Algebra general exam. January 18th 2011, 9am-2pm

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.

1. Problem 1

   Let $G$ be a group of order 30 .
   (a) ( 8 pts) Prove that $G$ has a normal subgroup of order 3 or a normal subgroup of order 5.
   (b) ( 6 pts) Prove that $G$ has a normal subgroup of order 15 .

2. Problem 2

   Let $p$ be a prime and $G$ a finite abelian group of $p$-power order. Denote by

• $e(G)$ the number of elements of order $p$ in $G$

• $s(G)$ the number of subgroups of order $p$ in $G$

• $t(G)$ the number of subgroups of index $p$ in $G$

• $h(G)$ the number of non-trivial homomorphisms from $G$ to $\mathbb{Z} / p \mathbb{Z}$
  (a) (7 pts) Prove that $s(G)=e(G) /(p-1)$ and $t(G)=h(G) /(p-1)$
  (b) ( 7 pts) Prove that $s(G)=t(G)$. Hint: By (a) this is equivalent to showing that $h(G)=e(G)$. Prove the latter by expressing both quantities in terms of $r$, the number of summands in the elementary divisor decomposition of $G$.

3. Problem 3

   Prove that the following polynomials are irreducible in $\mathbb{Q}[x]$ :
   (a) $(5 \mathrm{pts}) f(x)=x^{3}-3 x^{2}+15 x-7$
   (b) ( 7 pts) $f(x)=x^{5}-3 x^{2}+15 x-7$

Hint for (b): Use reduction mod 3.

Problem 4

Let $R$ be a commutative ring with 1 , and let $\Omega$ be the set of all ideals $I$ of $R$ such that every element of $I$ is 0 or a zero divisor.
(a) ( 5 pts) Prove that $\Omega$ has a maximal element (with respect to inclusion) and moreover any element of $\Omega$ is contained in a maximal element.
(b) ( 8 pts) Let $I$ be a maximal element of $\Omega$. Prove that $I$ must be prime.

Problem 5

For a field $F$ and a positive integer $n \geq 2$ denote by $M_{n}(F)$ the set of all $n \times n$ matrices over $F$.
(a) ( 8 pts) Assume that $F$ is algebraically closed of characteristic NOT equal to 2 . Prove that for every invertible matrix $A \in M_{n}(F)$ there exists $B \in M_{n}(F)$ such that $B^{2}=A$.
(b) ( 8 pts) Let $F$ be a field of characteristic 2 . Find an invertible matrix $A \in M_{n}(F)$ which cannot be written as $B^{2}$ for any $B \in M_{n}(F)$.
Hint (for both parts): Use the Jordan Canonical Form.

Problem 6

Let $R$ be a commutative ring with 1 . Let $M, N$ and $P$ be $R$-modules, and let $\phi: M \rightarrow N$ be an $R$-module homomorphism.
(a) ( 4 pts) Prove that there exists unique $R$-module homomorphism $\Phi$ : $M \otimes_{R} P \rightarrow N \otimes_{R} P$ such that $\Phi(m \otimes p)=\phi(m) \otimes p$ for all $m \in M$ and $p \in P$.
(b) ( 3 pts) Assume that $\phi$ is surjective. Prove that $\Phi$ from part (b) is also surjective.
(c) ( 4 pts) Show by example that if $\phi$ is injective, $\Phi$ need not be injective.

Problem 7

Let $p$ be a prime, let $\zeta_{p} \in \mathbb{C}$ be a primitive $p^{\text {th }}$ root of unity and $K=\mathbb{Q}\left(\zeta_{p}\right)$.
(a) (7 pts) Prove that the extension $K / \mathbb{Q}$ is Galois and $\operatorname{Gal}(K / \mathbb{Q})$ is cyclic of order $p-1$.
(b) ( 5 pts) Prove that $K$ contains a unique subfield $L$ of the form $\mathbb{Q}(\sqrt{m})$ where $m \in \mathbb{Z}$ and $m$ is not a perfect square.
In parts (c)-(e) of this problem let $p=7$, and let $L$ denote the subfield found in part (b).
(c) (2 pts) Find (explicitly) a generator for the group $\operatorname{Gal}(K / L)$.
(d) ( 3 pts) Find an element $\alpha \in L \backslash \mathbb{Q}$. You may express $\alpha$ as a polynomial in $\zeta_{7}$. Hint: use (c).
(e) ( 3 pts) Find $m$ from part (b) explicitly (it is well defined up to multiplication by a perfect square).