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The theory that gives a description of all Abelian extensions (finite Galois extensions having Abelian Galois groups) of a field  $  K $
that belongs to one of the following types: 1)  $  K $
is an algebraic number field, i.e. a finite extension of the field  $  \mathbf Q $; 
2)  $  K $
is a finite extension of the field of rational  $  p $-adic numbers  $  \mathbf Q _ {p} $; 
3)  $  K $
is a field of algebraic functions in one variable over a finite field; and 4)  $  K $
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|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  $  L/K $
with Galois group  $  G( L/K  ) $
is put into correspondence with the norm subgroup  $  N _ {L/K} ( L  ^ {*} ) $
of the multiplicative group  $  K  ^ {*} $
of  $  K $. 
The group  $  N _ {L/K} ( L  ^ {*} ) $
completely determines the field  $  L $, 
and there exists a canonical isomorphism  $  \phi :  G( L/K  ) \simeq K  ^ {*} /N _ {L/K} ( L  ^ {*} ) $ (the main isomorphism of class field theory). The theory of formal groups (see [[#References|[1]]]) gives an explicit form of this isomorphism. Conversely, any open subgroup of finite index in  $  K  ^ {*} $
is realized as a norm subgroup for a certain Abelian extension  $  L $ (the existence theorem).

If  $  L $
and  $  L _ {1} $
are finite Abelian extensions of a field  $  K $, 
$  M = L \cap L _ {1} $
and  $  N = L \cdot L _ {1} $, 
then

$$ \tag{1 }
\left .
\begin{array}{c}
N _ {M/K} ( M  ^ {*} )  = \ 
N _ {L/K} ( L  ^ {*} ) N _ {L _ {1}  /K } ( L _ {1}  ^ {*} ) ,
\\

N _ {N/K} ( N  ^ {*} )  = \ 
N _ {L/K} ( L  ^ {*} ) \cap N _ {L _ {1}  /K } ( L _ {1}  ^ {*} ). 
\end{array}

\right \}
$$

The inclusion  $  L _ {1} \supseteq L $
holds if and only if

$$ 
N _ {L/K} ( L  ^ {*} )  \supset  N _ {L _ {1}  /K } ( L _ {1}  ^ {*} ),
$$

and in that case the diagram

$$ \tag{2 }

\begin{array}{ccc}
G( L _ {1} /K  )  &{ \simeq } ^  \phi     &K  ^ {*} /N _ {L _ {1}  /K } ( L  ^ {*} )  \\
{size - 3 \alpha } \downarrow  &{}  &\downarrow {size - 3 \beta }  \\
G( L/K  )  &{ \simeq } _  \phi    &K  ^ {*} /N _ {L/K} ( L  ^ {*} )  \\
\end{array}

$$

is commutative, where  $  \alpha $
is obtained by restricting the automorphism from  $  L _ {1} $
to  $  L $, 
while  $  \beta $
is induced by the identity mapping  $  K  ^ {*} \rightarrow K  ^ {*} $. 
In particular, if  $  K  ^ {ab} $
is the maximal Abelian extension of  $  K $, 
then the Galois group  $  G( K  ^ {ab} /K  ) $
is canonically isomorphic to the profinite completion of the group  $  K  ^ {*} $.

The isomorphism  $  \phi $
also gives a description of the sequence of ramification subgroups in  $  G( L/K  ) $. 
For example, the extension  $  L/K $
is unramified if and only if the group of units  $  U( K  ) $
of  $  K $
is contained in  $  N _ {L/K} ( L  ^ {*} ) $. 
In that case the isomorphism  $  \phi $
is completely determined by the fact that the Frobenius automorphism that generates the group  $  G( L/K  ) $
corresponds to the class  $  \pi \cdot N _ {L/K} ( L  ^ {*} ) $, 
where  $  \pi $
is a prime element of  $  K $.

