) } \cdot h\left( {x}^{-1}\right) , \] (9.22) is the generator polynomial of \( {C}^{ \bot } \) . Write \[ {h}^{

Example 9.1.8. As an illustration of Theorem 9.1.6 we consider a Goppa code \[ {\left. \Gamma \left( L, g\left( z\right) \right) = {C}_{\Omega }\left( {D}_{L},{G}_{0} - {P}_{\infty }\right) \right| }_{{\mathbb{F}}_{q}} \] (notation as in Definition 2.3.10 and Proposition 2.3.11). Let \( {g}_{1}\left( z\right) \in {\mathbb{F}}_{{q}^{m}}\left\lbrack z\right\rbrack \) be the polynomial of maximal degree such that \( {g}_{1}{\left( z\right) }^{q} \) divides \( g\left( z\right) \) . We set \( {G}_{1} \mathrel{\text{:=}} {\left( {g}_{1}\left( z\right) \right) }_{0} - {P}_{\infty } \) where \( {\left( {g}_{1}\left( z\right) \right) }_{0} \) is the zero divisor of \( {g}_{1}\left( z\right) \), and obtain from (9.12) the estimate \[ \dim \Gamma \left( {L, g\left( z\right) }\right) \geq n - m\left( {\deg g\left( z\right) - \deg {g}_{1}\left( z\right) }\right) . \]

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Proposition 13.5. Let \( {S}^{r} \) denote the \( r \) -sphere. Then \( {\pi }_{1}\left( {S}^{1}\right) \cong \mathbb{Z} \), while \( {S}^{r} \) is simply-connected if \( r \geq 2 \) . Proof. We may identify the circle \( {S}^{1} \) with the unit circle in \( \mathbb{C} \) . Then \( x \mapsto \) \( {\mathrm{e}}^{2\pi ix} \) is a covering map \( \mathbb{R} \rightarrow {S}^{1} \) . The space \( \mathbb{R} \) is contractible and hence simply-connected, so it is the universal covering space. If we give \( {S}^{1} \subset {\mathbb{C}}^{ \times } \) the group structure it inherits from \( {\mathbb{C}}^{ \times } \), then this map \( \mathbb{R} \rightarrow {S}^{1} \) is a group homomorphism, so by Theorem 13.2 we may identify the kernel \( \mathbb{Z} \) with \( {\pi }_{1}\left( {S}^{1}\right) \) . To see that \( {S}^{r} \) is simply connected for \( r \geq 2 \), let \( p : \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{r} \) be a path. Since it is a mapping from a lower-dimensional manifold, perturbing the path slightly if necessary, we may assume that \( p \) is not surjective. If it omits one point \( P \in {S}^{r} \), its image is contained in \( {S}^{r} - \{ P\} \), which is homeomorphic to \( {\mathbb{R}}^{r} \) and hence contractible. Therefore \( p \), is path-homotopic to a trivial path. Proposition 13.6. The group \( \mathrm{{SU}}\left( 2\right) \) is simply-connected. The group \( \mathrm{{SO}}\left( 3\right) \) is not. In fact \( {\pi }_{1}\left( {\mathrm{{SO}}\left( 3\right) }\right) \cong \mathbb{Z}/2\mathbb{Z} \) . Proof. Note that \( \mathrm{{SU}}\left( 2\right) = \left\{ {\left. \left( \begin{matrix} a & b \\ - \bar{b} & \bar{a} \end{matrix}\right) \right| \;{\left| a\right| }^{2} + {\left| b\right| }^{2} = 1}\right\} \) is homeomorphic to the 3 sphere in \( {\mathbb{C}}^{2} \) . As such, it is simply connected. We have a homomorphism \( \mathrm{{SU}}\left( 2\right) \rightarrow \mathrm{{SO}}\left( 3\right) \), which we constructed in Example 7.1. Since this mapping induced an isomorphism of Lie algebras, its image is an open subgroup of \( \mathrm{{SO}}\left( 3\right) \), and since \( \mathrm{{SO}}\left( 3\right) \) is connected, this homomorphism is surjective. The kernel \( \{ \pm I\} \) of this homomorphism is finite, so this is a covering map. Because \( \mathrm{{SU}}\left( 2\right) \) is simply connected, it follows from the uniqueness of the simply connected covering group that it is the universal covering group of \( \mathrm{{SO}}\left( 3\right) \) . The kernel of this homomorphism \( \mathrm{{SU}}\left( 2\right) \rightarrow \mathrm{{SO}}\left( 3\right) \) is therefore the fundamental group, and it has order 2 . Let \( G \) and \( H \) be topological groups. By a local homomorphism \( G \rightarrow H \) we mean the following data: a neighborhood \( U \) of the identity and a continuous map \( \phi : U \rightarrow H \) such that \( \phi \left( {uv}\right) = \phi \left( u\right) \phi \left( v\right) \) whenever \( u, v \), and \( {uv} \in U \) . This implies that \( \phi \left( {1}_{G}\right) = {1}_{H} \), so if \( u,{u}^{-1} \in U \) we have \( \phi \left( {u}^{-1}\right) = \phi \left( u\right) \) . We may as well replace \( U \) by \( U \cap {U}^{-1} \) so this is true for all \( u \in U \) . Theorem 13.3. Let \( G \) and \( H \) be topological groups, and assume that \( G \) is simply connected. Let \( U \) be a neighborhood of the identity in \( G \) . Then any local homomorphism \( U \rightarrow H \) can be extended to a homomorphism \( G \rightarrow H \) . Proof. Let \( g \in G \) . Let \( p : \left\lbrack {0,1}\right\rbrack \rightarrow G \) be a path with \( p\left( 0\right) = {1}_{G}, p\left( 1\right) = g \) . (Such a path exists because \( G \) is path-connected.) We first show that there exists a unique path \( q : \left\lbrack {0,1}\right\rbrack \rightarrow H \) such that \( q\left( 0\right) = {1}_{H} \), and \[ q\left( v\right) q{\left( u\right) }^{-1} = \phi \left( {p\left( v\right) p{\left( u\right) }^{-1}}\right) \] (13.1) when \( u, v \in \left\lbrack {0,1}\right\rbrack \) and \( \left| {u - v}\right| \) is sufficiently small. We note that when \( u \) and \( v \) are sufficiently close, \( p\left( v\right) p{\left( u\right) }^{-1} \in U \), so this makes sense. To construct a path \( q \) with this property, find \( 0 = {x}_{0} < {x}_{1} < \cdots < {x}_{n} = 1 \) such that when \( u \) and \( v \) lie in an interval \( \left\lbrack {{x}_{i - 1},{x}_{i + 1}}\right\rbrack \), we have \( p\left( v\right) p{\left( u\right) }^{-1} \in U\left( {1 \leq i < n}\right) \) . Define \( q\left( {x}_{0}\right) = {1}_{H} \), and if \( v \in \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \) define \[ q\left( v\right) = \phi \left( {p\left( v\right) p{\left( {x}_{i}\right) }^{-1}}\right) q\left( {x}_{i}\right) . \] (13.2) This definition is recursive because here \( q\left( {x}_{i}\right) \) is defined by (13.2) with \( i \) replaced by \( i - 1 \) if \( i > 0 \) . With this definition,(13.2) is actually true for \( v \in \left\lbrack {{x}_{i - 1},{x}_{i + 1}}\right\rbrack \) if \( i \geq 1 \) . Indeed, if \( v \in \left\lbrack {{x}_{i - 1},{x}_{i}}\right\rbrack \) (the subinterval for which this is not a definition), we have \[ q\left( v\right) = \phi \left( {p\left( v\right) p{\left( {x}_{i - 1}\right) }^{-1}}\right) q\left( {x}_{i - 1}\right) , \] so what we need to show is that \[ q\left( {x}_{i}\right) q{\left( {x}_{i - 1}\right) }^{-1} = \phi {\left( p\left( v\right) p{\left( {x}_{i}\right) }^{-1}\right) }^{-1}\phi \left( {p\left( v\right) p{\left( {x}_{i - 1}\right) }^{-1}}\right) . \] It follows from the fact that \( \phi \) is a local homomorphism that the right-hand side is \[ \phi \left( {p\left( {x}_{i}\right) p{\left( {x}_{i - 1}\right) }^{-1}}\right) . \] Replacing \( i \) by \( i - 1 \) in (13.2) and taking \( v = {x}_{i} \), this equals \( q\left( {x}_{i}\right) q{\left( {x}_{i - 1}\right) }^{-1} \) . Now (13.1) follows for this path by noting that if \( \epsilon = \frac{1}{2}\min \left| {{x}_{i + 1} - {x}_{i}}\right| \), then when \( \left| {u - v}\right| < \epsilon, u, v \in \left\lbrack {0,1}\right\rbrack \), there exists an \( i \) such that \( u, v \in \left\lbrack {{x}_{i - 1},{x}_{i + 1}}\right\rbrack \) , and (13.1) follows from (13.2) and the fact that \( \phi \) is a local homomorphism. This proves that the path \( q \) exists. To show that it is unique, assume that (13.1) is valid for \( \left| {u - v}\right| < \epsilon \), and choose the \( {x}_{i} \) so that \( \left| {{x}_{i} - {x}_{i + 1}}\right| < \epsilon \) ; then for \( v \in \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \) ,(13.2) is true, and the values of \( q \) are determined by this property. Next we indicate how one can show that if \( p \) and \( {p}^{\prime } \) are path-homotopic, and if \( q \) and \( {q}^{\prime } \) are the corresponding paths in \( H \), then \( q\left( 1\right) = {q}^{\prime }\left( 1\right) \) . It is sufficient to prove this in the special case of a path-homotopy \( t \mapsto {p}_{t} \), where \( {p}_{0} = p \) and \( {p}_{1} = {p}^{\prime } \), such that there exists a sequence \( 0 = {x}_{1} \leq \cdots \leq {x}_{n} = 1 \) with \( {p}_{t}\left( u\right) {p}_{{t}^{\prime }}{\left( v\right) }^{-1} \in U \) when \( u, v \in \left\lbrack {{x}_{i - 1},{x}_{i + 1}}\right\rbrack \) and \( t \) and \( {t}^{\prime } \in \left\lbrack {0,1}\right\rbrack \) . For although a general path-homotopy may not satisfy this assumption, it can be broken into steps, each of which does. In this case, we define \[ {q}_{t}\left( v\right) = \phi \left( {{p}_{t}\left( v\right) p{\left( {x}_{i}\right) }^{-1}}\right) q\left( {x}_{i}\right) \] when \( v \in \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \) and verify that this \( {q}_{t} \) satisfies \[ {q}_{t}\left( v\right) {q}_{t}{\left( u\right) }^{-1} = \phi \left( {{p}_{t}\left( v\right) {p}_{t}{\left( u\right) }^{-1}}\right) \] when \( \left| {u - v}\right| \) is small. In particular, this is satisfied when \( t = 1 \) and \( {p}_{1} = \) \( {p}^{\prime } \), so \( {q}_{1} = {q}^{\prime } \) by definition. Now \( {q}^{\prime }\left( 1\right) = \phi \left( {{p}^{\prime }\left( 1\right) p{\left( 1\right) }^{-1}}\right) q\left( 1\right) = q\left( 1\right) \) since \( p\left( 1\right) = {p}^{\prime }\left( 1\right) \), as required. We now define \( \phi \left( g\right) = q\left( 1\right) \) . Since \( G \) is simply connected, any two paths from the identity to \( g \) are path-homotopic, so this is well-defined. It is straightforward to see that it agrees with \( \phi \) on \( U \) . We must show that it is a homomorphism. Given \( g \) and \( {g}^{\prime } \) in \( G \), let \( p \) be a path from the identity to \( g \), and let \( {p}^{\prime } \) be a path from the identity to \( {g}^{\prime } \), and let \( q \) and \( {q}^{\prime } \) be the corresponding paths in \( H \) defined by (13.1). We construct a path \( {p}^{\prime \prime } \) from the identity to \( g{g}^{\prime } \) by \[ {p}^{\prime \prime }\left( t\right) = \left\{ \begin{matrix} {p}^{\prime }\left( {2t}\right) \;\text{ if }0 \leq t \leq 1/2, \\ p\left( {{2t} - 1}\right) {g}^{\prime }\text{ if }1/2 \leq t \leq 1. \end{matrix}\right. \] Let \[ {q}^{\prime \prime }\left( t\right) = \left\{ \begin{matrix} {q}^{\prime }\left( {2t}\right) \;\text{ if }0 \leq t \leq 1/2 \\ q\left( {{2t} - 1}\right) {q}^{\prime }\left( 1\right) \text{ if }1/2 \leq t \leq 1 \end{matrix}\right. \] Then it is easy to check that \( {q}^{\prime \prime } \) is related to \( {p}^{\prime \prime } \) by (13.1), and taking \( t = 1 \) , we see that \( \phi \left( {g{g}^{\prime }}\right) = {q}^{\prime \prime }\left( 1\right) = q\left( 1\right) {q}^{\prime }\left( 1\right) = \phi \left( g\right) \phi \left( {g}^{\prime }\right) \) . We turn next to the computation of the fundamental groups of some noncom-pact Lie groups. As usual, we call a square complex matrix \( g \) Hermitia

Proposition 13.5. Let \( {S}^{r} \) denote the \( r \) -sphere. Then \( {\pi }_{1}\left( {S}^{1}\right) \cong \mathbb{Z} \), while \( {S}^{r} \) is simply-connected if \( r \geq 2 \) .

We may identify the circle \( {S}^{1} \) with the unit circle in \( \mathbb{C} \). Then \( x \mapsto {\mathrm{e}}^{2\pi ix} \) is a covering map \( \mathbb{R} \rightarrow {S}^{1} \). The space \( \mathbb{R} \) is contractible and hence simply-connected, so it is the universal covering space. If we give \( {S}^{1} \subset {\mathbb{C}}^{\times} \) the group structure it inherits from \( {\mathbb{C}}^{\times} \), then this map \( \mathbb{R} \rightarrow {S}^{1} \) is a group homomorphism, so by Theorem 13.2 we may identify the kernel \( \mathbb{Z} \) with \( {\pi }_{1}\left( {S}^{1}\right) \). To see that \( {S}^{r} \) is simply connected for \( r \geq 2 \), let \( p : \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{r} \) be a path. Since it is a mapping from a lower-dimensional manifold, perturbing the path slightly if necessary, we may assume that \( p \) is not surjective. If it omits one point \( P \in {S}^{r} \), its image is contained in \( {S}^{r} - \{ P\} \), which is homeomorphic to \( {\mathbb{R}}^{r} \) and hence contractible. Therefore \( p \) is path-homotopic to a trivial path.