Local field

In mathematics, a field like K is called local field if it is complete with respect to a topology induced by a discrete valuation like v and if its residue field k is finite.[1] Equivalently, local field is a locally compact topological field with respect to a non-discrete topology.[2] Given such a 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.[3] Local fields arise naturally in number theory as completions of global fields.[4]

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:[3]

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.[5] This article uses the former definition.

Given such an absolute value on a field K, the following topology can be defined on K: for a positive real number m, define the subset Bm of K by

Conversely, a topological field with a non-discrete locally compact topology has an absolute value defining its topology. It can be constructed using the Haar measure of the additive group of the field.

For a non-Archimedean local field F (with absolute value denoted by |·|), the following objects are important:

An equivalent and very important definition of a non-Archimedean local field is that it is a field that is complete with respect to a discrete valuation and whose residue field is finite.

The multiplicative group of non-zero elements of a non-Archimedean local field F is isomorphic to

where q is the order of the residue field, and μq−1 is the group of (q−1)st roots of unity (in F). Its structure as an abelian group depends on its characteristic:

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.[9]

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.[5] 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.