Math 772
Spring 2002
Homework 4
Due Wednesday, April 3, 2002. Do not forget to mark your
“favorite” problem.
- Let
be a field. Show that a polynomial of positive degree 
has a multiple
root if and only if the ideal generated by 
and 
is proper. Use
this to show that if the polynomial 
has a multiple root 
then 
must divide either 
or 
- Consider
the ring
List all of the
possibilities for the decomposition of a rational prime 
as a product of prime ideals in 
For all
possibilities, give an example of a prime with that particular
decomposition or explain why that decomposition cannot occur. (Hint: look
at the “generating” polynomial 
).
- Consider
the rings
for 
a) Which of these are UFDs?
b) Which of these are HFDs?
c) For the ones that are not HFDs, determine the structure of the class group.