formal
stringlengths 51
351
| informal
stringlengths 3
474
|
|---|---|
theorem exercise_2_92 {Ξ± : Type*} [topological_space Ξ±]
{s : β β set Ξ±}
(hs : β i, is_compact (s i))
(hs : β i, (s i).nonempty)
(hs : β i, (s i) β (s (i + 1))) :
(β i, s i).nonempty := | Give a direct proof that the nested decreasing intersection of nonempty covering compact sets is nonempty. |
theorem exercise_3_1 {f : β β β}
(hf : β x y, |f x - f y| β€ |x - y| ^ 2) :
β c, f = Ξ» x, c := | Assume that $f \colon \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t)-f(x)| \leq|t-x|^{2}$ for all $t, x$. Prove that $f$ is constant. |
theorem exercise_3_63a (p : β) (f : β β β) (hp : p > 1)
(h : f = Ξ» k, (1 : β) / (k * (log k) ^ p)) :
β l, tendsto f at_top (π l) := | Prove that $\sum 1/k(\log(k))^p$ converges when $p > 1$. |
theorem exercise_4_15a {Ξ± : Type*}
(a b : β) (F : set (β β β)) :
(β (x : β) (Ξ΅ > 0), β (U β (π x)),
(β (y z β U) (f : β β β), f β F β (dist (f y) (f z) < Ξ΅)))
β
β (ΞΌ : β β β), β (x : β), (0 : β) β€ ΞΌ x β§ tendsto ΞΌ (π 0) (π 0) β§
(β (s t : β) (f : β β β), f β F β |(f s) - (f t)| β€ ΞΌ (|s - t|)) := | A continuous, strictly increasing function $\mu \colon (0, \infty) \rightarrow (0, \infty)$ is a modulus of continuity if $\mu(s) \rightarrow 0$ as $s \rightarrow 0$. A function $f \colon [a, b] \rightarrow \mathbb{R}$ has modulus of continuity $\mu$ if $|f(s) - f(t)| \leq \mu(|s - t|)$ for all $s, t \in [a, b]$. Prove that a function is uniformly continuous if and only if it has a modulus of continuity. |
theorem exercise_2_1_18 {G : Type*} [group G]
[fintype G] (hG2 : even (fintype.card G)) :
β (a : G), a β 1 β§ a = aβ»ΒΉ := | If $G$ is a finite group of even order, show that there must be an element $a \neq e$ such that $a=a^{-1}$. |
theorem exercise_2_1_26 {G : Type*} [group G]
[fintype G] (a : G) : β (n : β), a ^ n = 1 := | If $G$ is a finite group, prove that, given $a \in G$, there is a positive integer $n$, depending on $a$, such that $a^n = e$. |
theorem exercise_2_2_3 {G : Type*} [group G]
{P : β β Prop} {hP : P = Ξ» i, β a b : G, (a*b)^i = a^i * b^i}
(hP1 : β n : β, P n β§ P (n+1) β§ P (n+2)) : comm_group G := | If $G$ is a group in which $(a b)^{i}=a^{i} b^{i}$ for three consecutive integers $i$, prove that $G$ is abelian. |
theorem exercise_2_2_6c {G : Type*} [group G] {n : β} (hn : n > 1)
(h : β (a b : G), (a * b) ^ n = a ^ n * b ^ n) :
β (a b : G), (a * b * aβ»ΒΉ * bβ»ΒΉ) ^ (n * (n - 1)) = 1 := | Let $G$ be a group in which $(a b)^{n}=a^{n} b^{n}$ for some fixed integer $n>1$ for all $a, b \in G$. For all $a, b \in G$, prove that $\left(a b a^{-1} b^{-1}\right)^{n(n-1)}=e$. |
theorem exercise_2_3_16 {G : Type*} [group G]
(hG : β H : subgroup G, H = β€ β¨ H = β₯) :
is_cyclic G β§ β (p : β) (fin : fintype G), nat.prime p β§ @card G fin = p := | If a group $G$ has no proper subgroups, prove that $G$ is cyclic of order $p$, where $p$ is a prime number. |
theorem exercise_2_5_23 {G : Type*} [group G]
(hG : β (H : subgroup G), H.normal) (a b : G) :
β (j : β€) , b*a = a^j * b:= | Let $G$ be a group such that all subgroups of $G$ are normal in $G$. If $a, b \in G$, prove that $ba = a^jb$ for some $j$. |
theorem exercise_2_5_31 {G : Type*} [comm_group G] [fintype G]
