Algebra Preliminary Exam, Fall 1985

Do ALL problems

1. 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.

2. Prove that there is no simple group of order 56.
3. Show that a finite field cannot be algebraically closed.
4. Find all finitely generated abelian groups A with the property that for any subgroups B and C, either or .
5. Let F be the splitting field of (x³-2)(x²-3) over . Describe the Galois group in as much detail as possible.
6. 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.
7. Prove that and are isomorphic as -modules. (Here denotes the rational numbers and denotes the integers.)
8. 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.

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