Editing Local field (section)

{{Short description|Locally compact topological field}}
In [[mathematics]], a [[Field (mathematics)|field]] ''K'' is called a non-Archimedean '''local field''' if it is [[Complete metric space|complete]] with respect to a [[Metric space|metric]] induced by a [[discrete valuation]] ''v'' and if its [[residue field]] ''k'' is finite.{{sfn|Cassels|Fröhlich|1967|loc=Ch. VI, Intro.|p=129}} In general, a local field is a [[locally compact]] [[topological field]] with respect to a [[Discrete space|non-discrete topology]].{{sfn|Weil|1995|p=20}} The [[real numbers]] '''R''', and the [[complex numbers]] '''C''' (with their standard topologies) are Archimedean local fields. Given a local field, the [[Valuation (algebra)|valuation]] defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is [[Archimedean property|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'''.{{sfn|Milne|2020|loc=Remark 7.49|p=127}} Local fields arise naturally in [[number theory]] as completions of [[global field]]s.{{sfn|Neukirch|1999|loc=Sec. 5|p=134}}

While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local fields, the fields of [[p-adic number|''p''-adic number]]s 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:{{sfn|Milne|2020|loc=Remark 7.49|p=127}}

*Archimedean local fields ([[Characteristic (algebra)|characteristic]] zero): the [[real numbers]] '''R''', and the [[complex numbers]] '''C'''.
*Non-Archimedean local fields of characteristic zero: [[finite extension]]s of the [[p-adic number|''p''-adic number]]s '''Q'''<sub>''p''</sub> (where ''p'' is any [[prime number]]).
*Non-Archimedean local fields of characteristic ''p'' (for ''p'' any given prime number): the field of [[formal Laurent series]] '''F'''<sub>''q''</sub>((''T'')) over a [[finite field]] '''F'''<sub>''q''</sub>, where ''q'' is a [[Exponentiation|power]] of ''p''.

In particular, of importance in number theory, classes of local fields show up as the completions of [[algebraic number field]]s with respect to their discrete valuation corresponding to one of their [[maximal ideal]]s. Research papers in modern number theory often consider a more general notion, requiring only that the residue field be [[Perfect field|perfect]] of positive characteristic, not necessarily finite.{{sfn|Fesenko|Vostokov|2002|loc=Def. 1.4.6}} This article uses the former definition.