Algebra general exam. January 25 2021, 9am -1pm

Your UVa ID Number:

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.

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

Problem 1

1. Let $F$ be a field. Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree $n$, let $G$ be the Galois group of $f$ over $F$, and assume that $G$ is abelian.
   (a) ( 7 pts) Prove that if $g \in G$ is any non-trivial element, then $g$ does not fix any of the roots of $f$.
   (b) ( 3 pts) Now assume that $F \subseteq \mathbb{R}$ and $n$ is odd. Prove that all roots of $f$ must be real.
   (c) ( 4 pts) Now let $F=\mathbb{Q}$. Prove that there are infinitely many $n$ for which there exists $f$ as above with a non-real root and cyclic $G$.

Problem 2

2. (12 pts) Classify conjugacy classes of matrices $A \in \mathrm{GL}_{7}(\mathbb{Q})$ such that $A^{3}=-I d$.

Problem 3

3. (12 pts) Let $R$ be a commutative ring with 1 , and assume that $|R|>1$. Prove that $R$ has at least one minimal prime ideal. Make sure to include all the details. Hint: Use Zorn’s lemma "backwards".

Note: A prime ideal $P$ of $R$ is called a minimal prime ideal if $R$ has no prime ideals strictly contained in $P$. In particular, the zero ideal is a minimal prime whenever it is prime.

Problem 4

( 12 pts) Let $G$ be a finite abelian group and let $p$ be a prime. Prove that the number of elements of order $p$ in $G$ is equal to the number of nontrivial homomorphisms from $G$ to $\mathbb{Z}_{p}$.

Hint: Calculate both numbers separately.

Problem 5

Let $n \geq 2$ be an integer, let $g=(1,2,3, \ldots, n) \in S_{n}$ and $H=\langle g\rangle$.
(a) ( 6 pts ) Prove that the centralizer of $g$ in $S_{n}$ is equal to $H$.
(b) ( 7 pts) Now assume that $n$ is prime, and let $N$ be the normalizer of $H$ in $S_{n}$. Prove that $|N|=n(n-1)$.

Problem 6

In both parts of this problem, $R$ is a commutative ring with 1 , assume that $|R|>1$, and $M$ is a flat $R$-module, that is, assume that

Whenever we have an injective homomorphism $N \xrightarrow{f} N^{\prime}$ of $R$-modules, the induced map $f \otimes 1_{M}: N \otimes_{R} M \rightarrow N^{\prime} \otimes_{R} M$ is also injective.

(1) ( 4 pts) Assume that $R$ is a domain. Prove that $M$ must be torsionfree.

(2) ( 8 pts) Now let $R$ be arbitrary. Prove that the following are equivalent:
(a) for every nonzero $R$-module $N \neq 0$ we have $N \otimes_{R} M \neq 0$
(b) for every maximal ideal $\mathfrak{m}$ of $R$ we have $\mathfrak{m} M \neq M$.

Note: You may use without proof standard isomorphisms of the form $(R / I) \otimes_{R} M \cong \ldots$ where $I$ is an ideal of $R$.

Problem 7

(13 pts) Recall that a field $F$ is called perfect if either $F$ has characteristic zero or $F$ has characteristic $p>0$ and every element of $F$ is equal to $a^{p}$ for some $a \in F$. Prove that $F$ admits a finite inseparable extension if and only if $F$ is not perfect.

Problem 8

(12 pts) Let $p \neq 3$ be a prime and $R=\mathbb{F}_{p}[x] /\left(x^{3}-1\right)$. Describe the multiplicative group $R^{\times}$as a direct product of cyclic groups.

Note: Your answer can (and should) involve cases, but should be expressed explicitly in terms of $p$.