Algebra Preliminary Exam, Fall 1985
Do ALL problems
Define what is meant by a prime ideal in a commutative ring.
Prove that a nonzero prime ideal in a principal ideal domain is always a maximal ideal.
Prove that there is no simple group of order 56.
Show that a finite field cannot be algebraically closed.
Find all finitely generated abelian groups
A
with the property that for any subgroups
B
and
C
, either
or
.
Let
F
be the splitting field of (
x
^{3}
-2)(
x
^{2}
-3) over
. Describe the Galois group in as much detail as possible.
Let
R
be an integral domain and let
be an element of the quotient field of
R
. Set
.
Prove that
I
is an ideal of
R
.
Show that either
or there exists a maximal ideal
of
R
such that
.
Conclude that
, where the intersection is taken over all the maximal ideals of
R
.
Prove that
and
are isomorphic as
-modules. (Here
denotes the rational numbers and
denotes the integers.)
Let
G
be a group. Suppose that
H
and
K
are nilpotent groups,
there are homomorphisms
and
, and
, the center of
G
.
Prove that
G
is nilpotent.
Return to
Algebra Prelims Index
Peter Linnell's Homepage
About this document ...
Peter Linnell
Wed Jul 31 16:55:41 EDT 1996