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Math 113 Section 5
Homework #10
Fall 2010
Due: Monday, November 22
1. Section 8.1, Exercise 3: Let
R
be a Euclidean domain. Let
m
be the minimum integer
if the set of nonzero elements of
R
(i.e., let
m
=
min
{
N
(
r
)

r
∈
R
{
0
}}
). Prove that
every nonzero element of
R
of norm
m
is a unit. Deduce that a nonzero element of
norm zero (if such an element exists) is a unit.
2. Section 8.1, Exercise 7: Find a generator for the ideal
(85
,
1+13
i
)
in
Z
[
i
]
, i.e., a greatest
common divisor for
85
and
1 + 13
i
, by the Euclidean Algorithm. Do the same for the
ideal
(47

13
i,
53 + 56
i
)
.
3. Section 8.2, Exercise 3: Prove that the quotient of a principal ideal domain by a prime
ideal is again a principal ideal domain.
4. Section 8.2, Exercise 5: Let
R
be the quadratic integer ring
Z
[
√

5]
. Deﬁne the ideals
I
2
= (2
,
1 +
√

5)
,I
3
= (3
,
2 +
√

5)
, and
I
Í
3
= (3
,
2

√

5)
.
(a) Prove that
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 Fall '08
 OGUS
 Math, Algebra

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