Local class field theory

In mathematics, local class field theory, introduced by Hasse (1930), is the study of abelian extensions of local fields; here, "local field means : a field which is complete with respect to a discrete valuation, with a finite residue field. It is the analogue for local fields of global class field theory.

Local class field 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 K^×/N(L^×) of K^× by the norm group N(L^×) of the extension L^× to the Galois group Gal(L/K) of the extension.

The absolute Galois group G of K 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, the group K^× is the product of a compact group with an infinite cyclic group Z. The main topological operation is to replace K^× by its profinite completion, which is roughly the same as replacing the factor Z by its profinite completion Z^{^}. The profinite completion of K^× is 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. There are also two different normalizations of the reciprocity map: in the case of an unramified extension, one of them asks that the (arithmetic) Frobenius element corresponds to the elements of "K" of valuation 1; the other one is the opposit.

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.