Math 721
Spring 2001
Final Exam
In
this assignment, if 
is a field, its algebraic closure is denoted by 
Also 
will denote a field
with 
elements.
- Show that
if and only if 
divides 
- Consider an algebraic
closure,
, of a field with 
elements.
a)
Show
that 
b)
Show
that 
is Galois over 
and 
is an infinite,
abelian, torsion-free group.
- Let
be integral
domains. An element 
is said to be
integral over 
if 
is a root of a monic
polynomial in 
We also define 
to be the set of
elements of 
that are
integral over 
If every element
of 
is integral over
we say that 
is an integral
extension of 
a)
Show
that 
is integral over 
if and only if the
ring 
is a finite 
module.
b)
Show
that if 
is an integral extension of 
and 
is an element of a
ring containing 
that is integral over
then 
is integral over 
c)
Use
the previous results to show that the integral closure of 
in 
, (
) is a ring containing 
d)
Let
be a square-free integer. Construct the integral closure of
the ordinary integers in the field 