Algebra general exam. August 19, 2019, 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

(a) (6 pts) Let $G$ be a group of order 12. Prove that $G$ has a normal Sylow subgroup.
(b) ( 9 pts) Prove that there are at least 5 pairwise non-isomorphic groups of order 12 (in fact, 5 is the exact number of isomorphism classes, but you are not asked to prove this). Partial credit for exhibiting fewer than 5 non-isomorphic groups (with proof) will be given.

Problem 2

( 10 pts ) Let $n \geq 3$ be an integer and let $S_{n}$ be the symmetric group on $\{1,2, \ldots, n\}$ ). Let $H$ be a subgroup of $S_{n}$ with $\left[S_{n}: H\right]=n$. Prove that

$$H \cong S_{n-1} .$$

Hint: Start by constructing a suitable action of $S_{n}$ associated to $H$. You may use the description of normal subgroups of $S_{n}$ without proof.

Problem 3

(10 pts) Let $m$ and $n$ be positive integers with $m \mid n$, and let $f: \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / m \mathbb{Z}$ be the natural projection. Proved that the associated map of the groups of units $f:(\mathbb{Z} / n \mathbb{Z})^{\times} \rightarrow(\mathbb{Z} / m \mathbb{Z})^{\times}$is surjective.

Problem 4

Let $R$ be a commutative ring with 1 , let $S$ be a subring of $R$ with 1 , and let $M$ and $N$ be $R$-modules
(a) ( 5 pts) Prove that there exists a surjective $S$-module homomorphism $\varphi: M \otimes_{S} N \rightarrow M \otimes_{R} N$ such that $\varphi\left(m \otimes_{S} n\right)=m \otimes_{R} n$ for all $m \in M$ and $n \in N$. Also prove that such $\varphi$ is unique.
(b) ( 6 pts) Assume that $R$ and $S$ are both fields, $R \neq S$ and $M$ and $N$ are both nonzero. Prove that $\varphi$ in part (a) is not injective. Warning: $M$ and $N$ are not assumed to be finitely generated.
(c) ( 4 pts) Suppose that we only know that $R$ is a field. Does the conclusion of (b) remain true?

Problem 5

Let $F$ be a field, let $V$ be a vector space over $F$ of finite dimension $n$, and let $T: V \rightarrow V$ be an $F$-linear map.
(a) ( 5 pts) Assume that $F$ is algebraically closed. Prove that $V$ has at least $n+1 T$-invariant subspaces (including 0 and $V$ )
(b) ( 2 pts) Give an example showing that if $F$ is not algebraically closed, the conclusion of (a) may be false.
(c) ( 7 pts) Now assume that $T$ is diagonalizable over $F$ and has $n$ distinct eigenvalues. Prove that the number of $T$-invariant subspaces depends only on $n$ and find that number.

Problem 6

Let $R=\mathbb{Z}[x]$.
(a) ( 6 pts) Let $I$ be an ideal of $R$ which contains a MONIC polynomial of degree $n$, call it $p(x)$. Prove that $I$ can be generated (as an ideal) by at most $n+1$ elements. Hint: Consider the quotient $I /(p(x))$.
(b) ( 6 pts) Now let $M=(2, x)$ and $I=M^{2}=\left(4,2 x, x^{2}\right)$. Prove that $I$ cannot be generated (as an ideal) by less than 3 elements. Hint: Consider the quotient $M^{2} / M^{3}$.

Problem 7

Let $p$ and $q$ be distinct primes, let $F=\mathbb{Q}(\sqrt[3]{p}, \sqrt[5]{q})$ and let $K$ be the Galois closure of $F$ over $\mathbb{Q}$.
(a) ( 3 pts ) Prove that $[F: \mathbb{Q}]=15$.
(b) (4 pts) Prove that $[K: \mathbb{Q}]=120$.
(c) ( 4 pts ) Prove that $\operatorname{Gal}(K / \mathbb{Q})$ has a normal subgroup of order 15.
(d) ( 3 pts) Prove that $\operatorname{Gal}(K / \mathbb{Q})$ has no normal subgroup of order 8 .

Problem 8

(10 pts) Let $q$ be a prime power, let $\mathbb{F}_{q}$ be a field of order $q$, and let $a \in \mathbb{F}_{q}$ be a nonzero element. Consider the polynomial

$$f(x)=x^{q+1}-a$$

Let $L$ be the splitting field of $f(X)$ over $K=\mathbb{F}_{q}$. Prove that $[L: K]=2$.
Note: Make sure to prove that the degree is equal to 2 , not just $\leq 2$.
Hint: Let $\alpha$ be a root of $f$. What can you say about the (multiplicative) order of $\alpha$ ?