Theorem 5.8
For a ring
R
and
a
∈
R
[
T
]
and
x
∈
R
,
a
(
x
) = 0
R
if and only if
(
T

x
)
divides
a
.
Proof.
Let us write
a
= (
T

x
)
q
+
r
, with
q, r
∈
R
[
T
] and deg(
r
)
<
1, which means that
r
∈
R
.
Then we have
a
(
x
) = (
x

x
)
q
(
x
) +
r
=
r
. Thus,
a
(
x
) = 0 if and only if
T

x
divides
a
.
2
With
R, a, x
as in the above theorem, we say that
x
is a
root
of
a
if
a
(
x
) = 0
R
.
Theorem 5.9
Let
D
be an integral domain, and let
a
∈
D
[
T
]
, with
deg(
a
) =
k
≥
0
. Then
a
has
at most
k
roots.
Proof.
We can prove this by induction. If
k
= 0, this means that
a
is a nonzero element of
D
,
and so it clearly has no roots.
Now suppose that
k >
0. If
a
has no roots, we are done, so suppose that
a
has a root
x
. Then
we can write
a
=
q
(
T

x
), where deg(
q
) =
k

1. Now, for any root
y
of
a
with
y
6
=
x
, we have
0
D
=
a
(
y
) =
q
(
y
)(
y

x
), and using the fact that
D
is an integral domain, we must have
q
(
y
) = 0.
Thus, the only roots of
a
are
x
and the roots of
q
. By induction,
q
has at most
k

1 roots, and
hence
a
has at most
k
roots.
2
Theorem 5.10
Let
D
be an infinite integral domain, and let
a
∈
D
[
T
]
. If
a
(
x
) = 0
D
for all
x
∈
D
,
then
a
= 0
D
.
Proof.
Exercise.
2
With this last theorem, one sees that for an infinite integral domain
D
, there is a onetoone
correspondence between polynomials over
D
and polynomial functions on
D
.
5.3
Ideals and Quotient Rings
Throughout this section, let
R
denote a ring.
Definition 5.11
An
ideal
of
R
is a additive subgroup
I
of
R
that is closed under multiplication
by element of
R
, that is, for all
x
∈
I
and
a
∈
R
,
xa
∈
I
.
Clearly,
{
0
}
and
R
are ideals of
R
.
Example 5.12
For
m
∈
Z
, the set
m
Z
is not only an additive subgroup of
Z
, it is also an ideal of
the ring
Z
.
2
Example 5.13
For
m
∈
Z
, the set
m
Z
n
is not only an additive subgroup of
Z
n
, it is also an ideal
of the ring
Z
n
.
2
If
d
1
, . . . , d
k
∈
R
, then the set
d
1
R
1
+
· · ·
+
d
k
R
:=
{
d
1
a
1
+
· · ·
+
d
k
a
k
:
a
1
, . . . , a
k
∈
R
}
is clearly an ideal, and contains
d
1
, . . . , d
k
. It is called the
ideal generated by
d
1
, . . . , d
k
. Clearly,
any ideal
I
that contains
d
1
, . . . , d
k
must contain
d
1
R
1
+
· · ·
+
d
k
R
. If an ideal
I
is equal to
dR
for
some
d
∈
R
, then we say that
I
is a
principal ideal
.
Note that if
I
and
J
are ideals, then so are
I
+
J
:=
{
x
+
y
:
x
∈
I, y
∈
J
}
and
I
∩
J
.
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