Editing Field (mathematics) (section)

==== Local fields ====
The following topological fields are called ''[[local field]]s'':<ref>{{harvp|Serre|1979}}</ref>{{efn|Some authors also consider the fields {{math|'''R'''}} and {{math|'''C'''}} to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point that {{harvtxt|Cassels|1986|loc=p. vi}} calls them "completely anomalous".}}
* finite extensions of {{math|'''Q'''<sub>''p''</sub>}} (local fields of characteristic zero)
* finite extensions of {{math|'''F'''<sub>''p''</sub>((''t''))}}, the field of Laurent series over {{math|'''F'''<sub>''p''</sub>}} (local fields of characteristic {{math|''p''}}).

These two types of local fields share some fundamental similarities. In this relation, the elements {{math|''p'' ∈ '''Q'''<sub>''p''</sub>}} and {{math|''t'' ∈ '''F'''<sub>''p''</sub>((''t''))}} (referred to as [[uniformizer]]) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in {{math|'''F'''<sub>''p''</sub>}}. (However, since the addition in {{math|'''Q'''<sub>''p''</sub>}} is done using [[carry (arithmetic)|carry]]ing, which is not the case in {{math|'''F'''<sub>''p''</sub>((''t''))}}, these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper:
* Any [[first-order logic|first-order]] statement that is true for almost all {{math|'''Q'''<sub>''p''</sub>}} is also true for almost all {{math|'''F'''<sub>''p''</sub>((''t''))}}. An application of this is the [[Ax–Kochen theorem]] describing zeros of homogeneous polynomials in {{math|'''Q'''<sub>''p''</sub>}}.
* [[Splitting of prime ideals in Galois extensions|Tamely ramified extension]]s of both fields are in bijection to one another.
* Adjoining arbitrary {{math|''p''}}-power roots of {{math|''p''}} (in {{math|'''Q'''<sub>''p''</sub>}}), respectively of {{math|''t''}} (in {{math|'''F'''<sub>''p''</sub>((''t''))}}), yields (infinite) extensions of these fields known as [[perfectoid field]]s. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields:<ref>{{harvp|Scholze|2014}}</ref> <math display="block">\operatorname {Gal}\left(\mathbf Q_p \left(p^{1/p^\infty} \right) \right) \cong \operatorname {Gal}\left(\mathbf F_p((t))\left(t^{1/p^\infty}\right)\right).</math>