Algebra general exam. January 13th 2012, 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

1. Let $F=\mathbb{F}_{q}$ be a finite field, where $q=p^{r}$ is a power of a prime $p$. Let $G=G L_{n}(F)$ be the group of all $n \times n$ invertible matrices with entries in $F$. Once you pick an ordered basis of $V:=F^{n}$, you may find it useful to identify $G$ with the group of invertible linear operators on $V$.
   (a) ( 6 pts) Calculate the order of $G$. Explain your answer carefully and write it in the simplest form as you can.
   (b) ( 3 pts) Determine the order of a Sylow $p$-subgroup of $G$, and explicitly exhibit a Sylow $p$-subgroup $U$ of $G$.
   (c) ( 2 pts) What is the normalizer in $G$ of the Sylow $p$-subgroup $U$ that you exhibited in (b)? An answer is sufficient.
   (d) ( 3 pts) How many Sylow $p$-subgroups of $G$ are there? Explain how your answer in (d) is consistent with Sylow’s theorem.

Problem 2

2. Let $G$ be a subgroup of the symmetric group $S_{n}$ for some integer $n>1$. Assume that $G$ acts transitively on $\mathbf{n}:=\{1,2, \cdots, n\}$, that is, for any $i, j \in \mathbf{n}$ there exists $g \in G$ s.t. $g(i)=j$.

A partition of $\mathbf{n}$ is a decomposition $\mathbf{n}=X_{1} \cup \cdots \cup X_{m}$ into a disjoint union of nonempty subsets. There are two trivial partitions: $\mathbf{n}=\mathbf{n}$ and $\mathbf{n}=X_{1} \cup \cdots \cup X_{n}$ (so each $X_{i}$ has just one element). Otherwise the partition is said to be nontrivial. The group $G$ is called imprimitive if there is a nontrivial partition $\mathbf{n}=X_{1} \cup \cdots \cup X_{m}$ such that, for $g \in G$ and $1 \leq i \leq m$, $g\left(X_{i}\right)=X_{j}$ for some $j$. (That is, $G$ permutes the partition members among themselves.) The set $\left\{X_{i}\right\}$ is called a system of imprimitivity for the action of $G$ on $\mathbf{n}$. The group $G$ is called primitive if it is not imprimitive.
(a) ( 3 pts) Let $n=6$ and consider the cyclic subgroup $G:=\langle(1,2,3,4,5,6)\rangle$ of $S_{6}$. There are two non-trivial systems of imprimitivity for the action of $G$ on $\mathbf{n}$. Find them.
(b) ( 3 pts) Prove that if $X_{1} \cup \cdots \cup X_{m}$ is a system of imprimitivity for the action of $G$ on $\mathbf{n}$, then all subsets $X_{i}$ have the same size $n / m$.
(c) ( 4 pts) $G$ is said to be doubly transitive if given elements $a, b, c, d \in \mathbf{n}$, with $a \neq b$ and $c \neq d$, there exists $g \in G$ such that $g(a)=c$ and $g(b)=d$. Show that a doubly transitive group $G$ is primitive.
(d) ( 4 pts ) Show that if $n \geq 3$, the alternating subgroup $G=A_{n}$ of $S_{n}$ is primitive.

Problem 3

Let $R=\mathbb{Z}[\sqrt{-2}]$.
(a) ( 7 pts) Prove that $R$ is a Euclidean domain. Hint: Use the square of the usual complex norm.
(b) ( 8 pts ) Write 7 and 11 as products of irreducible elements of $R$. Justify your answer.

Problem 4

Let $R$ be a ring with 1 . The opposite ring $R^{o p}$ is defined as follows: as a set $R^{o p}=R$, the addition on $R^{o p}$ coincides with the addition on $R$ and the multiplication * on $R^{o p}$ is the multiplication on $R$ in reverse order, that is, $a * b=b a$ (where $b a$ is the product in $R$ ). Let $e \in R$ be an idempotent element, that is, $e^{2}=e$.
(a) (2 pts) Prove that $e R e=\{$ ere $: r \in R\}$ is a subring of $R$.
(b) ( 6 pts) Consider the left $R$-module $M=R e$. Prove that its endomorphism ring $E n d_{R}(M)=H o m_{R}(M, M)$ is isomorphic to $(e R e)^{o p}$, the opposite ring of $e R e$.

