Instructions: do all problems.

- Let
*G*be a group.- (a)
- Show that if
*A*and*B*are normal subgroups of*G*, then*A**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*.

- Let
*A*be a finite noncyclic abelian group with two generators. Let*p*be a prime. Assume that for all primes*q*with*q**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**p*. - 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. - Prove that every group of order 441 has a quotient which is
isomorphic to
**Z**/(3). - Let
*R*be a commutative ring with a 1 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.

- (a)
- Prove that
**Q**() and**Q**(*i*) are isomorphic fields. - (b)
- Prove that
**Q**(*i*,) is a Galois extension of**Q**. - (c)
- Find the Galois group
Gal(
**Q**(*i*,)/**Q**) and prove your claim.

- Recall that a group
*G*of permutations of a set*S*is called*transitive*if given*s*,*t**S*, then there exists s*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*). - An ideal in a commutative ring
*R*with a 1 is called*primary*if*I**R*and if*ab**I*and*a**I*, then*b*^{n}*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*=*P*^{n}for some prime ideal*P*of*R*.