Math 793
Summer 2001
Homework 2
Turn
in at least two by Monday, June 25, 2001.
- Let
be a PID and
consider the formal power series ring 
For this exercise, you may assume that 
is a PID implies
that 
is a UFD.
a)
Suppose
that 
can be written in the
form 
with 
relatively prime.
Show that 
can be factored into
two series, one of which has constant term 
and the other of
which has constant term 
b)
Give
examples to show that the assumptions “PID”, “relatively prime”, and “power
series ring” (as opposed to “polynomial ring”) are all needed.
c)
Combine
the above steps to show that if 
(
) is a power series in 
then 
is irreducible
implies that 
where 
is prime and 
d)
Show
that if 
where 
are as above and 
is a unit in 
then 
is irreducible in 
e)
Is
the converse to either c) or d) true? Prove or give a counterexample.
- Consider the ring
a)
Is
this ring integrally closed? If not, find its integral closure.
b)
Repeat
part a) for the ring 
where 
denotes the real numbers and 
is the complex
numbers.
- Let
be rings. Show
that 
is a ring between 
and 