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 $\subseteq$ A and R $\subseteq$ B. Prove that if A is an integral extension of R and B is an integral extension of R, then the ring A $\otimes_{R}^{}$ 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 $\in$ K such that H coincides with
   {s $$\in$$ 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 = $\bigcap$N_G(P) where the intersection is taken over all Sylow p-subgroups P of G and N_G($\_$) denotes the normalizer. Show that

   (a)

   K$\lhd$G.

   (b)

   G and G/K have the same number of Sylow p-subgroups.

5. Suppose A is an abelian group (written additively) of order p^M for some prime p. Prove that if n is a positive integer such that pⁿA = 0, then
   |{a $$\in$$ A | pa = 0}|$$\ge$$p^M/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 $\mathbb {Z}$₆ denote the ring of integers modulo 6. Then every projective $\mathbb {Z}$₆-module is free.