Algebra general exam

August 15, 2017

Your name:

Please show all your work and justify any statements that you make.
State any theorem you use clearly and fully.
Vague statements and hand-waving arguments will not be appreciated.
You may assume the statement of an earlier question proven in order to solve a later one.

Sign below the pledge:
"On my honor, I pledge that I have neither given nor received help on this assignment."

( 10 points) Denote by $D_{2 n}$ the dihedral group of order $2 n, n=1$ being admitted. For which natural numbers $m$ and $n$ is $D_{4 n m}$ isomorphic to the direct product $D_{2 m} \times D_{2 n}$ ?

(10 points) Let $G$ be a group of order $16 \cdot 11 \cdot 13 \cdot 17$ . Assume that $G$ has a normal nonabelian Sylow 2-subgroup. Show that the center of $G$ is nontrivial.
Remark: The claim remains true if $G$ has an abelian normal Sylow 2 -subgroup but then it is a bit harder to prove.

( 16 points) Let $p$ be a prime, $n$ a natural number and $A \in G L_{n}\left(\mathbb{F}_{p}\right)$ diagonalizable over the algebraic closure $\overline{\mathbb{F}_{p}}$ .
(a) (4 points) Show that the order of $A$ in $G L_{n}\left(\mathbb{F}_{p}\right)$ is equal to the lcm of the orders of the eigenvalues of $A$ in ${\overline{\mathbb{F}_{p}}}^{\times}$ .
(b) ( 8 points) Prove that $G L_{n}\left(\mathbb{F}_{p}\right)$ has an element of order $p^{n}-1$ which is diagonalizable over $\overline{\mathbb{F}_{p}}$ .
(c) ( 4 points) Explicitly construct an element of order 8 of $G L_{2}\left(\mathbb{F}_{3}\right)$ .

( 12 points) Let $R$ be a commutative ring with 1 which is Artinian, i.e. for any descending chain $I_{1} \supseteq I_{2} \ldots \supseteq I_{n} \ldots$ of ideals of $R$ there exists an $n_{0}$ such that $I_{n}=I_{n_{0}}$ for all $n \geq n_{0}$ . Prove the following:
(a) ( 4 points) If $R$ is an integral domain, then it is a field.
(b) ( 4 points) Any prime ideal of $R$ is maximal.
(c) ( 4 points) $R$ has only finitely many maximal ideals.

(12 points) Let $R=\mathbb{Z}[x] /\left(x^{3}+x^{2}+1\right)$ be the quotient of the polynomial ring $\mathbb{Z}[x]$ modulo the principal ideal $\left(x^{3}+x^{2}+1\right)$ .
(a) ( 4 points) Is $R$ an integral domain?
(b) ( 8 points) Which of the principal ideals (2), (3), (5) of $R$ are prime ideals? And which of them are maximal?

(15 points) Let $M|K, M| L, K|F, L| F$ be finite field extensions. Assume that for $\alpha, \beta \in M, K=F(\alpha), L=F(\beta)$ and $M=F(\alpha, \beta)$ . Set $a=[K: F]$ and $b=[L: F]$ .
(a) ( 3 points) If $K \mid F$ and $L \mid F$ are Galois, show that also $M \mid F$ is Galois.
(b) (8 points) If $K \mid F$ and $L \mid F$ are Galois, prove that $[M: F]$ divides $a b$ .
(c) (4 points) Give an example of $M, K, L, F$ as above (but without the Galois assumption) such that [ $M: F$ ] does not divide $a b$ .

( 15 points) Consider the polynomial ring $R=F[x, y]$ in two variables over the field $F$ and the ideal $I=(x, y)$ of $R$ . Let $\phi: R \longrightarrow F$ be the $F$ -algebra homomorphism with $\phi(x)=\phi(y)=0$ , which turns $F$ into an $R$ -module.
(a) ( 4 points) Show that the $R$ -modules $F \otimes_{R} F$ and $F$ are isomorphic.
(b) (2 points) Define maps $s, t: I \longrightarrow F$ by $s(f)=c_{1,0}$ , respectively, $t(f)=c_{0,1}$ if $f=\sum_{i, j} c_{i, j} x^{i} y^{j} \in I$ (with $c_{i, j} \in F$ for all $i, j \in \mathbb{N}_{0}$ ). Verify that $s$ and $t$ are $R$ -module homomorphisms.
(c) ( 6 points) Prove that $x \otimes y-y \otimes x$ is not 0 in $I \otimes_{R} I$ .
(d) ( 3 points) Prove that $I$ is not a flat $R$ -module.

(10 points) Consider the $\mathbb{C}$ -vector space $V=M_{2}(\mathbb{C})$ of complex $2 \times 2$ matrices and the linear transformation $T: V \longrightarrow V$ defined by $T(X)= A X-X A$ for all $X \in M_{2}(\mathbb{C})$ , where $A$ is the matrix $A=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$ . Determine (as a $4 \times 4$ matrix) the Jordan canonical form of $T$ .