ALGEBRA PRELIMINARY EXAMINATION:
Winter 2001

Do all problems. All rings should be assumed to have a 1.

1. Let R, A and B be commutative rings with R A and R B. Prove that if A is an integral extension of R and B is an integral extension of R, then the ring A B is also an integral extension of R.

2. For this problem all fields have characteristic 0. Let K/L be a Galois extension with Galois group G and let H be a subgroup of G. Prove that there exists some b K such that H coincides with

{s G | s(b) = b}.

3. Let S be a semisimple ring. Prove that S X S is semisimple.

4. Let G be a finite group and assume that p is a fixed prime divisor of its order. Set K = NG(P) where the intersection is taken over all Sylow p-subgroups P of G and NG() denotes the normalizer. Show that
(a)
KG.
(b)
G and G/K have the same number of Sylow p-subgroups.

5. Suppose A is an abelian group (written additively) of order pM for some prime p. Prove that if n is a positive integer such that pnA = 0, then

|{a A | pa = 0}|pM/n.

6. Let G be a finite group. Prove that if H and K are normal nilpotent subgroups of G, then so is HK.

7. Prove or disprove: let 6 denote the ring of integers modulo 6. Then every projective 6-module is free.

Peter Linnell
2001-02-16