## ::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:

- Archimedean local fields (characteristic zero): the real numbers
**R**, and the complex numbers**C**. - Non-archimedean local fields of characteristic zero: finite extensions of the
*p*-adic numbers**Q**_{p}(where*p*is any prime number). - Non-archimedean local fields of characteristic
*p*(for*p*any given prime number): finite extensions of the field of formal Laurent series**F**_{q}((*T*)) over a finite field**F**_{q}(where*q*is a power of*p*).

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**

Intro Induced absolute value Higher-dimensional local fields See also Notes References Further reading External links

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