In the language of group cohomology the isomorphism  $  \phi $
is interpreted as an isomorphism between the Tate cohomology groups:

$$ 
H  ^ {- 2} ( G( L/K  ), \mathbf Z )  \simeq  G( L/K  )
$$

and

$$ 
H  ^ {0} ( G( L/K  ), L  ^ {*} )  =  K  ^ {*} /N _ {L/K} ( L  ^ {*} ).
$$

Moreover, let  $  L/K $
be an arbitrary finite Galois extension of local fields. Then for any integer  $  n $
there is a canonical isomorphism  $  \phi _ {n} $:

$$ 
H  ^ {n- 2} ( G( L/K  ), \mathbf Z )  \simeq  H  ^ {n} ( G( L/K  ), L  ^ {*} ).
$$

If a tower of Galois fields  $  M \supset L \supset K $
is given, then the inflation

$$ 
\inf :  H  ^ {2} ( G( L/K  ), L  ^ {*} )  \rightarrow  H  ^ {2} ( G( M/K  ), M  ^ {*} )
$$

preserves the invariant (see [[Brauer group|Brauer group]]) and the restriction

$$ 
 \mathop{\rm res} :  H  ^ {2} ( G( M/K  ), M  ^ {*} )  \rightarrow  H  ^ {2} ( G( M/L), M  ^ {*} )
$$

multiplies the invariant by  $  [ L :  K] $. 
If  $  \overline{K}\; $
is the separable closure of  $  K $, 
the invariant defines a canonical isomorphism between the Brauer group of  $  K $,

$$ 
 \mathop{\rm Br} ( K  )  \simeq  H  ^ {2} ( G ( \overline{K}\; / K  ), \overline{ {K  ^ {*} }}\; ) ,
$$

and  $  \mathbf Q / \mathbf Z $.

In global class field theory, the role of the multiplicative group is played by the idèle class group (cf. [[Idèle|Idèle]]). Let  $  L/K $
be a finite Galois extension of global fields and let  $  I _ {L} $
be the idèle group of the field  $  L $. 
The group  $  L  ^ {*} $
is imbedded in  $  I _ {L} $
as a discrete subgroup (it is called the group of principal idèles), while the quotient group  $  C _ {L} = I _ {L} / L  ^ {*} $, 
provided with the quotient topology, is called the idèle class group. It can be shown that  $  H  ^ {1} ( G( L/K  ), C _ {L} )= 1 $
and  $  H  ^ {2} ( G( L/K  ), C _ {L} ) \simeq \mathbf Z /n \mathbf Z $, 
where  $  n = [ L :  K] $. 
One has the canonical imbedding  $   \mathop{\rm inv} :  H  ^ {2} ( G( L/K  ), C _ {L} ) \rightarrow \mathbf Q / \mathbf Z $. 
As in local class field theory, for any integer  $  n $
there is an isomorphism (the main isomorphism of global class field theory):

$$ 
\psi _ {n} :  H  ^ {n- 2} ( G( L/K  ), \mathbf Z )  \simeq  H  ^ {n} ( G( L/K  ), C _ {L} ).
$$

For an Abelian extension  $  L/K $, 
the isomorphism  $  \psi _ {0} $
reduces to the isomorphism  $  \psi :  G( L/K  ) \simeq C _ {K} /N _ {L/K} ( C _ {L} ) $. 
The norm subgroup  $  N _ {L/K} ( C _ {L} ) $
uniquely determines the field  $  L $, 
and, conversely, any open subgroup of finite index in  $  C _ {K} $
is a norm subgroup for some finite Abelian extension  $  L $ (the global existence theorem). Relationships analogous to (1) and (2) are also valid for global fields. If  $  K  ^ {ab} $
is the maximal Abelian extension of a field  $  K $, 
then in the function field case the group  $  G( K  ^ {ab} /K  ) $
is isomorphic to the profinite completion of the group  $  C _ {K} $, 
while in the number field case the group  $  G( K  ^ {ab} /K  ) $
is isomorphic to the quotient group of the group  $  C _ {K} $
by the connected component.

