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{{Short description|Algebraic object with an ordered structure}}
In [[Mathematics|mathematics]], an '''ordered field''' is a [[Field (mathematics)|field]] together with a [[Total order|total order]]ing of its elements that is compatible with the field operations.  Basic examples of ordered fields are the [[Rational number|rational numbers]] and the real numbers, both with their standard orderings.

Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is [[Isomorphism|isomorphic]] to the [[Rational number|rational number]]s. Every Dedekind-complete ordered field is isomorphic to the reals. [[Square (algebra)|Squares]] are necessarily non-negative in an ordered field. This implies that the [[Complex number|complex number]]s cannot be ordered since the square of the [[Imaginary unit|imaginary unit]] ''i'' is {{num|−1}} (which is negative in any ordered field).  [[Finite field]]s cannot be ordered.

Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including [[Biography:David Hilbert|David Hilbert]], [[Biography:Otto Hölder|Otto Hölder]] and [[Biography:Hans Hahn (mathematician)|Hans Hahn]]. This grew eventually into the Artin–Schreier theory of ordered fields and [[Formally real field|formally real field]]s.

==Definitions==
There are two equivalent common definitions of an ordered field.  The definition of '''total order''' appeared first historically and is a [[First-order logic|first-order]] axiomatization of the ordering <math>\leq</math> as a binary predicate. Artin and Schreier gave the definition in terms of '''positive cone''' in 1926, which axiomatizes the subcollection of nonnegative elements.  Although the latter is higher-order, viewing positive cones as {{em|maximal}} prepositive cones provides a larger context in which field orderings are {{em|extremal}} partial orderings.

===Total order===
A [[Field (mathematics)|field]] <math>(F, +, \cdot\,)</math> together with a [[Total order#Strict total order|total order]] <math> \leq </math> on <math>F</math> is an '''{{visible anchor|ordered field}}''' if the order satisfies the following properties for all <math>a, b, c \in F:</math>
* if <math>a \leq b</math> then <math>a + c \leq b + c,</math> and
* if <math>0 \leq a</math> and <math>0  \leq b</math> then <math>0 \leq a \cdot b.</math>
As usual, we write <math>a < b</math> for <math>a\le b </math> and <math>a\ne b</math>. The notations <math>b\ge a</math> and <math>b> a</math> stand for <math>a\le b</math> and <math>a<b</math>, respectively. Elements <math>a\in F</math> with <math>a>0</math> are called positive.

===Positive cone===
A '''{{visible anchor|prepositive cone}}''' or '''preordering''' of a field <math>F</math> is a [[Subset|subset]] <math>P \subseteq F</math> that has the following properties:<ref name=Lam289>Lam (2005) p. 289</ref>
* For <math>x</math> and <math>y</math> in <math>P,</math> both <math>x + y</math> and <math>x \cdot y</math> are in <math>P.</math>
* If <math>x \in F,</math> then <math>x^2 \in P.</math>  In particular, <math>0 = 0^2 \in P</math>  and  <math>1 = 1^2 \in P.</math>
* The element <math>- 1</math> is not in <math>P.</math>

A '''{{visible anchor|preordered field}}''' is a field equipped with a preordering <math>P.</math>  Its non-zero elements <math>P^*</math> form a [[Subgroup|subgroup]] of the multiplicative group of <math>F.</math>

If in addition, the set <math>F</math> is the union of <math>P</math> and <math>- P,</math> we call <math>P</math> a '''positive cone''' of <math>F.</math>  The non-zero elements of <math>P</math> are called the '''positive''' elements of <math>F.</math>

An ordered field is a field <math>F</math> together with a positive cone <math>P.</math>

The preorderings on <math>F</math> are precisely the intersections of families of positive cones on <math>F.</math>  The positive cones are the maximal preorderings.<ref name=Lam289/>

===Equivalence of the two definitions===
Let <math>F</math> be a field. There is a bijection between the field orderings of <math>F</math> and the positive cones of <math>F.</math>

Given a field ordering ≤ as in the first definition, the set of elements such that <math>x \geq 0</math> forms a positive cone of <math>F.</math> Conversely, given a positive cone <math>P</math> of <math>F</math> as in the second definition, one can associate a total ordering <math>\leq_P</math> on <math>F</math> by setting <math>x \leq_P y</math> to mean <math>y - x \in P.</math> This total ordering <math>\leq_P</math> satisfies the properties of the first definition.

