General Exam: Algebra

August 15, 2011
Instructions. You have 4 hours for the exam. Each part of each question is worth 4 points. Do each question on a separate sheet of paper (one side only please), and staple the sheets together in the correct order.

On this exam, all rings $R$ have an identity 1 by fiat.

Problem 1

1. Let $M$ be a simple (left) module for a ring $R$. This means that $M$ has no submodules apart from 0 and $M$.
   (a) Prove that $M \cong R / I$ where $I$ is a maximal left ideal of $R$..
   (b) Show that $E:=\operatorname{End}_{R}(M)$ is a division ring, i. e., every nonzero element of $E$ is invertible.

Problem 2

2. The following question concerns symmetric groups. You can assume as given the fact that any permutation in $S_{n}$ can be written (uniquely up to order) as a (commuting) product of disjoint cycles (of varying lengths). Otherwise, your argument should be self-contained.
   (a) For $n \geq 2$, show that the symmetric group $S_{n}$ is generated by the transpositions $(i, j), 1 \leq i< j \leq n$.
   (b) For $n \geq 3$, show that the alternating group $A_{n}$ is generated by the 3-cycles $(1,2, i), 2<i \leq 3$.
   (c) Let $H$ be a subgroup of a group $G$ of index $n$. Show that $G$ has a normal subgroup $N$ which is contained in $H$ and which has index $\leq n!$.

Problem 3

3. (a) Let $V$ be an noetherian module for a ring $R$, so that $V$ satisfies the ascending chain condition on submodules. Let $T: V \rightarrow V$ be a surjective $R$-endomorphism. Prove that $T$ is an isomorphism.
   (b) In (a) suppose that $R$ is a field, so $V$ is a vector space. Give another explanation of (a) in terms of the rank and nullity of $T$.

Problem 4

4. (a) Find all the irreducible polynomials of degree 4 over the finite field $\mathbb{F}_{2}$.
   (b) Let $K / F$ be a finite extension of finite fields. Show the norm map $N_{K / F}: K \rightarrow F$ is surjective.

Problem 5

5. This problem tests some standard linear algebra facts. You can quote standard theorems.

Let $F$ be a field and let $T: F^{6} \rightarrow F^{6}$ be a linear operator with characteristic polynomial

$$\chi_{T}(t)=\left(t^{2}+t+1\right)\left(t^{2}-1\right) t^{2}$$

(a) If $F=\mathbb{R}$, what are the various possibilities for the minimal polynomial $\mu_{T}(t)$ of $T$ ?
(b) Fill in the blank:

$$\operatorname{det}(T)=\ldots ; \quad \operatorname{trace}(T)=$$

Be careful about signs!
(c) Write down the companion matrix $C$ of the polynomial $\chi_{T}(t)$. Calculate the minimal polynomial of $C$.
(d) When $F=\mathbb{C}$, when is $T$ diagonalizable (i. e., represented by a diagonal matrix w.r.t. some basis)? Some explanation in terms of $\chi_{T}(t)$ or $\mu_{T}(t)$ is required.
(e) When $F=\mathbb{R}$, when (if ever) is $T$ represented by a symmetric matrix? Why?
(f) Bonus ( +4 points): Let $S: \mathbb{C}^{m} \rightarrow \mathbb{C}^{m}$ be a nilpotent operator which is represented by a matrix in Jordan normal form having blocks of sizes $\lambda_{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{m} \geq 0$. Let $\lambda^{\prime}=\left(\lambda_{1}^{\prime}, \cdots, \lambda_{m}^{\prime}\right)$ be the partition of $m$ dual (or transpose) to the partition $\lambda=\left(\lambda_{1}, \cdots, \lambda_{m}\right)$ of $m$. What is the significance (in terms of $T$ ) of the integers $\lambda_{i}^{\prime}, i=1, \cdots, m$ ? (Note: your answer should be precisely one [short] sentence! No further explanation is wanted.)

Problem 6

(a) Suppose that $G=C_{p} \times \cdots C_{p}$ is a direct product of $n$ copies of the cyclic group $C_{p}$ of order $p$. How many subgroups does $G$ have of order $p$ ? How many does it have of order $p^{n-1}$ ? Explain.
(b) Now let $p_{1}, \cdots, p_{r}$ be distinct prime integers $>0$. Show that $F:=\mathbb{Q}\left[\sqrt{p_{1}}, \cdots, \sqrt{p_{r}}\right]$ is an abelian Galois extension of $\mathbb{Q}$.
(c) Suppose that $a=p_{i_{1}} \cdots p_{i_{m}}$ (with distinct factors) is a nontrivial product of some of the primes $p_{1}, \cdots, p_{r}$. Let $b \neq a$ be another such element. Show that $\mathbb{Q}[\sqrt{a}] \neq \mathbb{Q}[\sqrt{b}]$. Now use (a) to determine precisely the Galois group of $F / \mathbb{Q}$. Carefully justify your answer.
(d) Show that the numbers $\sqrt{p_{1}}, \cdots, \sqrt{p_{r}}$ are linearly independent over $\mathbb{Q}$, and that $\sqrt{p_{1}}+\cdots+\sqrt{p_{r}}$ is a primitive element of $F / \mathbb{Q}$.

Problem 7

(a) Let $G$ be a finite simple group of order 168. How many elements of order 7 does $G$ have? Why?
(b) How many conjugacy classes of elements of order 7 does $G$ have? Hint: By looking at Sylow 3-subgroups, show that $G$ has no cyclic subgroup of order 21. Use this to determine the centralizer in $G$ of an element of order $7 . \ldots$.
(c) Assume that you know that $G:=G L_{3}\left(\mathbb{F}_{2}\right)$ is a simple group. Explicitly exhibit two elements of $G$ of order 7 which are not conjugate in $G$. Explain.

Problem 8

Let $K$ be a field and let $A, B$ be commutative $K$-algebras. We do not assume $A$ or $B$ is finite dimensional over $K$.
(a) If the $K$-algebra $A \otimes_{K} B$ is a field, show that $A$ and $B$ must be fields, too. (Partial credit is given if you have to assume that $A$ or $B$ is finite dimensional over $K$.)
(b) Provide an example of two field extensions $A$ and $B$ of degree 2 over $K$ such that $A \otimes_{K} B$ is a field of degree 4 over $K$.
(c) Compute $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}$ explicitly.
(d) Suppose $K$ has characteristic $p>0$ and that $A / K$ is a field extension such that there exists $\xi \in A$ such that $\xi \notin K$, but $\xi^{p} \in K$. Prove $A \otimes_{K} A$ is not a field.