Жергілікті ықшам топологиялық өріс

Locally compact topological field
In mathematics, a field K is called a (non Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non discrete topology. Sometimes, the real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non Archimedean local fields, the fields of p adic numbers for positive prime integer p, were introduced by Kurt Hensel at the end of the 19th century. Every local field is isomorphic (as a topological field) to one of the following:

Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C.
Non Archimedean local fields of characteristic zero: finite extensions of the p adic numbers Qp (where p is any prime number). Non Archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq, where q is a power of p.

In particular, of importance in number theory, classes of local fields show up as the completions of algebraic number fields with respect to their discrete valuation corresponding to one of their maximal ideals. Research papers in modern number theory often consider a more general notion, requiring only that the residue field be perfect of positive characteristic, not necessarily finite. This article uses the former definition.

The p adic numbers: the ring of integers of Qp is the ring of p adic integers Zp. Its prime ideal is pZp and its residue field is Z/pZ. Every non zero element of Qp can be written as u pn where u is a unit in Zp and n is an integer, then v(u pn) = n for the normalized valuation. The formal Laurent series over a finite field: the ring of integers of Fq((T)) is the ring of formal power series Fq[[T]]. Its maximal ideal is (T) (i. e. the power series whose constant term is zero) and its residue field is Fq. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:
(where a−m is non zero). The formal Laurent series over the complex numbers is not a local field. For example, its residue field is C[[T]]/(T) = C, which is not finite.

This theory includes the study of types of local fields, extensions of local fields using Hensel's lemma, Galois extensions of local fields, ramification groups filtrations of Galois groups of local fields, the behavior of the norm map on local fields, the local reciprocity homomorphism and existence theorem in local class field theory, local Langlands correspondence, Hodge Tate theory (also called p adic Hodge theory), explicit formulas for the Hilbert symbol in local class field theory, see e. g.

A local field is sometimes called a one dimensional local field. A non Archimedean local field can be viewed as the field of fractions of the completion of the local ring of a one dimensional arithmetic scheme of rank 1 at its non singular point. For a non negative integer n, an n dimensional local field is a complete discrete valuation field whose residue field is an (n − 1) dimensional local field. Depending on the definition of local field, a zero dimensional local field is then either a finite field (with the definition used in this article), or a perfect field of positive characteristic. From the geometric point of view, n dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n dimensional arithmetic scheme.