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 $\displaystyle \in$ 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$NG(P) where the intersection is taken over all Sylow p-subgroups P of G and NG($ \_$) denotes the normalizer. Show that
    K$ \lhd$G.
    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 $\displaystyle \in$ A | pa = 0}|$\displaystyle \ge$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 $ \mathbb {Z}$6 denote the ring of integers modulo 6. Then every projective $ \mathbb {Z}$6-module is free.

Peter Linnell