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| In [[mathematics]], an '''ordered field''' is a [[field (mathematics)|field]] together with a [[total order]]ing of its elements that is compatible with the field operations. Historically, the [[axiomatization]] of an ordered field was abstracted gradually from the [[real number]]s, by mathematicians including [[David Hilbert]], [[Otto Hölder]] and [[Hans Hahn (mathematician)|Hans Hahn]]. In 1926, this grew eventually into the [[Artin–Schreier theory]] of ordered fields and [[formally real field]]s.
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| An ordered field necessarily has [[characteristic (algebra)|characteristic]] 0, all [[natural numbers]], i.e. the elements 0, 1, {{nowrap|1 + 1}}, {{nowrap|1 + 1 + 1}}, … are distinct. This implies that an ordered field necessarily contains an infinite number of elements: a [[finite field]] cannot be ordered.
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| 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]]s. Any [[Dedekind-complete]] ordered field is isomorphic to the real numbers. [[square (algebra)|Squares]] are necessarily non-negative in an ordered field. This implies that the [[complex number]]s cannot be ordered since the square of the [[imaginary unit]] ''i'' is {{num|−1}}. Every ordered field is a [[formally real field]].
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| ==Definitions==
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| There are two equivalent definitions of an ordered field. The definition of '''total order''' appeared first historically and is a first-order axiomatization of the ordering ≤ 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 ''maximal'' prepositive cones provides a larger context in which field orderings are ''extremal'' partial orderings.
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| ===Total order===
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| A [[Field_(mathematics)|field]] (''F'', + ,×) together with a [[total order]] ≤ on ''F'' is an '''ordered field''' if the order satisfies the following properties:
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| * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''
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| * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''a''×''b''
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| The symbol for [[multiplication]] will be henceforth omitted.
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| ===Positive cone===
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| A '''prepositive cone''' or '''preordering''' of a field ''F'' is a [[subset]] ''P'' ⊂ ''F'' that has the following properties:<ref name=Lam289>Lam (2005) p. 289</ref>
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| * For ''x'' and ''y'' in ''P'', both ''x''+''y'' and ''xy'' are in ''P''.
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| * If ''x'' is in ''F'', then ''x''<sup>2</sup> is in ''P''.
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| * The element −1 is not in ''P''.
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| A '''preordered field''' is a field equipped with a preordering ''P''. Its non-zero elements ''P''<sup>∗</sup> form a [[subgroup]] of the [[multiplication|multiplicative]] group of ''F''.
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| If in addition, the set ''F'' is the union of ''P'' and −''P'', we call ''P'' a '''positive cone''' of ''F''. The non-zero elements of ''P'' are called the '''positive''' elements of ''F''.
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| An '''ordered field''' is a field ''F'' together with a positive cone ''P''.
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| The preorderings on ''F'' are precisely the intersections of families of positive cones on ''F''. The positive cones are the maximal preorderings.<ref name=Lam289/>
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| === Equivalence of the two definitions ===
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| Let ''F'' be a field. There is a bijection between the field orderings of ''F'' and the positive cones of ''F''.
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| Given a field ordering ≤ as in Def 1, the elements such that ''x'' ≥ 0 forms a positive cone of ''F''. Conversely, given a positive cone ''P'' of ''F'' as in Def 2, one can associate a total ordering ≤<sub>''P''</sub> by setting ''x'' ≤<sub>''P''</sub> ''y'' to mean ''y'' − ''x'' ∈ ''P''. This total ordering ≤<sub>''P''</sub> satisfies the properties of Def 1.
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| === Fan ===
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| 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''-{0} and not containing −1 then ''S'' is an ordering (that is, ''S'' is closed under addition).<ref name=L8339>Lam (1983) p.39</ref>
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| ==Properties of ordered fields==
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| * If ''x'' < ''y'' and ''y'' < ''z'', then ''x'' < ''z''. ([[transitive property|transitivity]])
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| * If ''x'' < ''y'' and ''z'' > 0, then ''xz'' < ''yz''.
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| * If ''x'' < ''y'' and ''x'',''y'' > 0, then 1/''y'' < 1/''x''
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| For every ''a'', ''b'', ''c'', ''d'' in ''F'':
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| * Either −''a'' ≤ 0 ≤ ''a'' or ''a'' ≤ 0 ≤ −''a''.
