GCD for Polynomials Is Unique
As per title.
Note that gcd (denoted by ) of
f
and
g
, if it exists, is unique. If we had
d
′
which is another gcd, then since it also divides both polynomials we know that
d
′
∣
d
.
Similarly,
d
∣
d
′
if one regards
d
merely as a common divisor. By Exercise 4,
d
′
=
u
d
for some unit
u
∈
F
[
x
]
; that is,
d
′
=
u
d
for some nonzero constant
u
(prove why). Since
d
and
d
′
are both monic, however,
u
=
1
and
d
′
=
d
.
GCD as a Linear Combination
Let
F
be a field and let
f
(
x
)
,
g
(
x
)
∈
F
[
x
]
with
g
(
x
)
≠
0
. Then the
gcd
(
f
(
x
)
,
g
(
x
)
)
=
d
(
x
)
exists, and it is a linear combination of
f
(
x
)
and
g
(
x
)
; that is, there are polynomials
a
(
x
)
and
b
(
x
)
with
d
(
x
)
=
a
(
x
)
f
(
x
)
+
b
(
x
)
g
(
x
)
Every Polynomial Ideal is Principal
If
F
is a field, then every ideal in
F
[
x
]
is a principal ideal
Principal Ideal Domain
A crone
R
is called a principal ideal domain if it is a domain such that every ideal is a principal ideal.
Show that
Z
is a Principal Ideal Domain
As per title.
If
F
is a Field then
F
[
x
]
is a Principal Ideal Domain
As per title.
Let
I
be an ideal in
F
[
x
]
, if
I
=
{
0
}
then
I
=
(
0
)
⋄
, which shows that it is a principal ideal. Now suppose that
I
≠
{
0
}
, and let
m
(
x
)
∈
I
of smallest degree, we'll show that
I
=
(
m
(
x
)
)
⋄
.
We can see that
(
m
(
x
)
)
⋄
⊆
I
because for any element in an ideal, all of it's multiples are a part of it. Looking at
(
m
(
x
)
)
⋄
⊇
I
, then given some
f
(
x
)
∈
I
then by the division algorithm we have
q
(
x
)
,
r
(
x
)
such that
f
(
x
)
=
q
(
x
)
m
(
x
)
+
r
(
x
)
where either
r
(
x
)
=
0
or
deg
(
r
)
<
deg
(
m
)
but then
r
(
x
)
=
f
(
x
)
−
q
(
x
)
m
(
x
)
∈
I
. Now if
deg
(
r
)
<
deg
(
m
)
then we've found a polynomial with a smaller degree than
m
(
x
)
in
I
(namely
r
(
x
)
), so therefore we must have that
r
(
x
)
=
0
which means that
f
(
x
)
=
q
(
x
)
m
(
x
)
∈
(
m
(
x
)
)
⋄
.