The isomorphisms  $  \phi _ {n} $
and  $  \psi _ {n} $
are compatible. If  $  L/K $
is a finite Galois extension of global fields,  $  L _ {v} $
is the completion of  $  L $
with respect to some valuation  $  v $
and  $  K _ {v} $
is the completion of  $  K $
with respect to the restriction of  $  v $
on  $  K $, 
then there exists a commutative diagram

$$ \tag{3 }

\begin{array}{ccc}
H  ^ {n- 2} ( G( L/K  ), \mathbf Z )  & \simeq ^ { {\psi _ n} }   &H  ^ {n} ( G( L/K  ), C _ {L} )  \\
 \mathop{\rm cores} \uparrow  &{}  &\uparrow {size - 3 f }  \\
H  ^ {n- 2} ( G( L _ {v} / K _ {v} ), \mathbf Z )  & \simeq _ {\psi _ {n} }   &H  ^ {n} ( G( L _ {v} / K _ {v} ), L _ {v}  ^ {*} ) ,  \\
\end{array}

$$

where the mapping  $  f $
is induced by the imbedding  $  L _ {v}  ^ {*} \rightarrow I _ {L} \rightarrow C _ {L} $
and the co-restriction mapping cores. For  $  n = 0 $, 
(3) gives the commutative diagram

$$ \tag{4 }

\begin{array}{ccc}
G( L/K  )/[ G( L/K  ), G( L/K  )]  & \simeq ^  \psi     &C _ {K} /N _ {L/K} ( C _ {L} )  \\
\uparrow  &{}  &\uparrow  \\
G( L _ {v} / K _ {v} )/[ G( L _ {v} /K _ {v} ), G( L _ {v} / K _ {v} )]  & \simeq _  \phi    &K _ {v}  ^ {*} / N _ {L _ {v}  /K _ {v} } ( L _ {v}  ^ {*} ).  \\
\end{array}

$$

The diagram (4) enables one to obtain a decomposition law of prime divisors of the field  $  K $
in the Abelian extension  $  L/K $. 
That is, a prime divisor  $  \mathfrak c $
of  $  K $
is unramified (splits completely) in  $  L $
if and only if  $  U( K _ {\mathfrak c }  ) \subset  N _ {L/K} ( C _ {L} ) $(
correspondingly,  $  K  ^ {*} \subset  N _ {L/K} ( C _ {L} ) $).

If  $  \mathfrak c $
is a prime divisor of  $  K $
that is unramified in  $  L $, 
$  v $
is the valuation of  $  K $
corresponding to  $  \mathfrak c $
and  $  \pi $
is a prime element of  $  K _ {v} $, 
then the Artin symbol

$$ 
\left ( \frac{L/K}{\mathfrak c}
 \right )  =  \psi  ^ {- 1} ( \pi )  \in  G( L/K  )
$$

is defined and only depends on  $  \mathfrak c $. 
It is the Frobenius automorphism in the decomposition subgroup of  $  v $. 
According to Chebotarev's density theorem, any element of the group  $  G( L/K  ) $
has the form

$$ 
\left ( \frac{L/K}{\mathfrak c}
 \right )
$$

for an infinite number of prime divisors  $  \mathfrak c $
of  $  K $.

For example, the maximal unramified Abelian extension  $  F $
of a number field  $  K $ (called the Hilbert class field) is a field whose norm subgroup coincides with the image under the projection  $  I _ {K} \rightarrow C _ {K} $
of the group  $  K  ^ {*} \prod _ {v} U( K _ {v} ) $, 
where  $  v $
runs through all points of  $  K $. 
The group  $  I _ {K} /K  ^ {*} \prod _ {v} U( K _ {v} ) $
is canonically isomorphic to the class group  $   \mathop{\rm Cl} _ {K} $
of  $  K $, 
which gives the important isomorphism  $  G( F/K  ) \simeq  \mathop{\rm Cl} _ {K} $. 
In particular, there are no unramified Abelian extensions of  $  K $
if and only if  $  K $
has class number one.

