Graphics Reference
In-Depth Information
Definition.
Let f(X) Πk[X] be a polynomial of positive degree. An extension field K
of k is called a
splitting field
of f(X) if
(1) f(X) is a product of linear factors in K{X}, that is,
()
=-
(
)
(
)
◊◊◊
(
)
fX
aX
q
X
-
q
X
-
q
1
2
n
with a Πk, q
i
ΠK, and
(2) K = k(q
1
,q
2
,...,q
n
).
B.11.6. Theorem.
Every polynomial over a field k of positive degree has a splitting
field. Any two splitting fields of the polynomial are isomorphic over k.
Proof.
See [Jaco64].
Definition.
A polynomial f(X) Πk[X] is said to be
separable
if it is a product of irre-
ducible polynomials each of which has only simple roots in a splitting field of f(X).
B.11.7. Theorem.
Every polynomial over a field of characteristic 0 is separable.
Proof.
See [Jaco64].
Definition.
Let K be an extension of a field k. An element of K is said to be
separa-
ble
over k if it is algebraic over k and its minimum polynomial is separable. K is a
separable extension
of k if every element of K is separable over k.
Definition.
Let K be an extension of a field k. The elements a
1
, a
2
,..., a
n
in K
are said to be
algebraically dependent
over k if there exists a nonzero polynomial
f(X
1
,X
2
,...,X
n
) in k[X
1
,X
2
,...,X
n
] and f(a
1
,a
2
,...,a
n
) = 0. Otherwise, the elements
are said to be
algebraically independent
. An infinite set of elements of K is said to be
algebraically dependent
over k if each of its finite subsets is algebraically dependent.
Otherwise it is said to be
algebraically independent
.
Alternatively, the elements a
1
, a
2
,..., a
n
are algebraically independent over k if
the map
[
]
Æ
[
]
kX X
,
,...,
X
ka a
,
,...,
a
12
n
12
n
(
)
Æ
(
)
fX X
,
,...,
X
fa a
,
,...,
a
12
n
12
n
is an isomorphism over k.
Definition.
Let K be an extension of a field k. A maximal set of elements in K that
are algebraically independent over k is called a
transcendence basis
for K over k. The
number of elements in a transcendence basis is called the
transcendence degree
of K
over k and is denoted by tr
k
(K).
The notion of transcendence degree is justified by the following fact:
B.11.8. Theorem.
Any two transcendence bases of K over k have the same cardi-
nal number of elements.