{p m n : β} (hp : nat.prime p) (hp1 : Β¬ p β£ m) (hG : card G = p^n*m)
{H : subgroup G} [fintype H] (hH : card H = p^n) :
characteristic H := | Suppose that $G$ is an abelian group of order $p^nm$ where $p \nmid m$ is a prime. If $H$ is a subgroup of $G$ of order $p^n$, prove that $H$ is a characteristic subgroup of $G$. |
theorem exercise_2_5_43 (G : Type*) [group G] [fintype G]
(hG : card G = 9) :
comm_group G := | Prove that a group of order 9 must be abelian. |
theorem exercise_2_5_52 {G : Type*} [group G] [fintype G]
(Ο : G β* G) {I : finset G} (hI : β x β I, Ο x = xβ»ΒΉ)
(hI1 : (0.75 : β) * card G β€ card I) :
β x : G, Ο x = xβ»ΒΉ β§ β x y : G, x*y = y*x := | Let $G$ be a finite group and $\varphi$ an automorphism of $G$ such that $\varphi(x) = x^{-1}$ for more than three-fourths of the elements of $G$. Prove that $\varphi(y) = y^{-1}$ for all $y \in G$, and so $G$ is abelian. |
theorem exercise_2_7_7 {G : Type*} [group G] {G' : Type*} [group G']
(Ο : G β* G') (N : subgroup G) [N.normal] :
(map Ο N).normal := | If $\varphi$ is a homomorphism of $G$ onto $G'$ and $N \triangleleft G$, show that $\varphi(N) \triangleleft G'$. |
theorem exercise_2_8_15 {G H: Type*} [fintype G] [group G] [fintype H]
[group H] {p q : β} (hp : nat.prime p) (hq : nat.prime q)
(h : p > q) (h1 : q β£ p - 1) (hG : card G = p*q) (hH : card G = p*q) :
G β* H := | Prove that if $p > q$ are two primes such that $q \mid p - 1$, then any two nonabelian groups of order $pq$ are isomorphic. |
theorem exercise_2_10_1 {G : Type*} [group G] (A : subgroup G)
[A.normal] {b : G} (hp : nat.prime (order_of b)) :
A β (closure {b}) = β₯ := | Let $A$ be a normal subgroup of a group $G$, and suppose that $b \in G$ is an element of prime order $p$, and that $b \not\in A$. Show that $A \cap (b) = (e)$. |
theorem exercise_2_11_7 {G : Type*} [group G] {p : β} (hp : nat.prime p)
{P : sylow p G} (hP : P.normal) :
characteristic (P : subgroup G) := | If $P \triangleleft G$, $P$ a $p$-Sylow subgroup of $G$, prove that $\varphi(P) = P$ for every automorphism $\varphi$ of $G$. |
theorem exercise_3_2_21 {Ξ± : Type*} [fintype Ξ±] {Ο Ο: equiv.perm Ξ±}
(h1 : β a : Ξ±, Ο a = a β Ο a β a) (h2 : Ο β Ο = id) :
Ο = 1 β§ Ο = 1 := | If $\sigma, \tau$ are two permutations that disturb no common element and $\sigma \tau = e$, prove that $\sigma = \tau = e$. |
theorem exercise_4_1_34 : equiv.perm (fin 3) β* general_linear_group (fin 2) (zmod 2) := | Let $T$ be the group of $2\times 2$ matrices $A$ with entries in the field $\mathbb{Z}_2$ such that $\det A$ is not equal to 0. Prove that $T$ is isomorphic to $S_3$, the symmetric group of degree 3. |
theorem exercise_4_2_6 {R : Type*} [ring R] (a x : R)
(h : a ^ 2 = 0) : a * (a * x + x * a) = (x + x * a) * a := | If $a^2 = 0$ in $R$, show that $ax + xa$ commutes with $a$. |
theorem exercise_4_3_1 {R : Type*} [comm_ring R] (a : R) :
β I : ideal R, {x : R | x*a=0} = I := | If $R$ is a commutative ring and $a \in R$, let $L(a) = \{x \in R \mid xa = 0\}$. Prove that $L(a)$ is an ideal of $R$. |
theorem exercise_4_4_9 (p : β) (hp : nat.prime p) :
(β S : finset (zmod p), S.card = (p-1)/2 β§ β x : zmod p, x^2 = p) β§
