Algebra Prelim, Fall 1990
- Give a complete list of all non-isomorphic abelian groups of
- Show that a group of order cannot be
- Let G be a finite p-group with |G| > p2. Prove that G
contains a normal abelian subgroup of order p2.
- Show that if G is a subgroup of Sn, then
either or .
- Show that if and G is a normal subgroup of Sn,
then G=1, An or Sn.
- Show that if , then Sn has no subgroups of index 3.
- Let G be an abelian group with 54 elements. Suppose that
G cannot be generated by one element, but can be generated by two
elements. Prove that G is isomorphic to .
- Let K be an extension field of F with [K:F] = 14. Let
be a polynomial of degree 5. Suppose f(x) has
no roots in F but has a root in K. What can you say about the
factorization of f(x) into irreducibles in F[x] and K[x]?
- Let f(x) be irreducible over with splitting
field E, and let and be roots of f in E. If
has an abelian Galois group, prove that .
- Let R be a commutative ring with identity, and let
be the set of maximal ideals of R.
Let A be an ideal of the polynomial ring R[x] such that
. Show that for some . (Hint: consider the set is a coefficient of some polynomial in A}.)
Wed Jul 31 08:48:21 EDT 1996