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| * We are allowed to "add inequalities": If ''a'' ≤ ''b'' and ''c'' ≤ ''d'', then ''a'' + ''c'' ≤ ''b'' + ''d''
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| * We are allowed to "multiply inequalities with positive elements": If ''a'' ≤ ''b'' and 0 ≤ ''c'', then ''ac'' ≤ ''bc''.
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| * 1 is positive. (Proof: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) = 1 is positive, which is a contradiction)
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| * An ordered field has [[characteristic (algebra)|characteristic]] 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the field had characteristic ''p'' > 0, then −1 would be the sum of ''p'' − 1 ones, but −1 is not positive). In particular, finite fields cannot be ordered.
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| * Squares are non-negative. 0 ≤ ''a''<sup>2</sup> for all ''a'' in ''F''. (Follows by a similar argument to 1 > 0)
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| 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]] and contains [[infinitesimal]]s. For example, the [[real number]]s form an Archimedean field, but [[hyperreal numbers]] form a non-Archimedean field, because it [[field extension|extends]] real numbers with elements greater than any standard [[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 Liege]] | accessdate=2013-05-04 | author=Bair, Jaques; Henry, Valérie}}</ref>
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| An ordered field ''K'' is the real number field if it satisfies the [[axiom of Archimedes]] and every non-empty subset of ''K'' with an upper bound in ''K'' has a [[least upper bound]] in ''K''.
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| === Vector spaces over an ordered field ===
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| [[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 #Geometric properties and uses]] for discussion of those properties of '''R'''<sup>''n''</sup>, which can be generalized to vector spaces over other ordered fields.
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| == Examples of ordered fields ==
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| Examples of ordered fields are:
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| * the [[rational number]]s
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| * the real [[algebraic numbers]]
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| * the [[computable number]]s
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| * the [[real number]]s
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| * the field of real [[rational functions]] <math>\frac {p(x)} {q(x)}\,</math>, where ''p''(''x'') and ''q''(''x''), <math>q(x) \ne 0\,</math> are [[polynomial]]s with real coefficients, can be made into an ordered field where the polynomial ''p''(''x'') = ''x'' is greater than any constant polynomial, by defining that <math>\frac {p(x)} {q(x)} > 0\,</math> whenever <math>\frac {p_0} {q_0} > 0\,</math>, for <math>p(x) = p_0 x^n + \cdots</math> and <math>q(x) = q_0 x^m + \cdots\,</math>. This ordered field is not Archimedean.
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| * The field of [[formal power series|formal Laurent series]] with real coefficients <math>\Bbb{R}((x))</math>, where ''x'' is taken to be infinitesimal and positive
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| * [[real closed field]]s
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| * [[superreal number]]s
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| * [[hyperreal number]]s
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| 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.
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| ==Which fields can be ordered?==
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| Every ordered field is a [[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>
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| 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.)<ref name=Lam236>Lam (2005) p. 236</ref>
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| [[Finite field]]s and more generally fields of finite [[Characteristic (algebra)|characteristic]] cannot be turned into ordered fields, because in characteristic ''p'', the element −1 can be written as a sum of (''p'' − 1) squares 1<sup>2</sup>. The [[complex number]]s also cannot be turned into an ordered field, as −1 is a square (of the imaginary number ''i'') and would thus be positive. Also, the [[p-adic numbers]] cannot be ordered, since '''Q'''<sub>2</sub> contains a square root of −7 and '''Q'''<sub>''p''</sub> (''p'' > 2) contains a square root of 1 − ''p''.
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| == Topology induced by the order ==
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| If ''F'' is equipped with the [[order topology]] arising from the total order ≤, then the axioms guarantee that the operations + and × are [[continuous function (topology)|continuous]], so that ''F'' is a [[topological field]].
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| ==Harrison topology==
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| 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]] induces the [[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.1-2</ref>
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| ==Superordered fields==
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| 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.45</ref>
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| ==See also==
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| * [[Ordered ring]]
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| * [[Ordered vector space]]
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| * [[Preorder field]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation | last=Lam | first=T. Y. | authorlink=Tsit Yuen Lam | 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 }}
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| * {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 }}
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| * {{Lang Algebra|edition=3}}
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| {{DEFAULTSORT:Ordered Field}}
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| [[Category:Ordered algebraic structures]]
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| [[Category:Ordered groups]]
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| [[Category:Real algebraic geometry]]
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