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Proof.
See [ZarS60].
Definition.
A field k is said to be
algebraically closed
if every nonconstant polyno-
mial f(X) in k[X] has a zero in k.
The complex numbers
C
are the standard example of an algebraically closed field.
See Corollary E.5.2 for a proof.
Definition.
Let K be an extension of a field k. K is called an
algebraic closure
of k if
(1) K is algebraic over k and
(2) K is algebraically closed.
B.11.9. Theorem.
(1) Every field k has an algebraic closure.
(2) Any two algebraic closures of k are isomorphic over k.
Proof.
See [Jaco64].
B.11.10. Theorem.
If k is an algebraically closed field, then every polynomial in
k[X] factors into a product of linear factors.
Proof.
See [Dean66].
B.11.11. Theorem.
Every algebraically closed field k is infinite.
Proof.
We use Theorem B.8.2. If k has characteristic 0, then the prime subfield of k
is isomorphic to the reals. If k has characteristic p, p prime, then one needs to show
that the algebraic closure of
Z
p
is infinite.
B.11.12. Theorem.
If k is an algebraically closed field and f is a polynomial in
k[X
1
,X
2
,...,X
n
] that vanishes on all but a finite subset of k
n
, then f is the zero
polynomial.
Proof.
Because k is algebraically closed, it has an infinite number of elements by
Theorem B.11.11. If n = 1, then this follows from Theorem B.8.9 since f can only have
a finite number of zeros. The general case is proved by induction on n.
Definition.
If k is a field, then k((X)) will denote the field of quotients of k[[X]]. More
generally, k((X
1
,X
2
,...,X
n
)) will denote the field of quotients of k[[X
1
,X
2
,...,X
n
]].
There are natural inclusions
(
()
)
Ã
(
(
)
)
à ◊◊◊Ã
(
(
)
)
kkX
Ã
kXX
,
kXX
,
,...,
X
n
.
1
1
2
1
2
The multiplicative inverse of a formal power series f in k[[X]] is 1/f in k((X)). The
following fact about 1/f is often used and is therefore worth stating explicitly, namely,
1/f is itself a power series in k[[X]] if it has a nonzero constant term.
