# Class field theory

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

(1) |

The inclusion holds if and only if

and in that case the diagram

(2) |

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

(3) |

where the mapping is induced by the imbedding and the co-restriction mapping cores. For , (3) gives the commutative diagram

(4) |

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).

#### References

[1] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) pp. Chapt. VI |

[2] | A. Weil, "Basic number theory" , Springer (1973) |

[3] | H. Koch, "Galoissche Theorie der -Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970) |

[4] | 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) |

#### Comments

Let be the ring of integers of the global field . Then the class group is the divisor class group of ; i.e. it is the group of classes of ideals of modulo principal ideals.

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

Two excellent modern up-to-date books on class field theory are [a1] and [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 is contained in some where is a primitive -th root of unity (i.e. and for ). Kronecker also conjectured that every Abelian extension of an imaginary quadratic field , , is contained in an extension generated by the torsion points of an elliptic curve with complex multiplication. This was proved by T. Takagi [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 of a local field together with the maximal unramified extension of generate the maximal Abelian extension of [a4]. These formal groups can be used to give very explicit descriptions of the local reciprocity mappings , cf. also [a5]. The Lubin–Tate formal groups are analogous to elliptic curves with complex multiplication in that they have maximally large endomorphism rings.

The mapping sets up a one-to-one correspondence between the finite Abelian extensions and the closed subgroups of finite index in the idèle class group of (cf. Idèle for the topology on ). This is often called the existence theorem of class field theory. If is associated to a subgroup of , then is called the class field of . Let be a formal product of prime divisors of such that for all , for almost-all and or 1 for the infinite primes of . Such a formal product is called a positive divisor or cycle. For each prime divisor , let be the local field obtained by completion with respect to the valuation defined by and let be its ring of integers. For each finite prime, let for and , the group of units of . In addition, for the infinite primes define , the positive reals, if is real, if is real, and if is complex. Given a positive divisor , a corresponding subgroup of is defined by where is the subgroup of the group of idèles, , defined by

The subgroup is called a congruence subgroup; more precisely, it is the congruence subgroup of . The corresponding class field , i.e. the Abelian extension such that , is called the ray class field . Of particular interest is the ray class field , which is the Hilbert class field, since is clearly isomorphic to the ideal class group of all ideals modulo principal ideals.

#### References

[a1] | K. Iwasawa, "Local class field theory" , Oxford Univ. Press (1986) |

[a2] | J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, Sect. 8 |

[a3] | T. Takagi, "Ueber eine Theorie des relativ-abelschen Zahlkörpers" J. Coll. Sci. Imp. Univ. Tokyo , 41 (1920) pp. 1–132 |

[a4] | J. Lubin, J. Tate, "Formal complex multiplication in local fields" Ann. of Math. , 81 (1965) pp. 380–387 |

[a5] | M. Hazewinkel, "Local class field theory is easy" Adv. in Math. , 18 (1975) pp. 148–181 |

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Class field theory.

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