==Examples of ordered fields==
Examples of ordered fields are:
* the field <math>\Q</math> of [[Rational number|rational number]]s with its standard ordering (which is also its only ordering);
* the field <math>\R</math> of [[Real number|real number]]s with its standard ordering (which is also its only ordering);
* any subfield of an ordered field, such as the real algebraic numbers or the [[Computable number|computable number]]s, becomes an ordered field by restricting the ordering to the subfield;
* the field <math>\mathbb{Q}(x)</math> of rational functions <math>p(x)/q(x)</math>, where <math>p(x)</math> and <math>q(x)</math> are [[Polynomial|polynomial]]s with rational coefficients and <math>q(x) \ne 0</math>, can be made into an ordered field by fixing a real [[Transcendental number|transcendental number]] <math>\alpha</math> and defining <math>p(x)/q(x) > 0</math> if and only if <math>p(\alpha)/q(\alpha) > 0</math>.  This is equivalent to embedding <math>\mathbb{Q}(x)</math> into <math>\mathbb{R}</math> via <math>x\mapsto \alpha</math> and restricting the ordering of <math>\mathbb{R}</math> to an ordering of the image of <math>\mathbb{Q}(x)</math>. In this fashion, we get many different orderings of <math>\mathbb{Q}(x)</math>.
* the field <math>\mathbb{R}(x)</math> of rational functions <math>p(x)/q(x)</math>, where <math>p(x)</math> and <math>q(x)</math> are [[Polynomial|polynomial]]s with real coefficients and <math>q(x) \ne 0</math>, can be made into an ordered field by defining <math>p(x)/q(x) > 0</math> to mean that <math>p_n/q_m > 0</math>, where <math>p_n \neq 0</math> and <math>q_m \neq 0</math> are the leading coefficients of <math>p(x) = p_n x^n + \dots + p_0</math> and <math>q(x) = q_m x^m + \dots + q_0</math>, respectively. Equivalently: for rational functions <math>f(x), g(x)\in \mathbb{R}(x)</math> we have <math>f(x) < g(x)</math> if and only if <math>f(t) < g(t)</math> for all sufficiently large <math>t\in\mathbb{R}</math>. In this ordered field  the polynomial <math>p(x)=x</math> is greater than any constant polynomial and the ordered field is not Archimedean.
* The field <math>\mathbb{R}((x))</math> of [[Formal power series|formal Laurent series]] with real coefficients, where ''x'' is taken to be infinitesimal and positive
* the [[Transseries|transseries]]
* [[Real closed field|real closed field]]s
* the [[Superreal number|superreal number]]s
* the [[Hyperreal number|hyperreal number]]s

The surreal numbers form a [[Class (set theory)|proper class]] rather than a [[Set (mathematics)|set]], but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.

==Properties of ordered fields==
[[File:Invariance of less-than-relation by multiplication with positive number.svg|thumb|The property <math>a > 0 \land x < y \Rightarrow ax < ay</math>]]
[[File:Translation invariance of less-than-relation.svg|thumb|The property <math>x < y \Rightarrow a+x < a+y</math>]]

For every ''a'', ''b'', ''c'', ''d'' in ''F'':
* Either −''a'' ≤ 0 ≤ ''a'' or ''a'' ≤ 0 ≤ −''a''.
* One can "add inequalities": if ''a'' ≤ ''b'' and ''c'' ≤ ''d'', then ''a'' + ''c'' ≤ ''b'' + ''d''.
* One can "multiply inequalities with positive elements": if ''a'' ≤ ''b'' and 0 ≤ ''c'', then ''ac'' ≤ ''bc''.
* "Multiplying with negatives flips an inequality":  if ''a'' ≤ ''b'' and c ≤ 0, then ''ac'' ≥ ''bc''.
* If ''a'' < ''b'' and ''a'', ''b'' > 0, then 1/''b'' < 1/''a''.
* Squares are non-negative: 0 ≤ ''a''<sup>2</sup> for all ''a'' in ''F''. In particular, since 1=1<sup>2</sup>, it follows that 0 ≤ 1. Since 0 ≠ 1, we conclude 0 < 1.
* An ordered field has [[Characteristic (algebra)|characteristic]] 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc., and no finite sum of ones can equal zero.)  In particular, finite fields cannot be ordered.
* {{anchor|nontrivialSquareSum}}Every non-trivial sum of squares is nonzero.  Equivalently: <math>\textstyle \sum_{k=1}^n a_k^2 = 0 \; \Longrightarrow \; \forall k \; \colon a_k = 0 .</math><ref name=Lam41/><ref name=Lam232/>

Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is [[Isomorphism|isomorphic]] to the [[Rational number|rationals]] (as for any other field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves. 

If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be ''[[Archimedean property|Archimedean]]''. Otherwise, such field is a [[Non-Archimedean ordered field|non-Archimedean ordered field]] and contains [[Infinitesimal|infinitesimal]]s. For example, the [[Real number|real number]]s form an Archimedean field, but hyperreal numbers form a non-Archimedean field, because it extends real numbers with elements greater than any standard [[Natural number|natural number]].<ref name="BairHenry">{{cite web | url=http://orbi.ulg.ac.be/bitstream/2268/13591/1/ImplicitDiff.pdf | title=Implicit differentiation with microscopes | publisher=University of Liège | access-date=2013-05-04 |author1=Bair, Jaques |author2=Henry, Valérie }}</ref>

An ordered field ''F'' is isomorphic to the real number field '''R''' if and only if every non-empty subset of ''F'' with an upper bound in ''F'' has a least upper bound in&nbsp;''F''. This property implies that the field is Archimedean.

