Local Class Field Theory

In mathematics, local class field theory, introduced by Hasse (1930), is the study of abelian extensions of local fields. 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.

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