Algebra Preliminary Exam, Fall 1982
- Prove that a group of order has a normal
subgroup of order 15.
- For a positive integer n, show that every ideal in
- Explain how one determines the number of ideals of in terms of n.
- Calculate the Galois group of (x2-2)(x2+3) over .
- Explicitly state the correspondence between the subfields of
the splitting field K of (x2-2)(x2+3) over and the
subgroups of .
- Let V and W be vector spaces over the field k and let
W* be the space of linear functions from W to k. Prove that
the map defined by
- Well defined.
- Let K be a field of characteristic . Suppose f(x) =
p(x)/q(x) is a ratio of polynomials in K[x]. Prove that if f(x) =
f(-x), then there are polynomials p0(x2), q0(x2) such that
f(x) = p0(x2)/q0(x2). (HINT: look for a field automorphism of
K(x) that fixes f(x).)
- Let G be a solvable group. Prove that if is
normal in G and contains no other non trivial subgroups which are
normal in G, then N is abelian.
- Let A,B,C,D be finite abelian groups such that and . Prove that .
Wed Jul 31 20:47:24 EDT 1996