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 fields: those in which the absolute value 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 are quite well known in mathematics for 250 years and more, 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 to one of the following:
There is an equivalent definition of non-Archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. 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.

Induced absolute value

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
Then, the b+Bm make up a neighbourhood basis of b in K.
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.

Basic features of non-Archimedean local fields

For a non-Archimedean local field F, the following objects are important:
Every non-zero element a of F can be written as a = ϖnu with u a unit, and n a unique integer.
The normalized valuation of F is the surjective function v : F → Z ∪ defined by sending a non-zero a to the unique integer n such that a = ϖnu with u a unit, and by sending 0 to ∞. If q is the cardinality of the residue field, the absolute value on F induced by its structure as a local field is given by
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.

Examples

  1. 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 = n for the normalized valuation.
  2. The formal Laurent series over a finite field: the ring of integers of Fq) is the ring of formal power series FqT. Its maximal ideal is and its residue field is Fq. Its normalized valuation is related to the degree of a formal Laurent series as follows:
  3. ::.
  4. The formal Laurent series over the complex numbers is not a local field. For example, its residue field is CT/ = C, which is not finite.

    Higher unit groups

The nth higher unit group of a non-Archimedean local field F is
for n ≥ 1. The group U is called the group of principal units, and any element of it is called a principal unit. The full unit group is denoted U.
The higher unit groups form a decreasing filtration of the unit group
whose quotients are given by
for n ≥ 1.

Structure of the unit group

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 st roots of unity. 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, explicit formulas for the Hilbert symbol in local class field theory, see e.g.

Higher-dimensional local fields

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 -dimensional local field. Depending on the definition of local field, a zero-dimensional local field is then either a finite field, 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.