Prove that a group of order has a normal
subgroup of order 15.
For a positive integer n, show that every ideal in
is principal.
Explain how one determines the number of ideals of in terms of n.
Calculate the Galois group of (x^{2}-2)(x^{2}+3) over .
Explicitly state the correspondence between the subfields of
the splitting field K of (x^{2}-2)(x^{2}+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
is
Well defined.
Linear.
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 p_{0}(x^{2}), q_{0}(x^{2}) such that
f(x) = p_{0}(x^{2})/q_{0}(x^{2}). (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 .