Algebra Qualifying Exam, Fall 1989
- Compute the Galois group of 3x2 + 7x + 21 over
- By an ``N-group", we mean a finite group with the property
that every nonidentity homomorphic image has a nonidentity center.
Prove that maximal subgroups of N-groups are always normal.
- Let R be an integral domain and let K be its field of
fractions. Assume that if , then either or
. Prove that
- R is a local ring.
- R is integrally closed in K.
- Let R be a PID and let A, M be nonzero finitely generated
- Show that if A is torsion free, then .
- Provide a counterexample to the conclusion of (1)
in the case A is not torsion free.
- Assume that p and q are distinct primes. Show that a group
of order p2q cannot be simple.
- Let k be a field. If , define
Prove that if f1,f2, ... is a countable list of polynomials in
k[X1, ... ,Xn], then there is a positive integer T such that
- Let k be a field. Prove that if A and B are two -matrices with entries in k, both of which have minimal polynomial
Xn-1, then A and B are similar.
Wed Jul 31 09:34:40 EDT 1996