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
   $$\underbrace{M\oplus M\oplus \dots \oplus M}_{d \text{copies}}^{}\,$$
   is a free module.

4. Let B be a square matrix with rational entries. Show that if there is a monic polynomial f $\in$ $\mathbb { Z}$[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]/(X^m)  $$\otimes_{k[X]}^{}$$  k[X]/(Xⁿ)
   and prove your assertion.

6. In this problem X, Y, Z are indeterminates. Define s : $\mathbb {C}$(X, Y, Z) - > $\mathbb {C}$(X, Y, Z) by s(h(X, Y, Z)) = h(Y, Z, X) for every rational function h in three variables. Prove or disprove: every member of $\mathbb {C}$(X, Y, Z) which is left unchanged by s is a rational function of X + Y + Z, XY + YZ + XZ and XYZ.