Editing Field (mathematics) (section)

== Fields with additional structure ==
Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.

=== Ordered fields ===
{{Main|Ordered field}}
A field ''F'' is called an ''ordered field'' if any two elements can be compared, so that {{math|''x'' + ''y'' ≥ 0}} and {{math|''xy'' ≥ 0}} whenever {{math|''x'' ≥ 0}} and {{math|''y'' ≥ 0}}. For example, the real numbers form an ordered field, with the usual ordering&nbsp;{{math|≥}}. The [[Artin–Schreier theorem]] states that a field can be ordered if and only if it is a [[formally real field]], which means that any quadratic equation
: <math>x_1^2 + x_2^2 + \dots + x_n^2 = 0</math>
only has the solution {{math|1=''x''<sub>1</sub> = ''x''<sub>2</sub> = ⋯ = ''x''<sub>''n''</sub> = 0}}.<ref>{{harvp|Bourbaki|1988|loc=Chapter VI, §2.3, Corollary 1}}</ref> The set of all possible orders on a fixed field {{math|''F''}} is isomorphic to the set of [[ring homomorphism]]s from the [[Witt ring (forms)|Witt ring]] {{math|W(''F'')}} of [[quadratic form]]s over {{math|''F''}}, to {{math|'''Z'''}}.<ref>{{harvp|Lorenz|2008|loc=§22, Theorem 1}}</ref>

An [[Archimedean field]] is an ordered field such that for each element there exists a finite expression
: {{math|1 + 1 + ⋯ + 1}}
whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no [[infinitesimals]] (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of {{math|'''R'''}}.

[[File:Illustration of supremum.svg|thumb|300px|Each bounded real set has a least upper bound.]]
An ordered field is [[Dedekind-complete]] if all [[upper bound]]s, [[lower bound]]s (see ''[[Dedekind cut]]'') and limits, which should exist, do exist. More formally, each [[bounded set|bounded subset]] of {{math|''F''}} is required to have a least upper bound. Any complete field is necessarily Archimedean,<ref>{{harvp|Prestel|1984|loc=Proposition 1.22}}</ref> since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence {{math|1/2, 1/3, 1/4, ...}}, every element of which is greater than every infinitesimal, has no limit.

Since every proper subfield of the reals also contains such gaps, {{math|'''R'''}} is the unique complete ordered field, up to isomorphism.<ref>{{harvp|Prestel|1984|loc=Theorem 1.23}}</ref> Several foundational results in [[calculus]] follow directly from this characterization of the reals.

The [[hyperreals]] {{math|'''R'''<sup>*</sup>}} form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of [[non-standard analysis]].

=== 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>

=== Differential fields ===
[[Differential field]]s are fields equipped with a [[derivation (abstract algebra)|derivation]], i.e., allow to take derivatives of elements in the field.<ref>{{harvp|van der Put|Singer|2003|loc=§1}}</ref> For example, the field {{math|'''R'''(''X'')}}, together with the standard derivative of polynomials forms a differential field. These fields are central to [[differential Galois theory]], a variant of Galois theory dealing with [[linear differential equation]]s.