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6.5. FACTORIZATION
301
(a)
Every ideal in
R
is principal.
(b)
If an irreducible
p
divides the product of two nonzero elements,
then it divides one or the other of them.
6.5.3.
Let
R
be a Euclidean domain. Show that every nonzero, nonunit
element has a factorization into irreducibles, which is unique (up to units
and up to order of the factors).
6.5.4.
Let
Z
Œ
p
±
2Ł
be the subring of
C
generated by
Z
and
p
±
2
. Show
that
Z
Œ
p
±
2Ł
D f
a
C
b
p
±
2
W
a;b
2
Z
g
. Show that
Z
Œ
p
±
2Ł
is a Euclidean
domain.
Hint:
Try
N.z/
D j
z
j
2
for the Euclidean function.
6.5.5.
Let
!
D
exp
.2±i=3/
. Then
!
satisﬁes
!
2
C
!
C
1
D
0
. Let
Z
Œ!Ł
be the subring of
C
generated by
Z
and
!
. Show that
Z
Œ!Ł
D
f
a
C
b!
W
a;b
2
Z
g
. Show that
Z
Œ!Ł
is a Euclidean domain.
Hint:
Try
N.z/
D j
z
j
2
for the Euclidean function.
6.5.6.
Compute a greatest common divisor in
Z
ŒiŁ
of
14
C
2i
and
21
C
26i
.
6.5.7.
Compute a greatest common divisor in
Z
ŒiŁ
of
33
C
19i
and
18
±
16i
.
6.5.8.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra

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