Algebra general exam. August 20th 2010, 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.

Problem 1

Let $p$ be prime. Let $G$ be a finite group, $K$ a normal subgroup of $G$, and assume that $|G / K|$ is divisible by $p$. Let $P$ be a Sylow $p$-subgroup of $G$.
(a) (5 pts) Prove that $P K / K$ is a Sylow $p$-subgroup of $G / K$
(b) ( 4 pts ) Prove that $n_{p}(G / K)$ divides $n_{p}(G)$ where $n_{p}(\cdot)$ denotes the number of Sylow $p$-subgroups
(c) (3 pts) Prove that $n_{p}(G / K)=n_{p}(G)$ if and only if $P$ is normal in $P K$.

Problem 2

Let $F$ be a field, $M_{2}(F)$ the ring of $2 \times 2$ matrices over $F$ and $G L_{2}(F)$ the group of invertible elements of $M_{2}(F)$.
(a) ( 4 pts ) Prove that two matrices in $M_{2}(F)$ are similar if and only if they have the same minimal polynomial.
(b) ( 5 pts) Assume that $F$ is finite, and let $q=|F|$. Find the number of conjugacy classes in $G L_{2}(F)$.
(c) ( 6 pts) Again assume that $F$ is finite of order $q$. Find the number of nilpotent matrices in $M_{2}(F)$. Hint: If $A \in M_{2}(F)$ is nilpotent, what is its Jordan normal form? You may use without proof that $\left|G L_{2}(F)\right|=\left(q^{2}-1\right)\left(q^{2}-q\right)$.

Problem 3

Let $K=\mathbb{Q}(\sqrt[3]{3}, \sqrt[5]{5})$, the field obtained from $\mathbb{Q}$ by adjoining $\sqrt[3]{3}$ and $\sqrt[5]{5}$.
(a) (3 pts) Prove that $[K: \mathbb{Q}]=15$.
(b) ( 10 pts) Let $L \subseteq \mathbb{C}$ be the Galois closure of $K$ over $\mathbb{Q}$, that is, $L$ is the minimal Galois extension of $\mathbb{Q}$ which contains $K$. Describe $L$ explcitly in the form $\mathbb{Q}\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{t}\right)$, determine $[L: \mathbb{Q}]$ and describe the elements of the Galois group $\operatorname{Gal}(L / \mathbb{Q})$ by their actions on $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{t}$.
(c) (5 pts) Prove that $K=\mathbb{Q}[\sqrt[3]{3}+\sqrt[5]{5}]$.

Problem 4

Let $K$ be a field. Let $A$ be a finite-dimensional (possibly non-commutative) $K$-algebra (with 1 ), and assume that $A$ is a division ring.
(a) (5 pts) Prove that every $K$-subalgebra of $A$ is a division ring. Hint: Let $B$ be a $K$-subalgebra of $A$, take any $b \in B$, and consider the map $\mu_{b}: B \rightarrow B$ given by $\mu_{b}(x)=b x$.
(b) ( 5 pts ) Assume that $K$ is algebraically closed. Prove that $\operatorname{dim}_{K} A=1$.

Problem 5

Let $G$ be group. A subgroup $H$ of $G$ will be called essential if $H \cap K \neq\{1\}$ for every non-trivial subgroup $K$ of $G$.
(a) (2 pts) Let $p$ be a prime and $k \geq 2$. Prove that the group $\mathbb{Z} / p^{k} \mathbb{Z}$ has a proper essential subgroup.
(b) ( 7 pts) Assume that $H_{1}$ is an essential subgroup of $G_{1}$ and $H_{2}$ is an essential subgroup of $G_{2}$. Prove that $H_{1} \times H_{2}$ is an essential subgroup of $G_{1} \times G_{2}$.
(c) ( 6 pts) Let $G$ be a finite abelian group. Prove that $G$ does not have a proper essential subgroup if and only if $G$ is a direct product of groups of prime order.

Problem 6

Let $\mathbb{R}$ denote the real numbers. The purpose of this problem is to show that the ring $A=\mathbb{R}[x, y] /\left(x^{2}+y^{2}-1\right)$ is not a UFD. For an element $f \in \mathbb{R}[x, y]$ we denote its image in $A$ by $[f]$.
(a) ( 2 pts) Show that every element of $A$ can be uniquely represented in the form $[f(x)+g(x) y]$ where $f(x), g(x) \in F[x]$.
(b) ( 2 pts) Show that $A$ has an automorphism $\varphi$ of order 2 such that $\varphi([f(x)])=[f(x)]$ for each $f(x) \in F[x]$ and $\varphi([y])=-[y]$.
(c) ( 3 pts) Use (a) and (b) to construct a function $N: A \rightarrow F[x]$ such that $N(u v)=N(u) N(v)$ for all $u, v \in A$.
(d) ( 6 pts) Use the function $N$ from (c) to show that $[x]$ is an irreducible element of $A$ and that the only invertible elements of $A$ are (images of) nonzero constant polynomials. Hint: It is essential that you are working over $\mathbb{R}$, not over $\mathbb{C}$.
(e) ( 4 pts ) Now show that $A$ is not a UFD.

Problem 7

Recall that a ring $S$ is called graded if $S=\oplus_{n=0}^{\infty} S_{n}$ where each $S_{n}$ is an additive subgroup and $S_{n} \cdot S_{m} \subseteq S_{n+m}$ for all $n, m$. An element $s \in S$ is called homogeneous if $s \in S_{n}$ for some $n$. An ideal $I$ of $S$ is called a graded ideal if $I=\oplus_{n=0}^{\infty} I \cap S_{n}$.
(a) ( 5 pts) Let $S$ be a graded ring and $I$ an ideal of $S$. Prove that $I$ is a graded ideal if and only if $I$ is generated (as an ideal) by a set of homogeneous elements.
(b) ( 8 pts) Let $R$ be a commutative ring with 1 and $S=R[x]$. Then $S$ is naturally a graded ring where $S_{n}=\left\{r x^{n}: r \in R\right\}$. Assume that there exists $k \in \mathbb{N}$ such that every ideal of $R$ can be generated by at most $k$ elements.
Let $I$ be a graded ideal of $S$. Prove that $I$ can be written as

$$I=J \oplus M$$

where $J$ is an ideal of $S$ generated by at most $k$ elements and $M$ is a finitely generated $R$-module. Hint: Adapt the proof of Hilbert’s basis theorem.