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Math 4124
Wednesday, April 27
Ninth Homework Solutions
1. Exercise 7.3.10. Decide which of the following are ideals of the ring
Z
[
x
]
.
Let
I
be the relevant subset of
Z
[
x
]
.
(a) The set of all polynomials whose constant term is a multiple of 3.
Yes. Obviously 0
∈
I
and
I
is an abelian group under addition. Finally suppose
f
=
a
0
+
a
1
x
+
···
+
a
m
x
m
∈
I
and
g
=
b
0
+
b
1
x
+
···
+
b
n
x
n
∈
Z
[
x
]
. Then 3 divides
a
0
and
fg
=
a
0
b
0
+ (
a
1
b
0
+
a
0
b
1
)
x
+
···
+
a
m
b
n
x
m
+
n
. Since 3 divides
a
0
b
0
, we
see that
fg
∈
I
and we have proven that
I
is an ideal.
(b) The set of all polynomials whose coefﬁcient of
x
2
is a multiple of 3.
No.
x
∈
I
yet
x
2
/
∈
I
, so
I
is not even a subring.
(c) The set of all polynomials whose constant term, coefﬁcient of
x
and coefﬁcient of
x
2
are zero.
Yes. This is just the ideal
x
3
Z
[
x
]
.
(d)
Z
[
x
2
]
.
No. 1
∈
I
but 1
x
/
∈
I
(on the other hand,
I
is a subring).
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 Spring '08
 Staff
 Algebra, Polynomials

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