(β S : finset (zmod p), S.card = (p-1)/2 β§ Β¬ β x : zmod p, x^2 = p) := | Show that $(p - 1)/2$ of the numbers $1, 2, \ldots, p - 1$ are quadratic residues and $(p - 1)/2$ are quadratic nonresidues $\mod p$. |
theorem exercise_4_5_23 {p q: polynomial (zmod 7)}
(hp : p = X^3 - 2) (hq : q = X^3 + 2) :
irreducible p β§ irreducible q β§
(nonempty $ polynomial (zmod 7) β§Έ ideal.span ({p} : set $ polynomial $ zmod 7) β+*
polynomial (zmod 7) β§Έ ideal.span ({q} : set $ polynomial $ zmod 7)) := | Let $F = \mathbb{Z}_7$ and let $p(x) = x^3 - 2$ and $q(x) = x^3 + 2$ be in $F[x]$. Show that $p(x)$ and $q(x)$ are irreducible in $F[x]$ and that the fields $F[x]/(p(x))$ and $F[x]/(q(x))$ are isomorphic. |
theorem exercise_4_6_2 : irreducible (X^3 + 3*X + 2 : polynomial β) := | Prove that $f(x) = x^3 + 3x + 2$ is irreducible in $Q[x]$. |
theorem exercise_5_1_8 {p m n: β} {F : Type*} [field F]
(hp : nat.prime p) (hF : char_p F p) (a b : F) (hm : m = p ^ n) :
(a + b) ^ m = a^m + b^m := | If $F$ is a field of characteristic $p \neq 0$, show that $(a + b)^m = a^m + b^m$, where $m = p^n$, for all $a, b \in F$ and any positive integer $n$. |
theorem exercise_5_3_7 {K : Type*} [field K] {F : subfield K}
{a : K} (ha : is_algebraic F (a ^ 2)) : is_algebraic F a := | If $a \in K$ is such that $a^2$ is algebraic over the subfield $F$ of $K$, show that a is algebraic over $F$. |
theorem exercise_5_4_3 {a : β} {p : β β β}
(hp : p = Ξ» x, x^5 + real.sqrt 2 * x^3 + real.sqrt 5 * x^2 +
real.sqrt 7 * x + 11)
(ha : p a = 0) :
β p : polynomial β , p.degree < 80 β§ a β p.roots β§
β n : p.support, β a b : β€, p.coeff n = a / b := | If $a \in C$ is such that $p(a) = 0$, where $p(x) = x^5 + \sqrt{2}x^3 + \sqrt{5}x^2 + \sqrt{7}x + \sqrt{11}$, show that $a$ is algebraic over $\mathbb{Q}$ of degree at most 80. |
theorem exercise_5_6_14 {p m n: β} (hp : nat.prime p) {F : Type*}
[field F] [char_p F p] (hm : m = p ^ n) :
card (root_set (X ^ m - X : polynomial F) F) = m := | If $F$ is of characteristic $p \neq 0$, show that all the roots of $x^m - x$, where $m = p^n$, are distinct. |
theorem exercise_2_3_2 {G : Type*} [group G] (a b : G) :
β g : G, b* a = g * a * b * gβ»ΒΉ := | Prove that the products $a b$ and $b a$ are conjugate elements in a group. |
theorem exercise_2_8_6 {G H : Type*} [group G] [group H] :
center (G Γ H) β* (center G) Γ (center H) := | Prove that the center of the product of two groups is the product of their centers. |
theorem exercise_3_2_7 {F : Type*} [field F] {G : Type*} [field G]
(Ο : F β+* G) : injective Ο := | Prove that every homomorphism of fields is injective. |
theorem exercise_3_7_2 {K V : Type*} [field K] [add_comm_group V]
[module K V] {ΞΉ : Type*} [fintype ΞΉ] (Ξ³ : ΞΉ β submodule K V)
(h : β i : ΞΉ, Ξ³ i β β€) :
(β (i : ΞΉ), (Ξ³ i : set V)) β β€ := | Let $V$ be a vector space over an infinite field $F$. Prove that $V$ is not the union of finitely many proper subspaces. |
theorem exercise_6_4_2 {G : Type*} [group G] [fintype G] {p q : β}
(hp : prime p) (hq : prime q) (hG : card G = p*q) :
is_simple_group G β false := | Prove that no group of order $p q$, where $p$ and $q$ are prime, is simple. |
theorem exercise_6_4_12 {G : Type*} [group G] [fintype G]
