Editing Local field (section)

== Higher-dimensional local fields ==
{{main|Higher local field}}
A local field is sometimes called a ''one-dimensional local field''.

A non-Archimedean local field can be viewed as the field of fractions of the completion of the [[local ring]] of a one-dimensional arithmetic scheme of rank 1 at its non-singular point.

For a [[non-negative integer]] ''n'', an ''n''-dimensional local field is a complete discrete valuation field whose residue field is an  (''n'' − 1)-dimensional local field.{{sfn|Fesenko|Vostokov|2002|loc=Def. 1.4.6}} Depending on the definition of local field, a ''zero-dimensional local field'' is then either a finite field (with the definition used in this article), or a perfect field of positive characteristic.

From the geometric point of view, ''n''-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an ''n''-dimensional arithmetic scheme.