Algebra Prelim, January 2005

Do all problems

1. If p and q are distinct primes and G is a finite group of order p²q, prove that G has a nontrivial normal Sylow subgroup.

2. Find the Galois group of K over the rationals Q where K is the splitting field of the polynomial x⁴ +4x² + 2.

3. Show that Q is not a projective Z-module.

4. 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 fⁿ.)

5. 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.

6. 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.

7. Let R be the ring Q + x²Q[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.