Prove that
Ι
[
x
]
=
{
a
0
+
a
1
x
+
...
+
a
n
x
n
|
a
0
=
2
k
f
o
r
k
∈
ℤ
}
, the set of all polynomials in
ℤ
[
x
]
with even constant term, is an ideal of
ℤ
[
x
]
.
Show that
Ι
[
x
]
is not a principal ideal; that is, show that there is no
f
(
x
)
∈
ℤ
[
x
]
such that
Ι
[
x
]
=
(
f
(
x
)
)
=
{
f
(
x
)
g
(
x
)
|
g
(
x
)
∈
ℤ
[
x
]
}
.
Show that
Ι
[
x
]
is an ideal generated by two elements in
ℤ
[
x
]
that is,
Ι
[
x
]
=
(
x
,
2
)
=
{
x
f
(
x
)
+
2
g
(
x
)
|
f
(
x
)
,
g
(
x
)
∈
ℤ
[
x
]
}
.