(hG : card G = 224) :
is_simple_group G β false := | Prove that no group of order 224 is simple. |
theorem exercise_10_1_13 {R : Type*} [ring R] {x : R}
(hx : is_nilpotent x) : is_unit (1 + x) := | An element $x$ of a ring $R$ is called nilpotent if some power of $x$ is zero. Prove that if $x$ is nilpotent, then $1+x$ is a unit in $R$. |
theorem exercise_10_6_7 {I : ideal gaussian_int}
(hI : I β β₯) : β (z : I), z β 0 β§ (z : gaussian_int).im = 0 := | Prove that every nonzero ideal in the ring of Gauss integers contains a nonzero integer. |
theorem exercise_10_4_7a {R : Type*} [comm_ring R] [no_zero_divisors R]
(I J : ideal R) (hIJ : I + J = β€) : I * J = I β J := | Let $I, J$ be ideals of a ring $R$ such that $I+J=R$. Prove that $I J=I \cap J$. |
theorem exercise_11_2_13 (a b : β€) :
(of_int a : gaussian_int) β£ of_int b β a β£ b := | If $a, b$ are integers and if $a$ divides $b$ in the ring of Gauss integers, then $a$ divides $b$ in $\mathbb{Z}$. |
theorem exercise_11_4_6a {F : Type*} [field F] [fintype F] (hF : card F = 7) :
irreducible (X ^ 2 + 1 : polynomial F) := | Prove that $x^2+x+1$ is irreducible in the field $\mathbb{F}_2$. |
theorem exercise_11_4_6c : irreducible (X^3 - 9 : polynomial (zmod 31)) := | Prove that $x^3 - 9$ is irreducible in $\mathbb{F}_{31}$. |
theorem exercise_11_13_3 (N : β):
β p β₯ N, nat.prime p β§ p + 1 β‘ 0 [MOD 4] := | Prove that there are infinitely many primes congruent to $-1$ (modulo $4$). |
theorem exercise_13_6_10 {K : Type*} [field K] [fintype KΛ£] :
β (x : KΛ£), x = -1 := | Let $K$ be a finite field. Prove that the product of the nonzero elements of $K$ is $-1$. |
theorem exercise_1_1_2a : β a b : β€, a - b β b - a := | Prove the the operation $\star$ on $\mathbb{Z}$ defined by $a\star b=a-b$ is not commutative. |
theorem exercise_1_1_4 (n : β) :
β (a b c : β), (a * b) * c β‘ a * (b * c) [ZMOD n] := | Prove that the multiplication of residue class $\mathbb{Z}/n\mathbb{Z}$ is associative. |
theorem exercise_1_1_15 {G : Type*} [group G] (as : list G) :
as.prodβ»ΒΉ = (as.reverse.map (Ξ» x, xβ»ΒΉ)).prod := | Prove that $(a_1a_2\dots a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}\dots a_1^{-1}$ for all $a_1, a_2, \dots, a_n\in G$. |
theorem exercise_1_1_17 {G : Type*} [group G] {x : G} {n : β}
(hxn: order_of x = n) :
xβ»ΒΉ = x ^ (n - 1 : β€) := | Let $x$ be an element of $G$. Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$. |
theorem exercise_1_1_20 {G : Type*} [group G] {x : G} :
order_of x = order_of xβ»ΒΉ := | For $x$ an element in $G$ show that $x$ and $x^{-1}$ have the same order. |
theorem exercise_1_1_22b {G: Type*} [group G] (a b : G) :
order_of (a * b) = order_of (b * a) := | Deduce that $|a b|=|b a|$ for all $a, b \in G$. |
theorem exercise_1_1_29 {A B : Type*} [group A] [group B] :
β x y : A Γ B, x*y = y*x β (β x y : A, x*y = y*x) β§
(β x y : B, x*y = y*x) := | Prove that $A \times B$ is an abelian group if and only if both $A$ and $B$ are abelian. |
theorem exercise_1_3_8 : infinite (equiv.perm β) := | Prove that if $\Omega=\{1,2,3, \ldots\}$ then $S_{\Omega}$ is an infinite group |
theorem exercise_1_6_11 {A B : Type*} [group A] [group B] :
A Γ B β* B Γ A := | Let $A$ and $B$ be groups. Prove that $A \times B \cong B \times A$. |
