Math Problems Vault

Problem 1.3.1

(a)(b)

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(a)

Prove the ring $\mathbb {Z}[\sqrt{-2}]$ is Euclidean.

(b)

Using (a), find all integer solutions to the equation $y^2 + 2 = x^3$ .

Problem 1.3.2

Let $p$ be a prime. Show that for any element $a \in \mathbb {Z}/p\mathbb {Z}$ , there exist $b, c \in \mathbb {Z}/p\mathbb {Z}$ such that $a = b^2 + c^2$ .

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Problem 1.3.3

For a ring $R$ , $R^*$ denotes its multiplicative group of units, $M_n(R)$ the ring of $n \times n$ matrices over $R$ , and $GL_n(R) = M_n(R)^*$ .

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(a)

If $a \in R$ is nilpotent ( $a^m = 0$ for some $m$ ) then $1 - a \in R^*$ .

(b)

Let $J$ be a nilpotent ideal of $R$ ( $J^m = 0$ for some $m$ ). Show that $GL_n(R) \to GL_n(R/J)$ is surjective, with kernel

I + M_n(J) = \left\{ A \in M_n(R) \mid A \equiv I \bmod M_n(J)\right\} .

(c)

Let $F_q$ denote the finite field with $q = p^r$ elements ( $p$ prime). What is the order of $GL_3(F_q)$ ?

(d)

Show that $U = \left\{ \begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{pmatrix} \, \middle |\, a, b, c \in F_q\right\}$ is a $p$ -Sylow subgroup of $GL_3(F_q)$ .

(e)

Show that $GL_3(F_q)$ contains an element of order $q^3 - 1$ .

(f)

Find the order of $GL_3(\mathbb {Z}/25\mathbb {Z})$ , and describe a $5$ -Sylow subgroup.

Problem 1.3.4

Describe an infinite set of integral solutions $(a, b)$ of the Pell's equation $x^2 - 2y^2 = 1$ (i.e., $a, b \in \mathbb {Z}$ such that $a^2 - 2b^2 = 1$ ).

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Problem 1.3.5

(a)(b)

Let $p$ be a prime. Let $R$ be a commutative ring in which $a^p = a$ for all $a \in R$ .

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(a)

If $R$ is an integral domain, prove that $R$ is isomorphic (as a ring) to $\mathbb {Z}/p\mathbb {Z}$ .

(b)

In the general case (i.e., $R$ not necessarily an integral domain), prove that $R$ is isomorphic (as a ring) to a subring of a direct product (not necessarily finite) of rings each of which is isomorphic to $\mathbb {Z}/p\mathbb {Z}$ .

Problem 1.3.6

Let $R$ be a commutative ring with $1$ . If $R$ satisfies the a.c.c. (ascending chain condition) for finitely generated ideals then show that $R$ satisfies the a.c.c. for all ideals. Give an example (without proof) of such a ring which is an integral domain but not a p.i.d.

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Problem 1.3.7

(a)(b)(c)(d)

Let $R$ be a commutative ring. Let $A$ be an ideal of $R$ .

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(a)

Show that $S = \left\{ 1 + a \mid a \in A\right\}$ is a multiplicatively closed subset of $R$ .

(b)

Show there is a one-to-one correspondence between the prime ideals of $S^{-1}R$ and those prime ideals $P$ of $R$ such that $P + A \neq R$ .

(c)

If $A$ is contained in every maximal ideal of $R$ , what is $S^{-1}R$ ?

(d)

If $A$ is a maximal ideal of $R$ , what can you say about the structure of $S^{-1}R$ ?

Problem 1.3.8

Let $I$ be a nilpotent ideal in a ring $R$ , let $M$ and $N$ be $R$ -modules, and let $f : M \to N$ be an $R$ -homomorphism. Show that if the induced map

\overline{f} : M/IM \to N/IN

is surjective, then $f$ is surjective.

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Problem 1.3.9

Let $F$ be a finite field containing $5$ elements. Let $t$ be transcendental over $F$ . Explicitly construct one non-archimedean absolute value $\left|\, \cdot \, \right|$ on $F(t)$ . If $\overline{F(t)}$ is the completion of $F(t)$ with respect to $\left|\, \cdot \, \right|$ , show that the set of $\alpha (t) \in \overline{F(t)}$ satisfying $\left|\alpha (t)\right| \leq 1$ is a local ring.

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Problem 1.3.10

Prove that the ideal generated by $X_1, \ldots , X_n$ in the polynomial ring $\mathbb {C}^{X_1, \ldots , X_n}$ cannot be generated by fewer than $n$ elements.

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Problem 1.3.11

Let $A$ be a commutative ring, and let $I$ and $J$ be two (proper) ideals such that every prime ideal of $A$ contains either $I$ or $J$ but no prime ideal contains both $I$ and $J$ . Prove that

A \simeq A_1 \times A_2

for some (nontrivial) rings $A_1$ and $A_2$ .

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Problem 1.3.12

Define $f : [0,1) \to [0,1)$ by

f(x) = \begin{cases} 2x, & \text{if } 0 \leq 2x < 1, \\ 2x - 1, & \text{if } 1 \leq 2x < 2. \end{cases}

Find all $x$ such that

f(f(f(f(f(f(x)))))) = x.

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Problem 1.3.13

Let $T$ be the ring of all real trigonometric polynomials

f(x) = a_0 + \sum _{n=1}^N a_n \cos nx + b_n \sin nx.

Define $\deg f(x) = N$ where $a_N$ or $b_N \neq 0$ . Show that $\deg f \cdot g = \deg f + \deg g$ . Use this result to prove that $T$ is an integral domain which is not a unique factorization domain.

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Problem 1.3.14

Let $A$ be a Noetherian integral domain integrally closed in its field of fractions $K$ . Let $L$ be a finite separable field extension of $K$ . If $B$ is the integral closure of $A$ in $L$ , outline a proof that $B$ is Noetherian.

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Problem 1.3.15

If $A$ is a commutative ring and $A[[x]]$ is the ring of formal power series over $A$ , show that if $f = \sum_{n=0}^\infty a_n x^n$ is nilpotent, then all the elements $a_i \in A$ are nilpotent. If $A$ is Noetherian, prove the converse, i.e., that if all the elements $a_i$ are nilpotent, then $f$ is nilpotent.

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Problem 1.3.16

(a)(b)

Let $F$ be a field, and for $n \geq 1$ let $R_n$ be the subring of the ring of polynomials $F[x]$ consisting of polynomials

f_0 + f_1 x + f_2 x^2 + \cdots + f_k x^k \in F[x]

such that $f_1 = f_2 = \cdots = f_n = 0$ .

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(a)

Show that $R_n$ is a Noetherian ring.

(b)

Show that the field of fractions of $R_n$ is $F(x)$ , and determine the integral closure of $R_n$ in its field of fractions.

Problem 1.3.17

Describe all subrings of $\mathbb {Q}$ . (A subring contains $1$ by definition.)

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