The theory that gives a description of all
Abelian extensions
(finite Galois extensions having Abelian Galois groups) of a field
that belongs to one of the following types: 1)
is an algebraic number field, i.e. a finite extension of the field
;
2)
is a finite extension of the field of rational
-adic
numbers
;
3)
is a field of algebraic functions in one variable over a finite field; and 4)
is the field of formal power series over a finite field.
The basic theorems in class field theory were formulated and proved in
particular cases by
L. Kronecker,
H. Weber,
D. Hilbert,
and others (see also
Algebraic number theory).
Fields of the types 2) and 4) are called
local,
while those of types 1) and 3) are called
global.
Correspondingly, one can speak of local and global class field theory.
In local class field theory, each finite Abelian extension
with Galois group
is put into correspondence with the norm subgroup
of the multiplicative group
of
.
The group
completely determines the field
,
and there exists a canonical isomorphism
(the
main isomorphism
of class field theory). The theory of formal groups (see
[1])
gives an explicit form of this isomorphism. Conversely,
any open subgroup of finite index in
is realized as a norm subgroup for a certain Abelian extension
(the
existence theorem).
If
and
are finite Abelian extensions of a field
,
and
, then
The inclusion
holds if and only if
and in that case the diagram
is commutative, where
is obtained by restricting the automorphism from
to
,
while
is induced by the identity mapping
.
In particular, if
is the maximal Abelian extension of
,
then the Galois group
is canonically isomorphic to the profinite completion of the group
.
The isomorphism
also gives a description of the sequence of ramification subgroups in
.
For example, the extension
is unramified if and only if the group of units
of
is contained in
.
In that case the isomorphism
is completely determined by the fact that
the Frobenius automorphism that generates the group
corresponds to the class
,
where
is a prime element of
.
In the language of group cohomology the isomorphism
is interpreted as an isomorphism between the Tate cohomology groups:
and
Moreover, let
be an arbitrary finite Galois extension of local fields. Then for any integer
there is a canonical isomorphism
:
If a tower of Galois fields
is given, then the inflation
preserves the invariant (see
Brauer group)
and the restriction
multiplies the invariant by
.
If
is the separable closure of
,
the invariant defines a canonical isomorphism between the Brauer group of
,
and
.
In global class field theory, the role of the multiplicative
group is played by the idèle class group (cf.
Idèle).
Let
be a finite Galois extension of global fields and let
be the idèle group of the field
.
The group
is imbedded in
as a discrete subgroup (it is called the
group of principal idèles),
while the quotient group
,
provided with the quotient topology, is called the
idèle class group.
It can be shown that
and
,
where
.
One has the canonical imbedding
.
As in local class field theory, for any integer
there is an isomorphism (the
main isomorphism of global class field theory):
 |
For an Abelian extension
,
the isomorphism
reduces to the isomorphism
.
The norm subgroup
uniquely determines the field
,
and, conversely, any open subgroup of finite index in
is a norm subgroup for some finite Abelian extension
(the
global existence theorem).
Relationships analogous to
(1)
and
(2)
are also valid for global fields. If
is the maximal Abelian extension of a field
,
then in the function field case the group
is isomorphic to the profinite completion of the group
,
while in the number field case the group
is isomorphic to the quotient group of the group
by the connected component.
The isomorphisms
and
are compatible. If
is a finite Galois extension of global fields,
is the completion of
with respect to some valuation
and
is the completion of
with respect to the restriction of
on
,
then there exists a commutative diagram
where the mapping
is induced by the imbedding
and the co-restriction mapping cores. For
,
(3)
gives the commutative diagram
The diagram
(4)
enables one to obtain a decomposition law of prime divisors of the field
in the Abelian extension
.
That is, a prime divisor
of
is unramified (splits completely) in
if and only if
(correspondingly,
).
If
is a prime divisor of
that is unramified in
,
is the valuation of
corresponding to
and
is a prime element of
,
then the Artin symbol
is defined and only depends on
.
It is the Frobenius automorphism in the decomposition subgroup of
.
According to
Chebotarev's density theorem,
any element of the group
has the form
for an infinite number of prime divisors
of
.
For example, the maximal unramified Abelian extension
of a number field
(called the
Hilbert class field)
is a field whose norm subgroup coincides with the image under the projection
of the group
,
where
runs through all points of
.
The group
is canonically isomorphic to the class group
of
,
which gives the important isomorphism
.
In particular, there are no unramified Abelian extensions of
if and only if
has class number one.
The type of decomposition for a prime divisor
of the field
in
is completely determined by the class of
in
.
In particular,
splits completely if and only if
is principal. All divisors of
become principal divisors in
.
Just as class field theory for unramified Abelian extensions can be
explained in terms of the divisor class group and its subgroups,
so can arbitrary Abelian extensions be characterized by means
of ray class groups with respect to suitable modules (see
Algebraic number theory).
There are also generalizations of class field theory
to the case of infinite Galois extensions
[4].
Although class field theory arose as a theory on Abelian extensions,
the results give important information also for non-Abelian Galois extensions. For example,
class field theory is used in proving the existence of infinite class field towers (see
Tower of fields).