theorem exercise_1_6_23 {G : Type*}
[group G] (Ο : mul_aut G) (hs : β g : G, Ο g = 1 β g = 1)
(hs2 : β g : G, Ο (Ο g) = g) :
β x y : G, x*y = y*x := | Let $G$ be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g)=g$ if and only if $g=1$. If $\sigma^{2}$ is the identity map from $G$ to $G$, prove that $G$ is abelian. |
theorem exercise_2_1_13 (H : add_subgroup β) {x : β}
(hH : x β H β (1 / x) β H):
H = β₯ β¨ H = β€ := | Let $H$ be a subgroup of the additive group of rational numbers with the property that $1 / x \in H$ for every nonzero element $x$ of $H$. Prove that $H=0$ or $\mathbb{Q}$. |
theorem exercise_2_4_16a {G : Type*} [group G] {H : subgroup G}
(hH : H β β€) :
β M : subgroup G, M β β€ β§
β K : subgroup G, M β€ K β K = M β¨ K = β€ β§
H β€ M := | A subgroup $M$ of a group $G$ is called a maximal subgroup if $M \neq G$ and the only subgroups of $G$ which contain $M$ are $M$ and $G$. Prove that if $H$ is a proper subgroup of the finite group $G$ then there is a maximal subgroup of $G$ containing $H$. |
theorem exercise_2_4_16c {n : β} (H : add_subgroup (zmod n)) :
β p : β, nat.prime p β§ H = add_subgroup.closure {p} β
H β β€ β§ β K : add_subgroup (zmod n), H β€ K β K = H β¨ K = β€ := | Show that if $G=\langle x\rangle$ is a cyclic group of order $n \geq 1$ then a subgroup $H$ is maximal if and only $H=\left\langle x^{p}\right\rangle$ for some prime $p$ dividing $n$. |
theorem exercise_3_1_22a (G : Type*) [group G] (H K : subgroup G)
[subgroup.normal H] [subgroup.normal K] :
subgroup.normal (H β K) := | Prove that if $H$ and $K$ are normal subgroups of a group $G$ then their intersection $H \cap K$ is also a normal subgroup of $G$. |
theorem exercise_3_2_8 {G : Type*} [group G] (H K : subgroup G)
[fintype H] [fintype K]
(hHK : nat.coprime (fintype.card H) (fintype.card K)) :
H β K = β₯ := | Prove that if $H$ and $K$ are finite subgroups of $G$ whose orders are relatively prime then $H \cap K=1$. |
theorem exercise_3_2_16 (p : β) (hp : nat.prime p) (a : β) :
nat.coprime a p β a ^ p β‘ a [ZMOD p] := | Use Lagrange's Theorem in the multiplicative group $(\mathbb{Z} / p \mathbb{Z})^{\times}$to prove Fermat's Little Theorem: if $p$ is a prime then $a^{p} \equiv a(\bmod p)$ for all $a \in \mathbb{Z}$. |
theorem exercise_3_3_3 {p : primes} {G : Type*} [group G]
{H : subgroup G} [hH : H.normal] (hH1 : H.index = p) :
β K : subgroup G, K β€ H β¨ H β K = β€ β¨ (K β H).relindex K = p := | Prove that if $H$ is a normal subgroup of $G$ of prime index $p$ then for all $K \leq G$ either $K \leq H$, or $G=H K$ and $|K: K \cap H|=p$. |
theorem exercise_3_4_4 {G : Type*} [comm_group G] [fintype G] {n : β}
(hn : n β£ (fintype.card G)) :
β (H : subgroup G) (H_fin : fintype H), @card H H_fin = n := | Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each positive divisor $n$ of its order. |
theorem exercise_3_4_5b {G : Type*} [group G] [is_solvable G]
(H : subgroup G) [subgroup.normal H] :
is_solvable (G β§Έ H) := | Prove that quotient groups of a solvable group are solvable. |
theorem exercise_4_2_8 {G : Type*} [group G] {H : subgroup G}
{n : β} (hn : n > 0) (hH : H.index = n) :
