Algebra general exam. August 17th 2009, 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.

1. Problem 1

   Let $G$ be a group of order 56 which does NOT have a normal subgroup of order 8 .
   (a) ( 8 pts) Prove that $G$ has a normal subgroup of order 7 .
   (b) ( 4 pts ) Prove that $G$ has a subgroup of order 14.
   (c) ( 4 pts) Prove that $G$ has a normal subgroup of order 14.

Remark: Of course, you may omit (b) if you correctly answered (c).

Problem 2

Let $G$ be a finite group and let $H$ and $K$ be subgroups of $G$. For each $x \in G$ define $H x K=\{h x k: h \in H, k \in K\}$.
(a) (3 pts) Prove that for any $x, y \in G$ either $H x K=H y K$ or $H x K \cap H y K=\emptyset$.
(b) (8 pts) Prove that $|H x K|=\frac{|H||K|}{\left|H \cap x K x^{-1}\right|}$. Hint: Use group actions: either a suitable action of $H \times K$ on $G$ or a suitable action of $H$ on $G / K$.

Problem 3

(a) ( 6 pts) Let $R$ be a principal ideal domain and $I \subset R$ a proper nonzero ideal. Prove that if the quotient ring $R / I$ is a domain, then it must be a field.
(b) ( 4 pts) Does the assertion of (a) remain true if $R$ is only assumed to be a unique factorization domain? Prove or a give a counterexample.

Problem 4

Let $F$ be a field and $R=F[x, y]$ the ring of polynomials over $R$ in two (commuting) variables $x$ and $y$. Let $I=x R$ be the principal ideal of $R$ generated by $x$ and $S=F+I=\{f+i: f \in F, i \in I\}$. Observe that $S$ is a subring of $R$ and $I$ is an ideal of $S$ (you need not justify these facts).
(a) ( 7 pts) Prove that $I$ is not finitely generated as an ideal of $S$.

Hint: Assume that $I$ is finitely generated as an ideal of $S$ and reach a contradiction by showing that there must exist a natural number $m$ such that any polynomial $p(x, y) \in I$ contains no monomials of the form $x y^{n}$, with $n>m$.
(b) ( 5 pts) Prove that $S$ is not finitely generated as a ring.

Hint: It is possible to answer (b) using (a) without doing any computations.

Problem 5

(a) (8 pts) Let $A=\left(\begin{array}{lll}2 & 1 & 3 \\ 1 & 2 & 4 \\ 0 & 0 & 1\end{array}\right) \in M a t_{3}(\mathbb{C})$. Find the minimal polynomial, the characteristic polynomial and the Jordan canonical form of $A$.
(b) ( 7 pts) Let $J_{n}(0) \in \operatorname{Mat}_{n}(\mathbb{C})$ be the Jordan block of size $n$ with 0 ’s on the diagonal. Prove that there exists no matrix $A \in M a t_{n}(\mathbb{C})$ such that $A^{2}=J_{n}(0)$.

Problem 6

Let $K / F$ be a finite extension of fields, and let $\alpha, \beta \in K$ be such that $K=F(\alpha, \beta)$. Let $n=[F(\alpha): F]$ and $m=[F(\beta): F]$, and assume that $n$ and $m$ are relatively prime.
(a) (4 pts) Prove that $[K: F]=n m$.
(b) ( 6 pts) Assume that $K / F$ is Galois. Let $\mu_{\alpha, F}(x)$ and $\mu_{\beta, F}(x)$ be the minimal polynomials of $\alpha$ and $\beta$ over $F$, respectively. Let $\alpha^{\prime} \in K$ be a root of $\mu_{\alpha, F}(x)$ and let $\beta^{\prime} \in K$ be a root of $\mu_{\beta, F}(x)$. Prove that there exists unique $\sigma \in \operatorname{Gal}(K / F)$ such that $\sigma(\alpha)=\alpha^{\prime}$ and $\sigma(\beta)=\beta^{\prime}$.
(c) ( 6 pts) Again assume that $K / F$ is Galois. Let $S$ be the set of all elementns $c \in F$ such that $F(\alpha+c \beta) \neq K$. Prove that $|S| \leq n m$.

Problem 7

Let $F$ be a field and $K$ a finite-dimensional vector space over $F$. Let $n=\operatorname{dim}_{F} K$, and assume that $n>1$.
(a) ( 6 pts) Is it always true that $K \otimes_{F} K \cong \operatorname{Mat}_{n}(F)$ as $F$-modules?
(b) ( 6 pts) Now assume that $K$ also has the structure of a commutative ring with 1 , so being an $F$-vector space, $K$ becomes an $F$-algebra. Recall that in this case $K \otimes_{F} K$ possesses unique $F$-algebra structure such that $(a \otimes b) \cdot(c \otimes d)=a c \otimes b d$ for $a, b, c, d \in K$. Prove that $K \otimes_{F} K$ cannot be a field.
Hint: Construct a non-trivial $F$-algebra homomorphism $K \otimes_{F} K \rightarrow K$.

Problem 8

( 8 pts) Let $F$ be a field. Prove that the additive and multiplicative groups of $F$ cannot be isomorphic. Hint: Look at the orders of elements in both groups.