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

Given a local field, an 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. Non-Archimedean local fields can also be defined as those fields which are complete with respect to a metric induced by a discrete valuation whose residue field is finite.

Every local field is isomorphic (as a topological field) to one of the following:

• Archimedean local fields (characteristic zero): the real numbers, and the complex numbers.

• Non-Archimedean local fields of characteristic zero: finite extensions of the p-adic numbers is any prime number).

• of formal Laurent series over a finite field is a power.

Module, absolute value, metric

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 by

for any measurable is continuous and satisfies

.

that induces a metric.

Basic features of non-Archimedean local fields

, the following objects are important:

• its ring of integers which is a discrete valuation ring, is the closed unit ball, and is compact;

  the units which form a group and is the unit sphere;

  the unique non-zero prime ideal;

  a generator called a uniformizer; and

  which is finite (since it is compact and discrete).

a unique integer. The normalized valuation is the surjective function is the cardinality 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

• p-adic numbers for the normalized valuation.

  Formal Laurent series over a finite field is the ring of formal power series (i.e. the set of power series whose constant terms. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:

• is non-zero.

• of formal Laurent series over the complex numbers is not, which is not finite.

Higher unit groups

The n-th higher unit group is

is called the group of principal units.

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

whose quotients are given by

." means a non-canonical isomorphism.)

Structure of the unit group

is isomorphic to

. Its structure as an abelian group depends on its characteristic:

• , then

• denotes the natural numbers;

• , then

• .

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.

Variant definitions

The definition for "local field" adopted in this article, as a locally compact Hausdorff non-discrete topological field, is common today. Some authors however reserve the term "local field" for what we have called "non-Archimedean local field".

Research papers in modern number theory often consider a more general notion of non-Archimedean local field, requiring only that they be complete with respect to a discrete valuation and that the residue field be perfect of positive characteristic, not necessarily finite.

In his book Local Fields, Serre defines "local fields" as fields that are complete with respect to a discrete valuation, without any restriction on the residue field, leading to a notion that is more general still.

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-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.

-dimensional local fields with last finite residue field are naturally associated to a complete flag-dimensional arithmetic scheme.

See also

• Hensel's lemma

• Ramification group

• Local class field theory

• Higher local field

Citations

References

External links