The type of decomposition for a prime divisor  $  \mathfrak c $
of the field  $  K $
in  $  F $
is completely determined by the class of  $  \mathfrak c $
in  $   \mathop{\rm Cl} _ {K} $. 
In particular,  $  \mathfrak c $
splits completely if and only if  $  \mathfrak c $
is principal. All divisors of  $  K $
become principal divisors in  $  F $.

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|Algebraic number theory]]). There are also generalizations of class field theory to the case of infinite Galois extensions [[#References|[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|Tower of fields]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)  pp. Chapt. VI</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,   "Basic number theory" , Springer  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Koch,   "Galoissche Theorie der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022370/c022370147.png" />-Erweiterungen" , Deutsch. Verlag Wissenschaft.  (1970)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.V. Kuz'min,   "Homotopy of profinite groups, the Schur multiplicator and class field theory"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''33''' :  6  (1969)  pp. 1220–1254  (In Russian)</TD></TR></table>

====Comments====
Let  $  A $
be the ring of integers of the global field  $  K $. 
Then the class group  $   \mathop{\rm Cl} _ {K} $
is the [[Divisor class group|divisor class group]] of  $  A $; 
i.e. it is the group of classes of ideals of  $  A $
modulo principal ideals.

The fact that the divisors of a field  $  K $
become principal divisors in its maximal unramified Abelian extension  $  F $
is called the principal ideal theorem.

Two excellent modern up-to-date books on class field theory are [[#References|[a1]]] and [[#References|[a2]]]. The latter also discusses the relations between the idèle-theoretic formulation of class field theory and ray class groups.

The Kronecker–Weber theorem states that every Abelian finite extension of  $  \mathbf Q $
is contained in some  $  \mathbf Q ( \xi _ {n} ) $
where  $  \xi _ {n} $
is a primitive  $  n $-th root of unity (i.e.  $  \xi _ {n}  ^ {n} = 1 $
and  $  \xi _ {n}  ^ {m} \neq 1 $
for  $  m < n $). 
Kronecker also conjectured that every Abelian extension of an imaginary quadratic field  $  \mathbf Q ( \sqrt{ - d} ) $, 
$  d \in \mathbf N $, 
is contained in an extension generated by the torsion points of an elliptic curve with complex multiplication. This was proved by T. Takagi [[#References|[a3]]]. Its analogue for local fields is the Lubin–Tate theorem, stating that the torsion points of a Lubin–Tate formal group over the ring of integers  $  A $
of a local field  $  K $
together with the maximal unramified extension of  $  K $
generate the maximal Abelian extension of  $  K $[[#References|[a4]]]. These formal groups can be used to give very explicit descriptions of the local reciprocity mappings  $  K  ^ {*} \rightarrow  \mathop{\rm Gal} ( K  ^ {ab} / K ) $, 
cf. also [[#References|[a5]]]. The Lubin–Tate formal groups are analogous to elliptic curves with complex multiplication in that they have maximally large endomorphism rings.

