Do all problems

- If
*p*and*q*are distinct primes and*G*is a finite group of order*p*^{2}*q*, prove that*G*has a nontrivial normal Sylow subgroup. - Find the Galois group of
*K*over the rationals**Q**where*K*is the splitting field of the polynomial*x*^{4}+4*x*^{2}+ 2. - Show that
**Q**is not a projective**Z**-module. - Let
*R*be a commutative Noetherian ring with a 1 and let*M*be a finitely generated*R*-module. Show that if*f*:*M*- >*M*is a surjective*R*-module homomorphism, then it must also be injective. (Hint: consider the kernels of*f*^{n}.) - Suppose
*R*is a principal ideal domain that is not a field, and that*M*is a finitely generated*R*-module. Suppose further that for every irreducible element*p*e*R*, the*R*/*pR*-module*M*/*pM*is cyclic (has a single generator). Show that*M*is cyclic. - Let
*G*be a finite group with a composition series of length 2. Prove that if*M*and*N*are distinct nonidentity proper normal subgroups of*G*, then*G*=*M*X*N*. - Let
*R*be the ring**Q**+*x*^{2}**Q**[*x*], the collection of all polynomials with rational coefficients that have no*x*term.- (a)
- Show that if
0 =/=
*f*e*R*, then*R*/*fR*is a finite dimensional vector space over**Q**. - (b)
- Use part (a) to prove that every nonzero prime ideal of
*R*is maximal.

Peter Linnell 2005-01-15