Algebra General Exam

August 19, 2013

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 viewed favorable

• You may assume the statement for any early part of a problem in order to do a later part

Do each problem on a separate sheet of paper

Problem 1

Let $p$ be an odd prime and $G$ a nonabelian group of order $p^{3}$.
(a) (4 points) Prove that $|Z(G)|=p$
(b) (4 points) Prove that $Z(G)=[G, G]$.

Problem 2

( 5 points) Let $K$ and $L$ be fields of characteristic 0 . Prove that $K \bigotimes_{\mathbb{Z}} L$ is nonzero.

Problem 3

If $G$ is a group, then there is a natural action of $\Sigma_{n}$ on $G^{\times n}$ given by permuting the factors. Define the wreath product $G \geq \Sigma_{n}$ to be

$$G \succ \Sigma_{n}=G^{n} \rtimes \Sigma_{n}$$

using this action of $\Sigma_{n}$ on $G^{n}$.
(a) ( 3 points) If $X$ is a $G$-set, show that $X^{n}$ is naturally a $G \succ \Sigma_{n}$ set by combining two actions: $G^{n}$ on $X^{n}$ via

$$\left(g_{1}, \ldots, g_{n}\right) \cdot\left(x_{1}, \ldots, x_{n}\right)=\left(g_{1} x_{1}, \ldots, g_{n} x_{n}\right)$$

for $\left(g_{1}, \ldots, g_{n}\right) \in G^{n}$ and $\left(x_{1}, \ldots, x_{n}\right) \in X^{n}$, and $\Sigma_{n}$ on $X^{n}$ via

$$\sigma \cdot\left(x_{1}, \ldots, x_{n}\right)=\left(x_{\sigma^{-1}(1)}, \ldots, x_{\sigma^{-1}(n)}\right)$$

where $\sigma \in \Sigma_{n}$.
(b) ( 3 points) Show that $\Sigma_{n} \prec \Sigma_{m}$ embeds into $\Sigma_{n m}$.
(c) ( 3 points) Identify $\Sigma_{2} l \Sigma_{2}$ with a more familiar group
(d) ( 2 points) Determine the order of $G$ c $\Sigma_{n}$ as a function of the orders of $G$ and $n$
(e) ( 2 points) Bonus: Determine (no proof needed) the $p$-Sylow subgroup of $\Sigma_{p^{k+1}}$ as a function of $k$. Provide no more than a sentence of justification.

Problem 4

Let $K$ be a field, and let $M_{n}(K)$ be the ring of $n \times n$ matrices with entries in $K$. For this problem, let $D \in M_{n}(K)$ be diagonalizable (over $K$ ) and, for each eigenvalue $\lambda$ of $D$, let

$$E_{\lambda}:=\left\{v \in K^{n} \mid D v=\lambda v\right\}$$

be the corresponding eigenspace.
(a) (4 points) For any $A \in M_{n}(K)$, show that $A D=D A$ if and only if $A\left(E_{\lambda}\right) \subseteq E_{\lambda}$ for all eigenvalues $\lambda$ of $D$.
(Hint: For the "if" part, you may use that $A D=D A$ if $A D v=D A v$ for all $v \in K^{n}$.)
(b) (4 points) If $A$ is also diagonalizable and $A D=D A$, show that $A$ and $D$ are simultaneously diagonalizable (that is, there is a matrix $P$ such that both $P A P^{-1}$ and $P D P^{-1}$ are diagonal). Provide a counter-example showing that this need not be the case if the matrices do not commute.
(c) (3 points) If $D$ is invertible, show that the centralizer of $D$ in $G L_{n}(K)$ is isomorphic to a direct product $G L_{n_{1}}(K) \times \ldots \times G L_{n_{r}}(K)$, where $n_{1}+\ldots+n_{r}=n$. Also show that each of these products can be realized as the centralizer of some (appropriately chosen) $D$, provided that $K$ has at least $n+1$ elements.

Problem 5

Let $F$ be a field and $f(x)=x^{4}+1 \in F[x]$.
(a) ( 3 points) Determine for which characteristic of $F f(x)$ is separable.
(b) ( 4 points) Assume that $f(x)$ is separable and irreducible over $F$, and denote by $K$ the splitting field of $f(x)$ over $F$. Determine the Galois group $\operatorname{Gal}(K \mid F)$.
(c) (4 points) If $f(x)$ is irreducible over $F$, prove first that $F$ is infinite, and then that the characteristic of $F$ is 0 .

Problem 6

Let $p$ be a prime and $\zeta$ a primitive $p^{\text {th }}$ root of unity (in $\mathbb{C}$ ). Set $R:=\mathbb{Z}[\zeta]$ and $K:=\mathbb{Q}(\zeta)$.
(a) ( 2 points) Show that $R$ is a free $\mathbb{Z}$-module and $R \cap \mathbb{Q}=\mathbb{Z}$.
(b) (2 points) Identify $\operatorname{Gal}(K \mid \mathbb{Q})$ and show that the natural action of $\operatorname{Gal}(K \mid \mathbb{Q})$ on $K$ sends elements of $R$ to itself (hence giving an action of $\operatorname{Gal}(K \mid \mathbb{Q})$ on $R$ ).
(c) ( 3 points) For any two integers $m, n$ which are not divisible by $p$, show that the quotient $\left(1-\zeta^{m}\right) /\left(1-\zeta^{n}\right)$ is an element of $R$.
Hint: Reduce to the case where $n$ divides $m$.
(d) ( 2 points) Verify that $p=(1-\zeta) \ldots\left(1-\zeta^{p-1}\right)$.

Hint: manipulate the cyclotomic polynomial associated to $\zeta$.
(e) ( 3 points) Prove that $1-\zeta$ is not a unit of $R$.
(f) (2 points) Prove (using norms) that $1-\zeta$ is an irreducible element of $R$. (It is true, but harder to prove, that $1-\zeta$ is in fact a prime element of $R$.)