Local classfield theory
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, local class field theory, introduced by , is the study of abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....

s of local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

s. It is the analogue for local fields of global class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

.

Local class field theory gives a description of the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

 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 number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

s Qp 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 pro-finite 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
Class formation
In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. They were invented by Emil Artin and John Tate.-Definitions:...

.

Higher local class field theory

For a higher dimensional local field  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 is an -dimensional local field then one uses or its separated quotient endowed with a suitable topology. When the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if . Higher dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and K. Kato
Kazuya Kato
is a Japanese mathematician. He grew up in the prefecture of Wakayama in Japan. He attended college at the University of Tokyo, from which he also obtained his master's degree in 1975, and his PhD in 1980. He was a professor at Tokyo University, Tokyo Institute of Technology and Kyoto University...

, I. Fesenko
Ivan Fesenko
Ivan Fesenko is a mathematician working in number theory and other areas of mathematics. In 1992 Fesenko won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on class field theory.- Work :...

, Sh. Saito in the general case.

See also

  • Lubin–Tate theory
  • Quasi-finite field
    Quasi-finite field
    In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually...

  • Local Langlands conjectures
    Local Langlands conjectures
    In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between representations of the Weil group of a local field and representations of algebraic groups over the local field, generalizing local class field theory from...

  • Norm group
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