Algebra general exam. August 19, 2020, 9am-1pm

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Directions.

• Please show all your work and justify any statements that you make.

• State clearly and fully any theorem you use.

• Vague statements and hand-waving arguments will not be appreciated.

• You may assume the statement in an earlier part proven in order to do a later part.
  DO EACH PROBLEM ON A SEPARATE SHEET OF PAPER, AND STAPLE THEM TOGETHER IN THE CORRECT ORDER BEFORE TURNING THE EXAM IN.

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"On my honor, I pledge that I have neither given nor received help on this assignment."

Problem 1

( 12 pts) Let $A$ be a finitely generated abelian group. Prove that the following are equivalent:
(a) $\operatorname{dim}_{\mathbb{Q}}\left(A \otimes_{\mathbb{Z}} \mathbb{Q}\right) \leq 1$
(b) $\operatorname{Aut}(A)$ is finite.

Hint: Use classification of finitely generated abelian groups. What does the condition $\operatorname{dim}_{\mathbb{Q}}\left(A \otimes_{\mathbb{Z}} \mathbb{Q}\right) \leq 1$ tell you about a standard decomposition of $A$ ?

Problem 2

Let $n \geq 3$ be an integer. Denote by $S_{n}$ the symmetric group on $\{1, \ldots, n\}$ and by $D_{2 n}$ the dihedral group of order $2 n$.
(a) ( 7 pts) Prove that for every $k \geq 3$, there exists an injective homomorphism $\varphi: D_{2 k} \rightarrow S_{k}$ whose image contains a $k$-cycle.
(b) ( 7 pts) Prove (using (a) or otherwise) that every element of $S_{n}$ can be written as a product of two elements of order $\leq 2$.

Problem 3

(10 pts) Let $R=\mathbb{Z}[\sqrt{3}]$. You may assume without proof that $R$ is a Euclidean domain. Let $p \in \mathbb{N}$ be a prime number with $p \equiv 5 \bmod 12$. Prove that $p$ is a prime element of $R$. Hint: $12=3 \cdot 4$.

Problem 4

Let $R$ be a commutative ring with 1 . An ideal $I$ of $R$ is called irreducible if $I$ cannot be written as $I=J \cap K$ where $J$ and $K$ are both ideals strictly containing $I$.
(a) ( 4 pts ) Prove that every prime ideal is irreducible.
(b) ( 8 pts) Assume that $R$ contains a nonzero nilpotent element (that is, a nonzero element $a$ such that $a^{n}=0$ for some $n$ ). Prove that $R$ contains an irreducible ideal which is not prime.

Hint: Fix a nonzero nilpotent element $a \in R$. Consider the set of all ideals NOT containing $a$ and show that this set has a maximal element.

Problem 5

Let $F$ be an algebraically closed field of $\operatorname{char} F \neq 2$.
(a) ( 7 pts) Let $J$ be a Jordan block of size $n$ with eigenvalue $\lambda$ over $F$. Determine the Jordan canonical form of the matrix $J^{2}$. Hint: Consider the cases $\lambda=0$ and $\lambda \neq 0$ separately.
(b) ( 7 pts) Let $A \in$ Mat $_{5}(F)$. Determine necessary and sufficient conditions for $A$ to have a square root, i.e. for there to exist a matrix $B \in \operatorname{Mat}_{5}(F)$ such that $A=B^{2}$. State your answer in the form: $A$ has a square root $\Longleftrightarrow J C F(A)$ satisfies certain conditions. Make sure to prove your answer.

Problem 6

Let $K$ be a field of characteristic 0 , and fix some algebraic closure $\bar{K}$ of $K$. Let $L / K$ be a finite extension, with $L \subseteq \bar{K}$, and let $\alpha$ be an element of $\bar{K}$.
(a) ( 7 pts) Prove that there is a surjective $K$-algebra homomorphism $\rho: L \otimes_{K} K(\alpha) \rightarrow L(\alpha)$.
(b) ( 7 pts) Prove that there exists an injective $K$-algebra homomorphism (not necessarily sending 1 to 1 ) $\iota: L(\alpha) \rightarrow L \otimes_{K} K(\alpha)$.

Hint: $K(\alpha)=\frac{K[x]}{\left(\mu_{\alpha, K}(x)\right)}$.

Problem 7

Let $p$ be an odd prime number and set $q=p^{2}$. Consider the finite field $\mathbb{F}_{q}$.
(a) ( 6 pts) Show that $\mathbb{F}_{q}$ contains a primitive 8-th root of unity $w$ and that the element $\alpha=w+w^{-1}$ satisfies $\alpha^{2}=2$.
(b) ( 6 pts) Show that $\alpha \in \mathbb{F}_{p}$ if and only if $p \equiv 1$ or $7 \bmod 8$. Hint: Check when is $\alpha^{p}=\alpha$.

Problem 8

Let $K$ be the splitting field of the polynomial $f(x)=\left(x^{3}-11\right)\left(x^{2}+x-1\right)$ over $\mathbb{Q}$.
(a) ( 6 pts) Find (with proof) the degree of the extension $K / \mathbb{Q}$.
(b) ( 6 pts) Determine the isomorphism class of the Galois group $\operatorname{Gal}(K / \mathbb{Q})$ and prove your answer. State your answer in the form $\operatorname{Gal}(K / \mathbb{Q}) \cong G$ where $G$ is a "familiar" group, e.g. $\mathrm{GL}_{2}\left(\mathbb{F}_{2}\right)$ or $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$.