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. 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 (as a topological field) 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. 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. A topological field with a non-discrete locally compact topology has an absolute value 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. Define |·| : K → R by 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). For a non-Archimedean local field F (with absolute value denoted by |·|), the following objects are important: