6.8. IRREDUCIBILITY CRITERIA
313
is an ideal in
R
, and that
A
k
±
A
k
C
1
for all
k
²
0
.
6.7.2.
Suppose that
R
is Noetherian and
'
W
R
!
S
is a surjective ring
homomorphism. Show that
S
is Noetherian.
6.7.3.
We know that
Z
Œ
p
³
5Ł
is not a UFD and, therefore, not a PID.
Show that
Z
Œ
p
³
5Ł
is Noetherian.
Hint:
Find a Noetherian ring
R
and a
surjective homomorphism
'
W
R
!
Z
Œ
p
³
5Ł
.
6.7.4.
Show that
Z
ŒxŁ
C
x
Q
ŒxŁ
is not Noetherian.
6.7.5.
Show that a Noetherian domain in which every irreducible element
is prime is a unique factorization domain.
6.8. Irreducibility Criteria
In this section, we will consider some elementary techniques for determin
ing whether a polynomial is irreducible.
We restrict ourselves to the problem of determining whether a polyno
mial in
Z
ŒxŁ
is irreducible. Recall that an integer polynomial factors over
the integers if, and only if, it factors over the rational numbers, according
to Lemma
6.6.9
and Corollary
6.6.10
.
A basic technique in testing for irreducibility is to reduce the polyno
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 Fall '08
 EVERAGE
 Algebra, Prime number, Integral domain, Ring theory, Polynomial ring, Noetherian

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