Problem 1 (yau-contest 2015)
Section: Algebra and Number Theory

Let $G$ be an finite $\mathbb{Z}$-module (i.e., a finite abelian group with additive group law) with a bilinear, (strongly) alternative, and non-degenerate pairing $$\ell : G \times G \to \mathbb{Q}/\mathbb{Z}.$$ Here “(strongly) alternating” means for every $a \in G$, $\ell(a,a)=0$; “non-degenerate” means for every nonzero $a \in G$ there is a $b \in G$ such that $\ell(a,b) \neq 0$. Show in steps the following statement:

(S) $:$ $G$ is isomorphic to $H_1 \oplus H_2$ for some finite abelian groups $H_1 \simeq H_2$ such that $\ell|_{H_i \times H_i}=0$.

(1.1) For every $a \in G$, write $o(a)$ for the order of $a$ and $\ell_a : G \to \mathbb{Q}/\mathbb{Z}$ for the homomorphism $\ell_a(b)=\ell(a,b)$. Show that the image of $\ell_a$ is $o(a)^{-1}\mathbb{Z}/\mathbb{Z}$.

(1.2) Show that $G$ has a pair of elements $a,b$ with the following properties:
(a) $o(a)$ is maximal in the sense that for any $x \in G$, $o(x) \mid o(a)$;
(b) $\ell(a,b)=o(a)^{-1} \bmod \mathbb{Z}$;
(c) $o(a)=o(b)$.

We call the subgroup $\langle a,b \rangle := \mathbb{Z}a + \mathbb{Z}b$ a maximal hyperbolic subgroup of $G$.

(1.3) Let $\langle a,b \rangle$ be a maximal hyperbolic subgroup of $G$. Let $G'$ be the orthogonal complement of $\langle a,b \rangle$ consisting of elements $x \in G$ such that $\ell(x,c)=0$ for all $c \in \langle a,b \rangle$. Show that $G$ is a direct sum as follows:
$$G = \mathbb{Z}a \oplus \mathbb{Z}b \oplus G'.$$

(1.4) Finish the proof of (S) by induction.

Problem 2 (yau-contest 2015)
Section: Algebra and Number Theory

Let $O_n(\mathbb{C})$ denote the group of $n\times n$ orthogonal complex matrices, and $M_{n\times k}(\mathbb{C})$ the space of $n\times k$ complex matrices, where $n$ and $k$ are two positive integers. For $i=0,1$, let $F_i$ be the space of rational function $f$ on $M_{n\times k}(\mathbb{C})$ such that
$$(*)\quad f(gx)=\det(g)^i f(x) \quad \text{for all } g\in O_n(\mathbb{C}) \text{ and } x\in M_{n\times k}(\mathbb{C}).$$
We want to study in steps the structures of $F_0$ and $F_1$.

(2.1) For each $x \in M_{n\times k}$, let $V_x$ denote the subspace of $\mathbb{C}^n$ generated by columns of $x$, and let $Q(x)=x^t\cdot x \in M_{k\times k}(\mathbb{C})$. Show the following are equivalent:
(a) the space $V_x$ has dimension $k$, and the Euclidean inner product $(\cdot,\cdot)$ is non-degenerate on $V_x$ in the sense that $V_x^{\perp} \cap V_x = 0$;
(b) $\det Q(x) \neq 0$.

(2.2) Show that $F_0$ is a field generated by entries of $Q(x)$.

(2.3) Assume $k<n$ and let $f \in F_1$. Show that $f=0$ by the following two steps:
(a) for any $x \in M_{n\times k}(\mathbb{C})$ with $\det Q(x) \neq 0$, construct a $g \in O_n(\mathbb{C})$ such that $g|_{V_x}=1$ and $\det g=-1$;
(b) Show that $f$ vanishes on a general point $x \in M_{n\times k}(\mathbb{C})$ with $\det Q(x) \neq 0$, thus $f \equiv 0$.

(2.4) Assume $k \ge n$. Show that $F_1$ is a free vector space of rank $1$ over $F_0$.

