Editing Field (mathematics) (section)

=== 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]].