Algebra Preliminary Exam, Fall 1985
- Define what is meant by a prime ideal in a commutative ring.
- Prove that a nonzero prime ideal in a principal ideal domain is
always a maximal ideal.
- Prove that there is no simple group of order 56.
- Show that a finite field cannot be algebraically closed.
- Find all finitely generated abelian groups A with the
property that for any subgroups B and C, either or
- Let F be the splitting field of (x3-2)(x2-3) over
. Describe the Galois group in as much detail as
- Let R be an integral domain and let be an element of
the quotient field of R. Set .
- Prove that I is an ideal of R.
- Show that either or there exists a maximal ideal
of R such that .
- Conclude that , where the
intersection is taken over all the maximal ideals of R.
- Prove that and
are isomorphic as -modules. (Here
denotes the rational numbers and denotes the integers.)
- Let G be a group. Suppose that
Prove that G is nilpotent.
- H and K are nilpotent groups,
- there are homomorphisms and , and
- , the center of
Wed Jul 31 16:55:41 EDT 1996