ALGEBRA PRELIMINARY EXAMINATION:
Fall 2000

Do all problems

1. Let

1 - - > A - - > G - - > P - - > 1

be a short exact sequence of groups such that A is abelian, | P| = 81, and | A| = 332. Show that G has a nontrivial center.

2. Let F be a finite field of odd characteristic. Prove that the rings F[X]/(X2 - a) as a ranges over all nonzero elements of F fall into exactly two isomorphism classes.

3. Let R be a finite-dimensional simple algebra and let M be a finite-dimensional left R-module. Prove that there is a positive integer d such that

is a free module.

4. Let B be a square matrix with rational entries. Show that if there is a monic polynomial f [T] such that f (B) = 0 then the trace of B is an integer.

5. Let k be a field. Compute the dimension over k of

k[X]/(Xm  k[X]/(Xn)