Math 793
Summer 2001
Homework 4
- Consider the ring of
integers
.
a)
Show
that the element 
is irreducible in
this ring.
b)
Use
the previous to construct an element that has one factorization with precisely
two irreducible factors and another with precisely 12 irreducible factors.
- Consider
where 
if 
and 
if 
(
is square-free and for this problem, 
).
a)
Show
that if 
is a UFD, then 
is necessarily prime (or 
).
b)
Show
that if 
is an HFD, then 
has at most two (distinct) prime factors.
c)
Is
the converse to either statement true? Prove or give a counterexample.
- Consider the quadratic
ring of integers
where 
if 
and 
if 
(
is square-free). Let 
be a nonzero
prime.
a)
Show
that in 
, 
factors into at most two irreducible elements.
b)
Let
be odd. Show that if 
and 
is not a square mod 
then 
is a prime element of 
c)
Let
be odd. Show that if 
and 
is a square mod 
then 
is a not prime element of 
d)
Show
that if 
then 
is not a prime element of 
e)
Determine
when 2 is a prime in 
f)
Give
examples of a prime that is not prime in 
that factors into two
irreducible elements and an example of a prime that is not prime in 
that is irreducible.
g)
Show
that if a prime factors into two irreducible elements, then the two elements
are primes of 