β K β€ H, K.normal β§ K.index β€ n.factorial := | Prove that if $H$ has finite index $n$ then there is a normal subgroup $K$ of $G$ with $K \leq H$ and $|G: K| \leq n!$. |
theorem exercise_4_2_9a {G : Type*} [fintype G] [group G] {p Ξ± : β}
(hp : p.prime) (ha : Ξ± > 0) (hG : card G = p ^ Ξ±) :
β H : subgroup G, H.index = p β H.normal := | Prove that if $p$ is a prime and $G$ is a group of order $p^{\alpha}$ for some $\alpha \in \mathbb{Z}^{+}$, then every subgroup of index $p$ is normal in $G$. |
theorem exercise_4_4_2 {G : Type*} [fintype G] [group G]
{p q : nat.primes} (hpq : p β q) (hG : card G = p*q) :
is_cyclic G := | Prove that if $G$ is an abelian group of order $p q$, where $p$ and $q$ are distinct primes, then $G$ is cyclic. |
theorem exercise_4_4_6b :
β (G : Type*) (hG : group G) (H : @subgroup G hG), @characteristic G hG H β§ Β¬ @subgroup.normal G hG H := | Prove that there exists a normal subgroup that is not characteristic. |
theorem exercise_4_4_8a {G : Type*} [group G] (H K : subgroup G)
(hHK : H β€ K) [hHK1 : (H.subgroup_of K).normal] [hK : K.normal] :
H.normal := | Let $G$ be a group with subgroups $H$ and $K$ with $H \leq K$. Prove that if $H$ is characteristic in $K$ and $K$ is normal in $G$ then $H$ is normal in $G$. |
theorem exercise_4_5_13 {G : Type*} [group G] [fintype G]
(hG : card G = 56) :
β (p : β) (P : sylow p G), P.normal := | Prove that a group of order 56 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order. |
theorem exercise_4_5_15 {G : Type*} [group G] [fintype G]
(hG : card G = 351) :
β (p : β) (P : sylow p G), P.normal := | Prove that a group of order 351 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order. |
theorem exercise_4_5_17 {G : Type*} [fintype G] [group G]
(hG : card G = 105) :
nonempty(sylow 5 G) β§ nonempty(sylow 7 G) := | Prove that if $|G|=105$ then $G$ has a normal Sylow 5 -subgroup and a normal Sylow 7-subgroup. |
theorem exercise_4_5_19 {G : Type*} [fintype G] [group G]
(hG : card G = 6545) : Β¬ is_simple_group G := | Prove that if $|G|=6545$ then $G$ is not simple. |
theorem exercise_4_5_21 {G : Type*} [fintype G] [group G]
(hG : card G = 2907) : Β¬ is_simple_group G := | Prove that if $|G|=2907$ then $G$ is not simple. |
theorem exercise_4_5_23 {G : Type*} [fintype G] [group G]
(hG : card G = 462) : Β¬ is_simple_group G := | Prove that if $|G|=462$ then $G$ is not simple. |
theorem exercise_4_5_33 {G : Type*} [group G] [fintype G] {p : β}
(P : sylow p G) [hP : P.normal] (H : subgroup G) [fintype H] :
β R : sylow p H, R.to_subgroup = (H β P.to_subgroup).subgroup_of H β§
nonempty (sylow p H) := | Let $P$ be a normal Sylow $p$-subgroup of $G$ and let $H$ be any subgroup of $G$. Prove that $P \cap H$ is the unique Sylow $p$-subgroup of $H$. |
theorem exercise_7_1_2 {R : Type*} [ring R] {u : R}
(hu : is_unit u) : is_unit (-u) := | Prove that if $u$ is a unit in $R$ then so is $-u$. |
theorem exercise_7_1_12 {F : Type*} [field F] {K : subring F}
(hK : (1 : F) β K) : is_domain K := | Prove that any subring of a field which contains the identity is an integral domain. |
theorem exercise_7_2_2 {R : Type*} [ring R] (p : polynomial R) :
p β£ 0 β β b : R, b β 0 β§ b β’ p = 0 := | Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in $R[x]$ if and only if there is a nonzero $b \in R$ such that $b p(x)=0$. |
theorem exercise_7_3_16 {R S : Type*} [ring R] [ring S]
{Ο : R β+* S} (hf : surjective Ο) :
Ο '' (center R) β center S := | Let $\varphi: R \rightarrow S$ be a surjective homomorphism of rings. Prove that the image of the center of $R$ is contained in the center of $S$. |
theorem exercise_7_4_27 {R : Type*} [comm_ring R] (hR : (0 : R) β 1)
{a : R} (ha : is_nilpotent a) (b : R) :
is_unit (1-a*b) := | Let $R$ be a commutative ring with $1 \neq 0$. Prove that if $a$ is a nilpotent element of $R$ then $1-a b$ is a unit for all $b \in R$. |
theorem exercise_8_2_4 {R : Type*} [ring R][no_zero_divisors R]
[cancel_comm_monoid_with_zero R] [gcd_monoid R]
(h1 : β a b : R, a β 0 β b β 0 β β r s : R, gcd a b = r*a + s*b)
(h2 : β a : β β R, (β i j : β, i < j β a i β£ a j) β
β N : β, β n β₯ N, β u : R, is_unit u β§ a n = u * a N) :
is_principal_ideal_ring R := | Let $R$ be an integral domain. Prove that if the following two conditions hold then $R$ is a Principal Ideal Domain: (i) any two nonzero elements $a$ and $b$ in $R$ have a greatest common divisor which can be written in the form $r a+s b$ for some $r, s \in R$, and (ii) if $a_{1}, a_{2}, a_{3}, \ldots$ are nonzero elements of $R$ such that $a_{i+1} \mid a_{i}$ for all $i$, then there is a positive integer $N$ such that $a_{n}$ is a unit times $a_{N}$ for all $n \geq N$. |
theorem exercise_8_3_5a {n : β€} (hn0 : n > 3) (hn1 : squarefree n) :
irreducible (2 :zsqrtd $ -n) β§
irreducible (β¨0, 1β© : zsqrtd $ -n) β§
irreducible (1 + β¨0, 1β© : zsqrtd $ -n) := | Let $R=\mathbb{Z}[\sqrt{-n}]$ where $n$ is a squarefree integer greater than 3. Prove that $2, \sqrt{-n}$ and $1+\sqrt{-n}$ are irreducibles in $R$. |
theorem exercise_8_3_6b {q : β} (hq0 : q.prime)
(hq1 : q β‘ 3 [ZMOD 4]) {R : Type*} [ring R]
(hR : R = (gaussian_int β§Έ ideal.span ({q} : set gaussian_int))) :
is_field R β§ β finR : fintype R, @card R finR = q^2 := | Let $q \in \mathbb{Z}$ be a prime with $q \equiv 3 \bmod 4$. Prove that the quotient ring $\mathbb{Z}[i] /(q)$ is a field with $q^{2}$ elements. |
theorem exercise_9_1_10 {f : β β mv_polynomial β β€}
(hf : f = Ξ» i, X i * X (i+1)):
infinite (minimal_primes (mv_polynomial β β€ β§Έ ideal.span (range f))) := | Prove that the ring $\mathbb{Z}\left[x_{1}, x_{2}, x_{3}, \ldots\right] /\left(x_{1} x_{2}, x_{3} x_{4}, x_{5} x_{6}, \ldots\right)$ contains infinitely many minimal prime ideals. |
theorem exercise_9_4_2a : irreducible (X^4 - 4*X^3 + 6 : polynomial β€) := | Prove that $x^4-4x^3+6$ is irreducible in $\mathbb{Z}[x]$. |
theorem exercise_9_4_2c : irreducible
(X^4 + 4*X^3 + 6*X^2 + 2*X + 1 : polynomial β€) := | Prove that $x^4+4x^3+6x^2+2x+1$ is irreducible in $\mathbb{Z}[x]$. |
theorem exercise_9_4_9 :
irreducible (X^2 - C sqrtd : polynomial (zsqrtd 2)) := | Prove that the polynomial $x^{2}-\sqrt{2}$ is irreducible over $\mathbb{Z}[\sqrt{2}]$. You may assume that $\mathbb{Z}[\sqrt{2}]$ is a U.F.D. |
theorem exercise_11_1_13 {ΞΉ : Type*} [fintype ΞΉ] :
(ΞΉ β β) ββ[β] β := | Prove that as vector spaces over $\mathbb{Q}, \mathbb{R}^n \cong \mathbb{R}$, for all $n \in \mathbb{Z}^{+}$. |