Algebra general exam. August 22nd 2012, 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

For a positive integer $n$ , denote by $S_{n}$ the symmetric group on $\{1,2, \ldots, n\}$ . Let $p>2$ be a prime number.
(a) ( 2 pts) Give an example of a non-cyclic group of order $2 p$ .
(b) ( 5 pts) Find the smallest $n$ for which $S_{n}$ contains a cyclic subgroup of order $2 p$ .
(c) ( 7 pts) Find the smallest $n$ for which $S_{n}$ contains some subgroup of order $2 p$ .
In both (b) and (c), if $n$ is your answer, explain clearly why $S_{n}$ contains a desired subgroup and why $S_{m}$ for $m<n$ does not contain such subgroup.

Problem 2

(8 pts) Let $G$ be a finite group and $p$ a prime divisor of $|G|$ . Assume that every element of $G$ of $p$ -power order is contained in a normal $p$ -subgroup of $G$ . Show that $G$ has only one Sylow $p$ -subgroup.

Problem 3

Let $k$ be a field and $R=k[x, y] /\left(x^{5}-y^{2}\right)$ .
(a) ( 5 pts) Prove that $R$ is isomorphic to the subring $k\left[t^{2}, t^{5}\right]$ of $k[t]$ (the polynomials in one variable over $k$ ).
(b) ( 8 pts) Prove that $R$ is not isomorphic to $k[t]$ (as a ring).

Problem 4

( 8 pts) Let $R$ be a commutative ring with 1 . Let $N$ be the nilradical of $R$ , that is, $N$ is the set of all nilpotent elements of $R$ (including 0 ). You may use without proof that $N$ is the intersection of all prime ideals of $R$ . Prove that the following conditions are equivalent:
(i) $R$ has just one prime ideal.
(ii) $R / N$ is a field.

Problem 5

Recall that if $R$ is a commutative ring with 1 and $A$ and $B$ are $R$ -algebras, then $A \otimes_{R} B$ also has the natural structure of an $R$ -algebra.
(a) ( 3 pts) Let $K$ and $L$ be fields of different characteristics. Prove that $K \otimes_{\mathbb{Z}} L=\{0\}$ .
(b) ( 6 pts) Let $K$ and $L$ be fields of the same positive characteristic $p$ . Prove that $K \otimes_{\mathbb{Z}} L$ can be provided in a natural way with the structure of an $\mathbb{F}_{p}$ -algebra, and that this $\mathbb{F}_{p}$ -algebra is isomorphic to $K \otimes_{\mathbb{F}_{p}} L$ . Deduce that $K \otimes_{\mathbb{Z}} L$ is nonzero.
(c) (5 pts) Find an example of commutative rings $A$ and $B$ which are NOT fields such that $A \otimes_{\mathbb{Z}} B$ is a field. Hint: Use a suitable property of tensor products involving direct sums.

Problem 6

Let $p$ be an odd prime number and $n$ an integer $\geq 2$ .
(a) ( 5 pts) Show that $G L_{n}(\mathbb{Q})$ has an element of order $p$ if and only if $n \geq p-1$ .
(b) ( 5 pts) Show that there is an $A \in G L_{n}(\mathbb{Q})$ of order $p$ which does NOT have 1 as an eigenvalue if and only if $p-1$ divides $n$ .
(c) ( 5 pts) Let $A \in G L_{4}(\mathbb{Q})$ be an element of order 5 . Prove that the complex JCF of $A$ is independent of such $A$ (up to permutation of blocks) and write it down.

Problem 7

Let $K / F$ be a field extension, let $\alpha, \beta \in K \backslash F$ be algebraic over $F$ , and let $p=\operatorname{deg}_{F}(\alpha)$ and $q=\operatorname{deg}_{F}(\beta)$ . Suppose that $p$ and $q$ are distinct primes and that $p>q$ .
(a) (2 pts) Prove that $[F(\alpha, \beta): F]=p q$ .
(b) ( 6 pts ) Prove that $\operatorname{deg}_{F}(\alpha \beta)=p$ or $p q$ .
(c) ( 4 pts) Give an example showing that it MAY happen that $\operatorname{deg}_{F}(\alpha \beta)= p$ .

Problem 8

In this problem you may use the following fact without proof: for any group $G$ there exists a Galois extension $M / L$ with $\operatorname{Gal}(M / L) \cong G$ .
(a) ( 8 pts) Prove that there exists a field extension $K / F$ such that $[K$ : $F]=4$ and there are no intermediate fields between $F$ and $K$ other than $F$ and $K$ . Hint: First reduce the question to a purely grouptheoretic problem. Partial credit will be given for such reduction.
(b) ( 3 pts) Is it possible to construct an extension satisfying (a) if $F$ is finite? Justify your answer.
(c) ( 5 pts) Is it possible to construct an extension satisfying (a) if $K$ is contained in a cyclotomic field $\mathbb{Q}\left(\zeta_{n}\right)$ for some $n$ (where $\zeta_{n}$ is a primitive $n^{\text {th }}$ root of unity)? Justify your answer.