Math 793
Summer 2001
Homework 2
Turn
in at least two by Monday, June 25, 2001.
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.
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.