Editing Field (mathematics) (section)

=== Topological fields ===
Another refinement of the notion of a field is a '''topological field''', in which the set {{math|''F''}} is a [[topological space]], such that all operations of the field (addition, multiplication, the maps {{math|''a'' ↦ −''a''}} and {{math|''a'' ↦ ''a''<sup>−1</sup>}}) are [[continuous map]]s with respect to the topology of the space.<ref>{{harvp|Warner|1989|loc=Chapter 14}}</ref>
The topology of all the fields discussed below is induced from a [[metric (mathematics)|metric]], i.e., a [[function (mathematics)|function]]
: {{math|''d'' : ''F'' × ''F'' → '''R''',}}
that measures a ''distance'' between any two elements of {{math|''F''}}.

The [[completion (metric space)|completion]] of {{math|''F''}} is another field in which, informally speaking, the "gaps" in the original field {{math|''F''}} are filled, if there are any. For example, any [[irrational number]] {{math|''x''}}, such as {{math|1=''x'' = {{radic|2}}}}, is a "gap" in the rationals {{math|'''Q'''}} in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers {{math|''p''/''q''}}, in the sense that distance of {{math|''x''}} and {{math|''p''/''q''}} given by the [[absolute value]] {{math|{{abs|''x'' − ''p''/''q''}}}} is as small as desired.
The following table lists some examples of this construction. The fourth column shows an example of a zero [[sequence]], i.e., a sequence whose limit (for {{math|''n'' → ∞}}) is zero.

{| class="wikitable"
! Field !! Metric !! Completion !! zero sequence
|-
| {{math|'''Q'''}} || {{math|{{abs|''x'' − ''y''}}}} (usual [[absolute value]]) || '''R''' || {{math|1/''n''}}
|-
| {{math|'''Q'''}}
|| obtained using the [[p-adic valuation|''p''-adic valuation]], for a prime number {{math|''p''}}
|| {{math|'''Q'''<sub>''p''</sub>}} ([[p-adic number|{{math|''p''}}-adic numbers]])
|| {{math|''p''<sup>''n''</sup>}}
|-
| {{math|''F''(''t'')}}<br /> ({{math|''F''}} any field)
|| obtained using the {{math|''t''}}-adic valuation
|| {{math|''F''((''t''))}}
|| {{math|''t''<sup>''n''</sup>}}
|}

The field {{math|'''Q'''<sub>''p''</sub>}} is used in number theory and [[p-adic analysis|{{math|''p''}}-adic analysis]]. The algebraic closure {{math|{{overline|'''Q'''}}<sub>''p''</sub>}} carries a unique norm extending the one on {{math|'''Q'''<sub>''p''</sub>}}, but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of [[Metric completions and algebraic closures|complex ''p''-adic number]]s and is denoted by {{math|'''C'''<sub>''p''</sub>}}.<ref>{{harvp|Gouvêa|1997|loc=§5.7}}</ref>

==== 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>