Problem 5

(9 pts) Let $F$ be a field, $n$ a positive integer and $M_{n}(F)$ the set of $n \times n$ matrices over $F$. Let $A \in \operatorname{Mat}_{n}(F)$ be such that $A^{2}=A$. Prove that $A$ is diagonalizable and classify all such $A$ up to similarity. (Recall that $A, B \in \operatorname{Mat}_{n}(F)$ are similar if there exists $C \in G L_{n}(F)$ s.t. $C^{-1} A C=B$.)

Problem 6

Let $R$ be a commutative ring with 1 . Recall that a left $R$-module $M$ is called Noetherian if it satisfies the ascending chain condition on submodules and Artinian if it satisfies the descending chain condition on submodules. Assume that an $R$-module $M$ is both Artinian and Noetherian. (For example, $R$ might be a field, and $M$ might be a finite-dimensional vector space over $R$ ). Let $T: M \rightarrow M$ be an $R$-module homomorphism.
(a) ( 3 pts) Prove that there exists $k \in \mathbb{N}$ s.t. $\operatorname{Ker}\left(T^{k}\right)=\operatorname{Ker}\left(T^{2 k}\right)$ and $\operatorname{Im}\left(T^{k}\right)=\operatorname{Im}\left(T^{2 k}\right)$.
(b) ( 4 pts) Prove that if $k$ is as in part (a), then $M=\operatorname{Ker}\left(T^{k}\right) \oplus \operatorname{Im}\left(T^{k}\right)$
(c) ( 2 pts) Deduce from (a) and (b) that there exist submodules $M_{0}$ and $M_{1}$ of $M$ s.t. $M=M_{0} \oplus M_{1}, T_{\mid M_{0}}$ is nilpotent and $T_{\mid M_{1}}$ is invertible (as a map from $M_{1}$ to $M_{1}$ ).
(d) ( 5 pts) Now assume that $R$ is a field of characteristic zero, $M$ is a finite-dimensional vector space over $R$ and $\operatorname{tr}\left(T^{n}\right)=0$ for every $n \in \mathbb{Z}_{>0}$. Prove that $T$ is nilpotent. Hint: Apply (c), assume that $M_{1} \neq 0$ and reach a contradiction by applying the Cayley-Hamilton theorem to $T_{\mid M_{1}}$.

Problem 7

If $q$ is a prime power, denote by $\mathbb{F}_{q}$ a finite field of order $q$.
(a) ( 6 pts) Find a monic irreducible polynomial of degree 3 over $\mathbb{F}_{5}$ and use it to construct a field of order 125 . Justify your answer.
(b) ( 6 pts) Find all $q$ for which the polynomial $p(x)=x^{2}+x+1$ is irreducible in $\mathbb{F}_{q}[x]$. Hint: What can you say about roots of $p(x)$ and what do you know about the multiplicative group $\mathbb{F}_{q}^{\times}$?

Problem 8

Let $F$ be a field of characteristic zero, let $K$ and $L$ be finite extensions of $F$ and $K L$ the compositum of $K$ and $L$.
(a) (4 pts) Prove that $[K L: F] \leq[K: F] \cdot[L: F]$.
(b) (2 pts) Assume that $[K: F]$ and $[L: F]$ are relatively prime. Prove that $[K L: F]=[K: F][L: F]$.
(c) (4 pts) Give an example where $K \cap L=F$ but $[K L: F] \neq[K$ : $F][L: F]$.
(d) (4 pts) Assume that $K / F$ and $L / F$ are both Galois. Prove that $\operatorname{Gal}(K L / F)$ is isomorphic to a subgroup of $\operatorname{Gal}(K / F) \times \operatorname{Gal}(L / F)$. (You need not prove that $K L / F$ is Galois).
Note: The assertions of (a),(b) and (d) remain valid for $F$ of positive characteristic, but part (a) has shorter proof in the case of characteristic zero.