Cryptography Reference
In-Depth Information
Theorem A.6.4
Let K/
be a finitely generated field extension. Then the transcendence
degree is well-defined (i.e., all transcendence bases have the same number of elements).
k
Proof
See Theorem 25 of Section II.12 of [
573
], Theorem VI.1.8 of [
271
] or Theorem
1.6.13 of [
568
].
Theorem
A.6.5
Let K/
k
and F/K be finitely generated field extensions. Then
trdeg(
F/
k
)
=
trdeg(
F/K
)
+
trdeg(
K/
k
)
.
Proof
See Theorem 26 of Section II.12 of [
573
].
Corollary A.6.6
Let K/
k
be finitely generated with transcendence degree 1 and let x
∈
K
be transcendental over
k
. Then K is a finite algebraic extension of
k
(
x
)
.
A
perfect field
is one for which every algebraic extension is separable. A convenient
equivalent definition is that a field
x
p
:
x
(see
Section V.6 of Lang [
329
]). We restrict to perfect fields for a number of reasons, one
of which is that the primitive element theorem does not hold for non-perfect fields, and
another is due to issues with fields of definition (see Remark
5.3.7
). Finite fields, fields of
characteristic zero, and algebraic closures of finite fields are perfect (see Exercise V.7.13
of [
271
] or Section V.6 of [
329
]).
k
of characteristic
p
is perfect if
{
∈ k}=k
k
/
Theorem A.6.7
(
Primitive element theorem
) Let
k
be a perfect field. If
k
is a finite,
∈ k
such that
k
= k
separable, extension then there is some α
(
α
)
.
Proof
Theorem V.6.15 of [
271
], Theorem 27 of [
14
], Theorem V.4.6 of [
329
].
A.7 Galois theory
For an introduction to Galois theory see Chapter V of Hungerford [
271
], Chapter 6 of
Lang [
329
]orStewart[
526
]. An algebraic extension
k
/
k
is Galois if it is normal (i.e.,
k
splits completely over
k
) and
every irreducible polynomial
F
(
x
)
∈ k
[
x
] with a root in
k
/
separable. The Galois group of
k
is
k
/
k
→ k
:
σ
is a field automorphism, and
σ
(
x
)
Gal(
k
)
={
σ
:
=
x
for all
x
∈ k}
.
k
/
Theorem A.7.1
Let
be a finite Galois extension. Then there is a one-to-one corre-
spondence between the set of subfields
k
{k
:
k ⊆ k
⊆ k
}
and the set of normal subgroups
k
/
H of
Gal(
k
)
,via
k
={
∈ k
:
σ
(
x
)
x
=
x for all σ
∈
H
}
.
Proof
See Theorem V.2.5 of [
271
].
If
k
is a perfect field then
k
is a separable extension and hence a
G
alois extension of
k
is any alg
eb
raic extension of
k
is Galois. The
k
.If
k
(not necessarily Galois) then
k
/
Galois group Gal(
) can be define
d
using the notion of an inverse limit (see Chapter 5
of [
116
]). Topological aspects of Gal(
k
/
k
k
/
k
) are important, but we do not discuss them.
