Algebra Qualifying Exam, Summer 2001

Instructions: do all problems.

1. Let G be a group.

   (a)

   Show that if A and B are normal subgroups of G, then A $\cap$ B is also a normal subgroup of G.

   (b)

   Suppose that N is a proper nontrivial normal simple subgroup of G and that G/N is also a simple group. Prove that either N is the only nontrivial proper normal subgroup of G or that G is isomorphic to N X G/N.

2. Let A be a finite noncyclic abelian group with two generators. Let p be a prime. Assume that for all primes q with q$\ne$p, there are no nonzero group homomorphisms from Z/(q) to A. Describe the structure of A and prove that there is no nonzero group homomorphism from A to Z/(q) for all primes q with q$\ne$p.

3. Let R be a ring with a 1 and let 0 - > P - > M - > Q - > 0 be a short exact sequence of R-modules. Show that if P and Q are projective R-modules, then M is a projective R-module.

4. Prove that every group of order 441 has a quotient which is isomorphic to Z/(3).

5. Let R be a commutative ring with a 1$\ne$ 0. Suppose that every ideal of R different from R is a prime ideal.

   (a)

   Prove that R is an integral domain.

   (b)

   Prove that R is a field.

6. (a)

   Prove that Q($\sqrt[4]{2}$) and Q(i$\sqrt[4]{2}$) are isomorphic fields.

   (b)

   Prove that Q(i,$\sqrt[4]{2}$) is a Galois extension of Q.

   (c)

   Find the Galois group Gal(Q(i,$\sqrt[4]{2}$)/Q) and prove your claim.

7. Recall that a group G of permutations of a set S is called transitive if given s, t $\in$ S, then there exists s $\in$ G such that s(s) = t. Let K be a field. Let f (x) be a separable polynomial in K[x] and let F be a splitting field of f over K. Prove that f (x) is irreducible over K if and only if the Galois group of F over K is a transitive group when viewed as a group of permutations of the roots of f (x).

8. An ideal in a commutative ring R with a 1 is called primary if I$\ne$R and if ab $\in$ I and a$\notin$I, then bⁿ $\in$ I for some positive integer n.

   (a)

   Prove that prime ideals are primary.

   (b)

   Prove that if R is a PID, then I is primary if and only if I = Pⁿ for some prime ideal P of R.