Class Field Theory - Formulation in Contemporary Language

Formulation in Contemporary Language

In modern language there is a maximal abelian extension A of K, which will be of infinite degree over K; and associated to A a Galois group G which will be a pro-finite group, so a compact topological group, and also abelian. The central aim of the theory is to describe G in terms of K. In particular to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in an appropriate object for K, such as the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields, as well as to describe those norm groups directly, e.g., such as open subgroups of finite index. The finite abelian extension corresponding to such a subgroup is called a class field, which gave the name to the theory.

The fundamental result of class field theory states that the group G is naturally isomorphic to the profinite completion of the idele class group C_K of K with respect to the natural topology on C_K related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism

Gal(L / K)ab → C_K / N_L/K C_L

of the maximal abelian quotient of the Galois group of the extension with the quotient of the idele class group of K by the image of the norm of the idele class group of L.

For some small fields, such as the field of rational numbers or its quadratic imaginary extensions there is a more detailed theory which provides more information. For example, the abelianized absolute Galois group G of is (naturally isomorphic to) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity. This is known as the Kronecker–Weber theorem, originally conjectured by Leopold Kronecker. In this case the reciprocity isomorphism of class field theory (or Artin reciprocity map) also admits an explicit description due to the Kronecker–Weber theorem. Let us denote with

the group of all roots of unity, i.e. the torsion subgroup. The Artin reciprocity map is given by

when it is arithmetically normalized, or given by

if it is geometrically normalized. However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory.

The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field. The latter property is called the global reciprocity law and is a far reaching generalization of the Gauss quadratic reciprocity law.

One of methods to construct the reciprocity homomorphism uses class formation.

There are methods which use cohomology groups, in particular the Brauer group, and there are methods which do not use cohomology groups and are very explicit and good for applications.