Problem 3 (yau-contest 2015)
Section: Algebra and Number Theory

Consider the equation $f(x) := x^3 + x + 1 = 0$. We want to show in steps that
for any prime $p$, if $\left(\dfrac{31}{p}\right) = -1$, then $x^3 + x + 1$ is solvable mod $p$.

Let $x_1, x_2, x_3$ be three roots of $f(x) := x^3 + x + 1 = 0$. Let $F = \mathbb{Q}(x_1)$, and $L = \mathbb{Q}(x_1, x_2, x_3)$, and $K = \mathbb{Q}(\sqrt{\Delta})$ where $\Delta$ is the discriminant of $f(x)$:
$$\Delta = \big[(x_1 - x_2)(x_2 - x_3)(x_3 - x_1)\big]^2.$$

(3.1) Show that $f$ is irreducible, that $\Delta = -31$, and that $F$ is not Galois over $\mathbb{Q}$.

(3.2) Show that $\operatorname{Gal}(L/\mathbb{Q}) \simeq S_3$, the permutation group of three letters, that $\operatorname{Gal}(L/K) \simeq \mathbb{Z}/3\mathbb{Z}$, and that $\operatorname{Gal}(L/F) \simeq \mathbb{Z}/2\mathbb{Z}$.

(3.3) Let $\mathcal{O}_F, \mathcal{O}_K, \mathcal{O}_L$ be rings of integers of $F, K, L$ respectively. Let $p$ be a prime such that $x^3 + x + 1 = 0$ is not soluble in $\mathbb{Z}/p\mathbb{Z}$. Show the following:
(a) $p\mathcal{O}_F$ is still a prime ideal in $\mathcal{O}_F$,
(b) $p\mathcal{O}_L$ is product of two prime ideals in $\mathcal{O}_L$, and
(c) $p\mathcal{O}_K$ is product of two primes ideals in $\mathcal{O}_K$, and
(d) $x^2 + 31 = 0$ is soluble in $\mathbb{F}_p$.

Problem 4 (yau-contest 2015)
Section: Algebra

Let $p$ be a prime and $Z_p$ the ring of $p$-adic integers with a $p$-adic norm normalized by $|p| = p^{-1}$. Let $\phi : Z_p \to Z_p$ be a map defined by a power series of the form

$$\phi(x) = x^p + p \sum a_n x^n, \quad a_n \in Z_p, \quad |a_n| \to 0.$$

Let $E$ be a field, and $F$ the $E$-vector space of locally constant $E$-valued functions on $Z_p$ with an operator $\phi^*$ defined by $\phi^* f = f \circ \phi$. We want to show in steps the following statement:

The set of eigenvalues of $\phi^*$ on $F$ is $\{0, 1\}$.

(4.1) Show that $\phi$ is a contraction map on each residue class $R \in Z_p/pZ_p$:

$$|\phi(x) - \phi(y)| \le p^{-1}|x - y|, \quad \forall x, y \in R.$$

(4.2) Show that there is a $\epsilon_R \in R$ for each residue class $R$ such that

$$\lim_n \phi^n(x) = \epsilon_R, \quad \forall x \in R.$$

Here $\phi^n$ is defined inductively by $\phi^1 = \phi$, $\phi^n = \phi^{n-1} \circ \phi$.

(4.3) Let $F_0$ (resp. $F_1$) be the subspace of functions $f$ vanishing on each $\epsilon_R$ (resp. constant on $R$) for all residue class $R$. Show that $\phi^* = 1$ on $F_1$, and that for each $f \in F_0$ $\phi^{*n} f = 0$ for some $n \in N$.

(4.4) Show that for any $a \in E$, $a \ne 0, 1$, the operator $\phi^* - a$ is invertible on $F$.

Problem 5 (yau-contest 2015)
Section: Algebra

Check if the following rings are UFD (unique factorization domain).

(5.1) $R_1 = Z[\sqrt{6}]$;

(5.2) $R_2 = Z[(1 + \sqrt{-11})/2]$;

(5.3) $R_3 = C[x, y]/(x^2 + y^2 - 1)$;

(5.4) $R_4 = C[x, y]/(x^3 + y^3 - 1)$.