Algebra Qualifying Exam

Algebra Qualifying Exam

March 2012

Z

=

integers. Q

=

rational numbers. C

=

complex numbers.

The level of difficulty of the problems is indicated by the number of

*

with one

*

indicates the easier problems.

1.

*

Determine whether each statement below is true or false. If true, give a brief explanation. If false, provide a counterexample.

(a) (5 points) All subgroups of a direct product G x H of groups G and H are of the form G^I x HI for G/:::;GandH/:::;H.

(b) (5 points) An Artinian integral domain is a field.

(c) (5 points) Every algebraic extension of a finite field is finite.

(d) (5 points) If R is a PID, then its Jacobson radical J(R) is zero. Recall that J(R) is the intersection of all maximal ideals of R.

2. (10 points)

*

Show that Z[.J=S]

=

{a

+ ibJ5I

a, bE Z} is not a unique factorization domain.

3.

*

Let E be the splitting field over Q of the equation ^X4 5.

ea) (10 points) Determine the Galois group of E over Q.

(b) (5 points) Final all the intermediate fields K between E and Q satisfying : K]

=

2.

4.

*

Let X be a topological space. Consider the ring ReX) of continuous real-valued functions on X. The ring structure is given by point-wise addition and multiplication.

(a) (5 points) Show that for each x E X the set

Mx = {f ER(X)

I

fex) = O}

is a maximal ideal in R(X).

(b) (5 points) Show that if X is compact, that is, every open covers of X has a finite subcover, then every maximal ideal in R(X) is equal to Mx for some x EX.

5. (15 points)

**

^{Let V} be a finite dimensional vector space over C. Let <I> : V x V -> C be a bilinear map satisfying <I>(x,x)

=

0 for all x E V. Assume that for any nonzero element x E V, there is an element y E V such that <I>(x,y)

1:

O. Show that the dimension of V is even.

6. (15 points)

**

^{Let IFq} be the finite field with q elements and let Mz(lFq) be the ring of 2 x 2 matrices over IF'q. Determine the number of nonzero nilpotent matrices in MilF'q) as a function of q.

7. (15 points)

* * *

^Let p

1:

q be prime numbers. Prove that no group of order pZq is simple.

This exam has 7 questions, for a total of 100 points.