Top Qs
Timeline
Chat
Perspective

Local field

Locally compact topological field From Wikipedia, the free encyclopedia

Remove ads

In mathematics, a local field is a certain type of field: a locally compact Hausdorff non-discrete topological field.[1] Local fields find many applications in algebraic number theory, where they arise naturally as completions of global fields.[2] Further, tools like integration and Fourier analysis are available for functions defined on local fields.

Given a local field, a natural absolute value can be defined on it which gives rise to a complete metric that generates its topology. There are two basic types of local field: those called Archimedean local fields in which the absolute value is Archimedean, and those called non-Archimedean local fields in which it is not. The non-Archimedean local fields can also be defined as those fields which are complete with respect to a metric induced by a discrete valuation v whose residue field is finite.[3]

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:[4]

Research papers in modern number theory often consider a more general notion of non-Archimedean local field, requiring only that the residue field be perfect of positive characteristic, not necessarily finite.[5] This article uses the former definition.

Remove ads

Induced absolute value

Given a local field K, an absolute value on K can be defined as follows. First, consider the additive group of the field. As a locally compact topological group, it has a unique (up to positive scalar multiple) Haar measure μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of K. Specifically, define |·| : KR by[6]

for any measurable subset X of K (with 0 < μ(X) < ). This absolute value does not depend on X nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the numerator and the denominator).

This absolute value induces a metric on K (via the standard d(x,y) = |x-y|), K is complete with respect to this metric, and the metric induces the given topology on K.

Remove ads

Basic features of non-Archimedean local fields

Summarize
Perspective

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

  • its ring of integers which is a discrete valuation ring, is the closed unit ball of F, and is compact;
  • the units in its ring of integers which forms a group and is the unit sphere of F;
  • the unique non-zero prime ideal in its ring of integers which is its open unit ball ;
  • a generator of called a uniformizer of ;
  • its residue field which is finite (since it is compact and discrete).

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 : FZ ∪ {∞} 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:[7]

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, with v(u pn) = n for the normalized valuation.
  2. 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 set of power series whose constant terms are zero) and its residue field is Fq. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:
    (where am is non-zero).
  3. 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.

Higher unit groups

The nth higher unit group of a non-Archimedean local field F is

for n  1. The group U(1) is called the group of principal units, and any element of it is called a principal unit. The full unit group is denoted U(0).

The higher unit groups form a decreasing filtration of the unit group

whose quotients are given by

for n  1.[8] (Here "" means a non-canonical isomorphism.)

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 (q−1)st roots of unity (in F). Its structure as an abelian group depends on its characteristic:

  • If F has positive characteristic p, then
where N denotes the natural numbers;
  • If F has characteristic zero (i.e. it is a finite extension of Qp of degree d), then
where a  0 is defined so that the group of p-power roots of unity in F is .[9]
Remove ads

Theory of local fields

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.[10]

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 (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.

Remove ads

See also

Citations

References

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads