In mathematics, a local field is a certain type of topological field: by definition, 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. Further, 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. 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.

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

Given a local field F, a "module function" on F 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 module of an element a of F is defined so as to measure the change in size of a set after multiplying it by a. Specifically, define mod_K : F → R by

for any measurable subset X of F (with 0 < μ(X) < ∞). This module does not depend on X nor on the choice of Haar measure μ (since the same scalar multiple ambiguity will occur in both the numerator and the denominator). The function mod_K is continuous and satisfies

for some constant A that only depends on F.

Using mod_K, one may then define an absolute value |.| on F that induces a metric on F (via the standard d(x,y) = |x-y|), such that F is complete with respect to this metric, and the metric induces the given topology on F.

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