Math 793
Summer 2001
Homework 3
Turn
in at least two by Monday, July 2, 2001.
- Consider the ring
where 
is a square-free
integer. Consider the norm map 
given by 
Prove the
following properties of the norm:
a) 
b) 
is a unit if and only
if 
c) 
if and only if 
- Consider the ring
where 
For this
problem, you may assume that 
is a UFD.
a)
Find
all of the units in 
and 
b)
Show
that for any element 
there is a unit in 
such that 
c)
Show
that 
is atomic (hint:
norm) and that any irreducible in 
is a prime element in
d)
Use
the previous to show that any two factorizations of a nonzero nonunit in 
have the same length
(such a ring is called is called a half-factorial domain or HFD).
e)
Show
that 
is not a UFD.
- Let
be a field.
a)
Show
that 
is a Euclidean
domain.
b)
Classify
the following rings as PIDs or UFDs or neither: 