# 2015 Algebra Prelim

September 14, 2015

INSTRUCTIONS: Do as many of the eight problems as you can.
Four completely correct solutions will be a pass; a few complete solutions will count more than many partial solutions.
Always carefully justify your answers.
If you skip a step or omit some details in a proof, point out the gap and, if possible, indicate what would be required to fill it in.

1. (a) Find an irreducible polynomial of degree 5 over the field $\mathbb { Z } _ { 2 }$ of two elements and use it to construct a field of order 32 as a quotient of the polynomial ring $\mathbb { Z } _ { 2 } [ x ]$

(b) Using the polynomial you found in part (a), find a $5 \times 5$ matrix M over $\mathbb { Z } _ { 2 }$ of order 31, so that $M ^ { 3 1 } = I$ but $M \ne I .$

2. Find the minimal polynomial of ${ \sqrt { 2 } } + { \sqrt { 3 } }$ over Q. Justify your answer.

3. (a) Let R be a commutative ring with no nonzero nilpotent elements.
   Show that the only units in the polynomial ring $R [ x ]$ are the units of $R ,$ regarded as constant polynomials.

(b) Find all units in the polynomial ring $\mathbb { Z } _ { 4 } [ x ]$

4. Let p and q be two distinct primes.
   Prove that there is at most one nonabelian group of order pq (up to isomorphisms) and describe the pairs $( p , q )$ such that there is no non-abelian group of order pq.

5. (a) Let L be a Galois extension of a field K of degree 4. What is the minimum number of subfields there could be strictly between K and L? What is the maximum number of such subfields?
   Give examples where these bounds are attained.

(b) How do these numbers change if we assume only that L is separable (but not necessarily Galois) over K?

$$
D : R  R
$$

$$
D ( 1 ) = 0
$$

$$
D ( a b ) = D ( a ) b + a D ( b )
$$

(a) Describe all derivations of the polynomial ring $\mathbb { C } [ x ]$

$$
a , b \in R
$$

(b) Let A be the subring (or C-subalgebra) of $\operatorname { E n d } _ { \mathbb { C } } ( \mathbb { C } [ x ] )$ ) generated by all derivations of $\mathbb { C } [ x ]$ and the left multiplications by x. Prove that $\mathbb { C } [ x ]$ is a simple left Amodule.
Note that the inclusion $A \to \operatorname { E n d } _ { \mathbb { C } } ( \mathbb { C } [ x ] )$ defines a natural left A-module structure on $\mathbb { C } [ x ]$ .

7. Let G be a non-abelian group of order $p ^ { 3 }$ with p a prime.

(a) Determine the order of the center Z of G.

(b) Determine the number of inequivalent complex 1-dimensional representations of G.

(c) Compute the dimensions of all the inequivalent irreducible representations of G and verify that the number of such representations equals the number of conjugacy classes of G.

8. Prove that every finitely generated projective module over a commutative noetherian local ring is free.