ALGEBRA GENERAL EXAM, AUGUST 18, 2014, 9AM-1PM

Directions.

• 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 hurt your grade.

• Do each problem on a separate one-sided sheet of paper, and staple them together.

• The exam is pledged.

Throughout we denote $\mathbb{Z}_{n}=\mathbb{Z} / n \mathbb{Z}$.

Problem 1

(10 points). Let $G$ be a finite nilpotent group, $Z(G)$ its center and $p$ a prime number. Prove that $p$ divides $|G|$ if and only if $p$ divides $|Z(G)|$.

Problem 2

(9 points). Compute
(a) $\mathbb{Q} / \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q} / \mathbb{Z}$;
(b) $\mathbb{Z}_{2014} \otimes_{\mathbb{Z}} \mathbb{Z}_{2013}$;
(c) $\operatorname{Hom}_{\mathbb{Z}}\left(\mathbb{Z}_{2014}, \mathbb{Z}_{10}\right)$ (Here $\mathbb{Z}_{n}$ is regarded as a $\mathbb{Z}$-module).

Problem 3

(10 points). Let $p$ be a a prime number.
(a) Determine the order of the automorphism group of $\mathbb{Z}_{p} \times \mathbb{Z}_{p}$;
(b) Prove that there exists a non-abelian group of order $p^{3}$.

Problem 4

(10 points). Denote by $J$ the $n \times n$ Jordan block with eigenvalue 0 . For a positive integer $k$, determine the Jordan canonical form of $J^{k}$. (Hint: you can start by playing with some small values of $k$ ).

Problem 5

(10 points). Let $K$ be a field and $a \in K$. Consider $K$ as a $K[x]$-module (denoted by $K_{a}$ ) via the homomorphism e $v_{a}: K[x] \rightarrow K$ which is the identity on $K$ and sends $x$ to $a$. Let $K[[x]]$ be the power series ring, which is regarded as a $K[x]$-algebra in a natural way. Determine with proof the tensor product $K_{a} \otimes_{K[x]} K[[x]]$. (Hint: keep in mind if one needs to divide into cases depending on $a$.)

Problem 6

(10 points). (a) Compute the order of the group $G$ of rigid motions of a regular octahedron $O$.
(b) The group $G$ acts on the set of vertices of $O$. Describe the stabilizer of a vertex of $O$.

Problem 7

(11 points). Let $f(x)=x^{5}-2 \in \mathbb{Z}[x]$.
(a) Determine the splitting field $F$ of $f(x)$ over $\mathbb{Q}$;
(b) Determine the Galois group of $f(x)$ over $\mathbb{Q}$;
(c) List all the subfields $K$ of $F$ such that $[K: \mathbb{Q}]=4$.

Problem 8

(10 points). Let $A=M_{n}(R)$ be the algebra of $n \times n$ matrices over a commutative ring $R$ with 1 . Fix $1 \leq i, j \leq n$. Determine
(a) the left ideal of $A$ generated by $E_{i j}$;
(b) the (two-sided) ideal of $A$ generated by $E_{i j}$;
(c) Are there possibly other nonzero ideals of $A$ besides those of the form (b)?
(Here as usual $E_{i j}$ denotes the matrix whose ( $i, j$ )th entry is 1 and 0 elsewhere.)