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
OE
p
2Ł
be the subring of
C
generated by
Z
and
p
2
. Show
that
Z
OE
p
2Ł
D f
a
C
b
p
2
W
a; b
2
Z
g
. Show that
Z
OE
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
OE!Ł
be the subring of
C
generated by
Z
and
!
.
Show that
Z
OE!Ł
D
f
a
C
b!
W
a; b
2
Z
g
.
Show that
Z
OE!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
OEiŁ
of
14
C
2i
and
21
C
26i
.
6.5.7.
Compute a greatest common divisor in
Z
OEiŁ
of
33
C
19i
and
18
16i
.
6.5.8.
Show that for two elements
a; b
in an integral domain
R
, the fol
lowing are equivalent:
(a)
a
divides
b
and
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 Fall '08
 EVERAGE
 Algebra, Greatest common divisor, Integral domain, Ring theory, Commutative ring, Principal ideal domain

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