ALGEBRA PRELIMINARY EXAMINATION:
Do all problems. All rings should be assumed to have a 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.
- 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
) = b
- Let S be a semisimple ring. Prove that S X S is semisimple.
- 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
NG() denotes the normalizer. Show that
- G and G/K have the same number of Sylow p-subgroups.
- 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
- Let G be a finite group. Prove that if H and K are normal
nilpotent subgroups of G, then so is HK.
- Prove or disprove: let
denote the ring of integers modulo 6. Then every projective
6-module is free.