Local class field theory

In mathematics, local class field theory is the study in number theory of abelian extensions of local fields. It is also related global class field theory.

The basic theory gives a description of the Galois group G of the maximal abelian extension of a local field K via the reciprocity map which acts from the multiplicative group K^×=K\{0}. For a finite abelian extension L of K the reciprocity map induces an isomorphism of the quotient group of K^× by the norm group of the extension and the Galois group of the extension.

The group G is compact and the group K^× is not compact. Taking the case where K is a finite extension of the p-adic numbers Q_p or formal power series over a finite field, we can say more precisely that K^× is the product of a compact group with an infinite cyclic group. The main topological operation is to replace the infinite cyclic group by its pro-finite completion Z^{^}. This is then the group isomorphic with G via the local reciprocity map.

The actual isomorphism used and the existence theorem is described in the theory of the norm residue symbol. There are several different approaches to the theory, using central division algebras or Tate cohomology or an explicit description of the reciprocity map.

For a description of the general case of local class field theory see class formation.

Higher local class field theory

For a higher dimensional local field $K$ there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the Milnor K-group of the field. Namely, if $K$ is an $n$ -dimensional local field then one uses $\mathrm {K} _{n}^{\mathrm {M} }(K)$ or its separated quotient endowed with a suitable topology. When $n=1$ the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if $n>1$ . Higher dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and K. Kato, I. Fesenko, Sh. Saito in the general case.