Determine, up to isomorphism, all groups of order .
Let G be a noncyclic nilpotent group. Show that there
is a normal subgroup N of G such that G/N is a noncyclic
abelian group.
Describe all finitely generated abelian groups G such that if
A and B are subgroups of G, then either or
.
Let R be a commutative ring with a 1. Recall that if I is
an ideal, then
is also an ideal.
Let P_{1}, ... ,P_{n} be distinct prime ideals in R.
Show that has no
nonzero nilpotent elements.
Prove
that .
Prove that is an integral domain
if and only if there exists an i such that P_{i} is contained
in every P_{j} for j=1, ... , n.
Suppose that K is a subfield of a field L.
Assume that , where .
Note that . Let such that
. Prove that .
Let R be a commutative ring with 1 and let
be non-nilpotent.
Let .
Prove that there is a prime ideal P not containing a.
Let K be the quotient field of R/P.
Prove that there exists a ring homomorphism .
Let F be a finite Galois extension of K and suppose that
.
Show that there are at least 9 different proper intermediate fields between
F and K.
Show that there is a proper Galois extension E of K
(in F) and describe the
Galois group of E over K.
Find the Jordan canonical form of viewed
as matrices over the complex numbers
. Find all -matrices with entries in
that commute with this canonical form.