Algebra general exam. January 11, 2019, 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

1. Let $X$ and $Y$ be non-abelian simple groups.
   (a) ( 7 pts) Let $G=X \times Y$. Prove that the only normal subgroups of $G$ are $G, X \times\{1\},\{1\} \times Y$ and the trivial subgroup.
   Hint: Show that if $N$ is a normal subgroup of $G$ not contained in $X \times\{1\}$ (respectively, $\{1\} \times Y$ ), then $N$ contains an element of the form ( $1, y$ ) with $y \neq 1$ (respectively, $(x, 1)$ with $x \neq 1$ ).
   (b) ( 7 pts) Use (a) to prove that $\operatorname{Aut}(X \times X)$ is isomorphic to a semidirect product of $\operatorname{Aut}(X) \times \operatorname{Aut}(X)$ and $\mathbb{Z}_{2}$ (a cyclic group of order $2)$.

Problem 2

2. Let $G$ be a finite group of order $n$.
   (a) ( 7 pts) Prove that there exists an injective homomorphism $\varphi: G \rightarrow S_{n}$ such that for every $g \in G$, the permutation $\varphi(g)$ is a product of $n / k$ disjoint cycles of length $k$ (for some $k$ depending on $g$ )
   (b) ( 6 pts) Now assume that $n$ is even and a Sylow 2-subgroup of $G$ is cyclic. Use (a) to prove that $G$ has a subgroup of index 2 .

Problem 3

3. (10 pts) Let $R=\mathbb{Z}[x]$. Find the number of maximal ideals of $R$ which contain $x^{2}+1$ and 15 and find explicit generators for each such ideal. Hint: Reduce to a question about Gaussian integers.

Problem 4

4. Let $F$ be a field with $\operatorname{char}(F) \neq 2$, let $V$ be a finite-dimensional vector space over $F$, and let $B$ be a symmetric bilinear form on $V$.
   (a) (4 pts) Prove that if $B \neq 0$, there exists $v \in V$ such that $B(v, v) \neq 0$.
   (b) (4 pts) Prove that for any $v \in V$ with $B(v, v) \neq 0$ there exists a subspace $W$ such that $V=F v \oplus W$ and $W \perp v$, that is, $B(w, v)=0$ for all $w \in W$.
   (c) ( 4 pts) Use (a) and (b) to prove that there is a basis $\left\{v_{n}\right\}$ of $V$ such that $B\left(v_{i}, v_{j}\right)=0$ for all $i \neq j$.

Problem 5

5. Let $F$ be an algebraically closed field, $n \in \mathbb{N}$ and $A \in \mathrm{GL}_{n}(F)$ an invertible $n \times n$ matrix over $F$.
   (a) ( 9 pts) Assume that $\operatorname{char}(F) \neq 2$. Prove that if $A^{2}$ is diagonalizable, then $A$ is also diagonalizable over $F$
   (b) ( 4 pts ) Give an example where $\operatorname{char}(F)=2, A^{2}$ is diagonalizable, but $A$ is not diagonalizable.

Problem 6

6. Let $R$ be a commutative ring with 1 , let $M$ be an $R$-module and $N$ a submodule of $R$.
   (a) ( 8 pts) Prove that if $N$ and $M / N$ are both finitely generated, then $M$ is finitely generated
   (b) ( 5 pts) Give an example where $M$ is finitely generated and $N$ is not.

Problem 7

7. Let $F=\mathbb{Q}(\sqrt[6]{3}, i)$.
   (a) (4 pts) Prove that $[F: \mathbb{Q}]=12$
   (b) (4 pts) Prove that the extension $F / \mathbb{Q}$ is Galois
   (c) ( 4 pts) Prove that the Galois group $\operatorname{Gal}(F / \mathbb{Q})$ is isomorphic to $D_{12}$, the dihedral group of order 12.

Problem 8

8. Let $p>2$ be a prime, let $\mathbb{F}_{p}$ be a field of order $p$ and let $\overline{\mathbb{F}_{p}}$ be an algebraic closure of $\mathbb{F}_{p}$. Let $f(x)=x^{m}+1$ for some $m \in \mathbb{N}$. Assume that $f$ is irreducible, and let $\alpha$ be a root of $f$ in $\overline{\mathbb{F}_{p}}$.
   (a) ( 5 pts ) Prove that the multiplicative order of $\alpha$ is equal to $2 m$.
   (b) ( 5 pts) Prove that $2 m$ divides $p^{m}-1$ and $2 m$ does not divide $p^{k}-1$ for any $0<k<m$.
   (c) ( 3 pts) Prove that $m \neq 4$.