Math 793
Summer 2001
Homework 4
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.
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.
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