Math 793
Summer 2001
Homework 3
Turn
in at least two by Monday, July 2, 2001.
a)
b) is a unit if and only
if
c) if and only if
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.
a)
Show
that is a Euclidean
domain.
b)
Classify
the following rings as PIDs or UFDs or neither: