Algebra general exam. January 9, 2013, 9am -1pm

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.

Problem 1

Let $p$ be a prime and let $S_{2 p}$ denote the symmetric group on $2 p$ elements.
(a) ( 2 pts) Find the order of a $p$-Sylow subgroup of $S_{2 p}$.
(b) ( 5 pts) Describe explicitly a $p$-Sylow subgroup of $S_{2 p}$ (providing a generating set counts as explicit description, but make sure to prove that your subgroup is indeed $p$-Sylow).
(c) ( 2 pts) Consider the set of elements of order $p$ in $S_{2 p}$ - clearly, it is a union of conjugacy classes. How many conjugacy classes does it consist of?
(d) ( 5 pts) Now consider the set of elements of order $p$ in the alternating group $A_{2 p}$. How many conjugacy classes (of $A_{2 p}$ ) does it consist of? Make sure to justify your answer.
Hint: Distinguish between the cases $p=2$ and $p>2$.

Problem 2

In both parts of this problem $R$ is a commutative domain with 1 and $K$ is the field of fractions of $R$.
(a) ( 5 pts) Let $R=\mathbb{Z}[t]$, the ring of polynomials over $\mathbb{Z}$ in one variable. Let $p(x)=x^{n}+r_{n-1} x^{n-1}+\ldots+r_{0} \in R[x]$ be a monic polynomial with coefficients in $R$, and suppose that $p(\alpha)=0$ for some $\alpha \in K$. Prove that $\alpha \in R$.
(b) (4 pts) Now let $R=\mathbb{Z}[\sqrt{-3}]$. Find a monic polynomial $p(x) \in R[x]$ which has a root in $K$, but has no root in $R$ (and prove that $p(x)$ has required properties). Hint: There actually exists a quadratic polynomial with integer coefficients with required property.

Problem 3

(6 pts) Let $F$ be a field, $d$ a positive integer, and $f_{1}, f_{2}, \ldots \in F\left[x_{1}, \ldots, x_{d}\right]$ an infinite sequence of polynomials in $F\left[x_{1}, \ldots, x_{d}\right]$. Given a positive integer $n$, let $S_{n}$ be the set of all $d$-tuples $\left(a_{1}, \ldots, a_{d}\right) \in F^{d}$ satisfying the following system of equations:

$$f_{i}\left(a_{1}, \ldots, a_{d}\right)=0 \text { for each } 1 \leq i \leq n-1 \text { and } f_{n}\left(a_{1}, \ldots, a_{d}\right)=1 .$$

Prove that there exists an integer $N$ such that the set $S_{n}$ is empty for all $n \geq N$. Hint: Noetherian rings.

Problem 4

Let $p$ be a prime, $\mathbb{F}_{p}$ a finite field of order $p$, and let $F$ be a fixed algebraic closure of $\mathbb{F}_{p}$. For $n \in \mathbb{N}$, denote by $\mathbb{F}_{p^{n}}$ the unique subfield of order $p^{n}$ inside $F$.
(a) ( 3 pts) Prove that $\mathbb{F}_{p^{n}} \cup \mathbb{F}_{p^{m}}$ is a subfield if and only if $m$ divides $n$ or $n$ divides $m$.
(b) ( 4 pts ) For a subset $S$ of $\mathbb{N}$, let

$$F(S)=\bigcup_{n \in S} \mathbb{F}_{p^{n}}$$

Give an example (with proof) of an infinite set $S$ for which $F(S)$ is a subfield and $F(S) \neq F$.

Problem 5

Let $\omega=e^{2 \pi i / 3}$ and consider the field $K=\mathbb{Q}(\sqrt[3]{2}, \omega)$.
(a) (2 pts) Prove that $[K: \mathbb{Q}]=6$.
(b) ( 2 pts) Prove that $K / \mathbb{Q}$ is a Galois extension.
(c) ( 3 pts) Let $M / L$ be any finite Galois extension. Prove that an element $\gamma \in M$ is primitive for $M / L$ (that is, $L(\gamma)=M$ ) if and only if $\sigma(\gamma) \neq \gamma$ for any $\sigma \in \operatorname{Gal}(M / L) \backslash\{1\}$.
(d) (4 pts) Now prove that $\gamma=\sqrt[3]{2}+\omega$ is a primitive element for $K / \mathbb{Q}$.
(e) ( 3 pts) Let $x^{6}+a_{5} x^{5}+\ldots+a_{0}$ be the minimal polynomial of $\gamma$ over $\mathbb{Q}$. Prove that $a_{5}=3$ without actually computing the minimal polynomial.

Problem 6

Let $F$ be an algebraically closed field and $A \in \operatorname{Mat}_{n}(F)$ an $n \times n$ matrix over $F$ for some $n \geq 2$.
(a) ( 6 pts) Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ (that is, $N^{k}=0$ for some $k \in \mathbb{N}$ ) such that $A=D+N$ and $D$ and $N$ commute, that is, $D N=N D$.
(b) ( 4 pts) Assume that $A$ itself is diagonalizable. Prove that if $D$ and $N$ satisfy the conditions of part (a), then $N=0$ (and hence $D=A$ ). Hint: You may use the following fact without proof: if two diagonalizable matrices $X$ and $Y$ commute, then they are simultaneously diagonalizable, that is, there exists an invertible matrix $Q$ such that $Q^{-1} X Q$ and $Q^{-1} Y Q$ are both diagonal.