Corollary 2.36
The ring
K
[
x
1
, x
2
, . . . , x
n
]
of polynomials in
n
independent
indeterminates
x
1
, x
2
, . . . , x
n
with coefficients in a field
K
is a unique factor-
ization domain.
Corollary 2.37
The ring
Z
[
x
1
, x
2
, . . . , x
n
]
of polynomials in
n
independent
indeterminates
x
1
, x
2
, . . . , x
n
with integer coefficients is a unique factoriza-
tion domain.
Example
Let
K
be a field, and let
K
[
x, y
] be the ring of polynomials in
two independent interminates
x
and
y
.
Any polynomial
f
(
x, y
) in
x
and
y
with coefficients in
K
can be represented in the form
g
0
(
x
) +
d
∑
j
=1
g
j
(
x
)
y
j
,
where
g
0
(
x
)
, g
1
(
x
)
, . . . , g
d
(
x
) are polynomials in the indeterminate
x
with
coefficients in
K
. These polynomials
g
0
, g
1
, . . . , g
d
are uniquely determined by
the polynomial
f
(
x
). There is thus a well-defined function
ϕ
:
K
[
x, y
]
→
K
[
x
],
where
ϕ
g
0
(
x
) +
d
X
j
=1
g
j
(
x
)
y
j
!
=
g
0
(
x
)
.
for all
g
0
, g
1
, . . . , g
d
∈
K
[
x
]. Moreover
ϕ
(
f
)(
x
) =
f
(
x,
0
K
) for all
f
∈
K
[
x, y
],
where
f
(
x,
0
K
) denotes the polynomial in the indeterminate
x
obtained by
substituting the zero element 0
K
of the field
K
for the indeterminate
y
. The
function
ϕ
:
K
[
x, y
]
→
K
[
x
] is a surjective ring homomorphism, and its kernel
is the ideal
P
of
K
[
x, y
] generated by the polynomial
y
. Now
K
[
x, y
]
/P
∼
=
K
[
x
], and
K
[
x
] is an integral domain. It follows that the principal ideal
P
of
K
[
x, y
] generated by the polynomial
y
is a prime ideal of
K
[
x, y
] (see
Lemma 2.14). The function
ε
:
K
[
x, y
]
→
K
that maps a polynomial
f
(
x, y
)
to its value
f
(0
K
,
0
K
) at (0
K
,
0
K
) is also a surjective ring homomorphism. It
satisfies
ε
d
x
X
j
=0
d
y
X
k
=0
a
j,k
x
j
y
k
!
=
a
0
,
0
for all coefficients
a
j,k
, where
a
j,k
∈
K
for
j
= 0
,
1
, . . . , d
x
and
k
= 0
,
1
, . . . , d
y
.
The kernel of this homomorphism is the ideal
M
generated by the polyno-
mials
x
and
y
. It follows that
K
[
x, y
]
/M
is a field isomorphic to the field
K
of coefficients, and thus
M
is a maximal ideal of
K
[
x, y
] (see Lemma 2.13).