The mapping  $  L \rightarrow N _ {L/K} C _ {L} \subset  C _ {K} $
sets up a one-to-one correspondence between the finite Abelian extensions  $  L / K $
and the closed subgroups of finite index in the idèle class group  $  C _ {K} $
of  $  K $ (cf. [[Idèle|Idèle]] for the topology on  $  C _ {K} $). 
This is often called the existence theorem of class field theory. If  $  L / K $
is associated to a subgroup  $  N $
of  $  C _ {K} $, 
then  $  L / K $
is called the class field of  $  N $. 
Let  $  \mathfrak m = \prod _ {\mathfrak p} \mathfrak p ^ {n _ {\mathfrak p} } $
be a formal product of prime divisors of  $  K $
such that  $  n _ {\mathfrak p} \geq  0 $
for all  $  \mathfrak p $, 
$  n _ {\mathfrak p} = 0 $
for almost-all  $  \mathfrak p $
and  $  n _ {\mathfrak p} = 0 $
or 1 for the infinite primes of  $  K $. 
Such a formal product is called a positive divisor or cycle. For each prime divisor  $  \mathfrak p $, 
let  $  K _ {\mathfrak p} $
be the local field obtained by completion with respect to the valuation defined by  $  \mathfrak p $
and let  $  A _ {\mathfrak p} $
be its ring of integers. For each finite prime, let  $  U _ {\mathfrak p}  ^ {n} = \{ {x \in A _ {\mathfrak p} } : {x \equiv 1   \mathop{\rm mod}  \mathfrak p  ^ {n} } \} $
for  $  n > 0 $
and  $  U _ {\mathfrak p}  ^ {0} = U _ {\mathfrak p} = A _ {\mathfrak p}  ^ {*} $, 
the group of units of  $  A _ {\mathfrak p} $. 
In addition, for the infinite primes  $  \mathfrak p $
define  $  U _ {\mathfrak p}  ^ {1} = \mathbf R  ^ {+} $, 
the positive reals, if  $  \mathfrak p $
is real,  $  U _ {\mathfrak p}  ^ {0} = \mathbf R  ^ {*} $
if  $  \mathfrak p $
is real, and  $  U _ {\mathfrak p}  ^ {0} = U _ {\mathfrak p}  ^ {1} = \mathbf C  ^ {*} $
if  $  \mathfrak p $
is complex. Given a positive divisor  $  \mathfrak n $, 
a corresponding subgroup  $  C _ {K}  ^ {\mathfrak n} $
of  $  C _ {K} $
is defined by  $  C _ {k}  ^ {\mathfrak n} = I _ {K}  ^ {\mathfrak n} K  ^ {*} / K  ^ {*} $
where  $  I _ {K}  ^ {\mathfrak n} $
is the subgroup of the group of idèles,  $  I _ {K} $, 
defined by

$$ 
I _ {K}  ^ {\mathfrak n} =
\{ {\alpha \in I _ {K} } : {\alpha _ {\mathfrak p} \in U _ {\mathfrak p} ^ {n _ {\mathfrak p} }  \textrm{ for  all  }  \mathfrak p } \}
$$

The subgroup  $  C _ {K}  ^ {\mathfrak n} $
is called a congruence subgroup; more precisely, it is the congruence subgroup  $   \mathop{\rm mod}  \mathfrak n $
of  $  C _ {K} $. 
The corresponding class field  $  K  ^ {\mathfrak n} / K $, 
i.e. the Abelian extension such that  $  N _ {K  ^ {\mathfrak n}  / K } C _ {K  ^ {\mathfrak n}  } = C _ {K}  ^ {\mathfrak n} $, 
is called the ray class field  $   \mathop{\rm mod}  \mathfrak n $. 
Of particular interest is the ray class field  $   \mathop{\rm mod}  1 $, 
which is the Hilbert class field, since  $  C _ {K} / C _ {K}  ^ {1} $
is clearly isomorphic to the ideal class group of all ideals modulo principal ideals.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Iwasawa,   "Local class field theory" , Oxford Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neukirch,   "Class field theory" , Springer  (1986)  pp. Chapt. 4, Sect. 8</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Takagi,   "Ueber eine Theorie des relativ-abelschen Zahlkörpers"  ''J. Coll. Sci. Imp. Univ. Tokyo'' , '''41'''  (1920)  pp. 1–132</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Lubin,   J. Tate,   "Formal complex multiplication in local fields"  ''Ann. of Math.'' , '''81'''  (1965)  pp. 380–387</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Hazewinkel,   "Local class field theory is easy"  ''Adv. in Math.'' , '''18'''  (1975)  pp. 148–181</TD></TR></table>