## ::Local field

### ::concepts

Field::local    ''F''::''p''    Group::finite    Times::residue    ''n''::''q''    Number::value

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.<ref>Page 20 of {{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> Given such a field, an absolute value can be defined on it. There are two basic types of local field: 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.

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. However, some authors consider a more general notion, requiring only that the residue field be perfect, not necessarily finite.<ref>See, for example, definition 1.4.6 of {{#invoke:Footnotes|harvard_citation_no_bracket}}</ref> This article uses the former definition.

Local field sections