===Vector spaces over an ordered field===
[[Vector space]]s (particularly, [[Examples of vector spaces#Coordinate space|''n''-spaces]]) over an ordered field exhibit some special properties and have some specific structures, namely: [[Orientation (vector space)|orientation]], [[Convex analysis|convexity]], and [[Inner product space|positively-definite inner product]]. See [[Real coordinate space]] for discussion of those properties of '''R'''<sup>''n''</sup>, which can be generalized to vector spaces over other ordered fields.

==Orderability of fields==
Every ordered field is a [[Formally real field|formally real field]], i.e., 0 cannot be written as a sum of nonzero squares.<ref name=Lam41>Lam (2005) p. 41</ref><ref name=Lam232>Lam (2005) p. 232</ref>

Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order need not be uniquely determined.) The proof uses [[Zorn's lemma]].<ref name=Lam236>Lam (2005) p. 236</ref>

[[Finite field]]s and more generally fields of positive [[Characteristic (algebra)|characteristic]] cannot be turned into ordered fields, as shown above. The [[Complex number|complex number]]s also cannot be turned into an ordered field, as −1 is a square of the imaginary unit ''i''. Also, the ''p''-adic numbers cannot be ordered, since according to [[Hensel's lemma#Examples|Hensel's lemma]] '''Q'''<sub>2</sub> contains a square root of −7, thus 1<sup>2</sup>&nbsp;+&nbsp;1<sup>2</sup>&nbsp;+&nbsp;1<sup>2</sup>&nbsp;+&nbsp;2<sup>2</sup>&nbsp;+&nbsp;{{radic|−7}})<sup>2</sup>&nbsp;=&nbsp;0, and '''Q'''<sub>''p''</sub> (''p''&nbsp;>&nbsp;2) contains a square root of 1&nbsp;−&nbsp;''p'', thus (''p''&nbsp;−&nbsp;1)&sdot;1<sup>2</sup>&nbsp;+&nbsp;({{radic|1&nbsp;−&nbsp;''p''}})<sup>2</sup>&nbsp;=&nbsp;0.<ref>The squares of the square roots {{radic|−7}} and {{radic|1&nbsp;−&nbsp;''p''}} are in '''Q''', but are <&nbsp;0, so that these roots cannot be in '''Q''' which means that their {{nowrap|''p''-adic}} expansions are not periodic.</ref>

==Topology induced by the order==
If ''F'' is equipped with the [[Order topology|order topology]] arising from the total order ≤, then the axioms guarantee that the operations + and × are continuous, so that ''F'' is a topological field.

==Harrison topology==
The '''Harrison topology''' is a topology on the set of orderings ''X''<sub>''F''</sub> of a formally real field ''F''.  Each order can be regarded as a multiplicative group homomorphism from ''F''<sup>∗</sup> onto ±1.  Giving ±1 the discrete topology and ±1<sup>''F''</sup> the [[Product topology|product topology]] induces the [[Subspace topology|subspace topology]] on ''X''<sub>''F''</sub>.  The '''Harrison sets''' <math>H(a) = \{ P \in X_F : a \in P \}</math> form a subbasis for the Harrison topology.  The product is a Boolean space ([[Compact space|compact]], [[Hausdorff space|Hausdorff]] and [[Totally disconnected space|totally disconnected]]), and ''X''<sub>''F''</sub> is a closed subset, hence again Boolean.<ref name=Lam271>Lam (2005) p. 271</ref><ref name=L8312>Lam (1983) pp.&nbsp;1–2</ref>

==Fans and superordered fields==

A '''fan''' on ''F'' is a preordering ''T'' with the property that if ''S'' is a subgroup of index 2 in ''F''<sup>∗</sup> containing ''T''&nbsp;−&nbsp;{0} and not containing&nbsp;−1 then ''S'' is an ordering (that is, ''S'' is closed under addition).<ref name=L8339>Lam (1983) p.&nbsp;39</ref>  A '''superordered field''' is a totally real field in which the set of sums of squares forms a fan.<ref name=L8345>Lam (1983) p.&nbsp;45</ref>

== See also ==

* {{annotated link|Linearly ordered group}}
* {{annotated link|Ordered ring}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Ordered vector space}}
* {{annotated link|Partially ordered ring}}
* {{annotated link|Partially ordered space}}
* {{annotated link|Riesz space}}

==Notes==
{{reflist}}

==References==

* {{citation | last=Lam | first=T. Y. |  title=Orderings, valuations and quadratic forms | series=CBMS Regional Conference Series in Mathematics | volume=52 | publisher=American Mathematical Society | year=1983 | isbn=0-8218-0702-1 | zbl=0516.12001 | url-access=registration | url=https://archive.org/details/orderingsvaluati0000lamt }}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=[[Graduate Studies in Mathematics]] | first=Tsit-Yuen | last=Lam |  publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 }}
* {{Lang Algebra|edition=3}}

{{DEFAULTSORT:Ordered Field}}

[[Category:Real algebraic geometry]]
[[Category:Ordered groups]]

{{Sourceattribution|Ordered field}}