Algebra Preliminary Exam, Fall 1985

Do ALL problems

1. Define what is meant by a prime ideal in a commutative ring.
2. Prove that a nonzero prime ideal in a principal ideal domain is always a maximal ideal.

1. Prove that there is no simple group of order 56.
2. Show that a finite field cannot be algebraically closed.
3. Find all finitely generated abelian groups A with the property that for any subgroups B and C, either or .
4. Let F be the splitting field of (x3-2)(x2-3) over . Describe the Galois group in as much detail as possible.
5. Let R be an integral domain and let be an element of the quotient field of R. Set .

1. Prove that I is an ideal of R.
2. Show that either or there exists a maximal ideal of R such that .
3. Conclude that , where the intersection is taken over all the maximal ideals of R.
6. Prove that and are isomorphic as -modules. (Here denotes the rational numbers and denotes the integers.)
7. Let G be a group. Suppose that

1. H and K are nilpotent groups,
2. there are homomorphisms and , and
3. , the center of G.
Prove that G is nilpotent.