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| {{dablink|"Artin–Schreier theorem" redirects here. For the branch of Galois theory, see [[Artin–Schreier theory]].}}
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| In [[mathematics]], a '''real closed field''' is a [[Field (mathematics)|field]] ''F'' that has the same first-order properties as the field of [[real numbers]]. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of [[hyperreal number]]s.
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| ==Definitions==
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| A real closed field is a field ''F'' in which any of the following equivalent conditions are true:
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| #''F'' is [[elementarily equivalent]] to the real numbers. In other words it has the same first-order properties as the reals: any sentence in the first-order language of fields{{clarify|reason=The article ((Field_(mathematics)#Alternative_axiomatizations)) lists axiomatizations with different signatures; Chang and Keisler (exm.1.4.9, p.40-41) use +,*,0,1; clarify which signature is meant, or state that the choice doesn't matter.|date=July 2013}} is true in ''F'' if and only if it is true in the reals.
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| #There is a [[total order]] on ''F'' making it an [[ordered field]] such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any [[polynomial]] of odd [[degree of a polynomial|degree]] with [[coefficients]] in ''F'' has at least one [[Root of a function|root]] in ''F''.
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| #''F'' is a [[formally real field]] such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' of ''F'' there is ''b'' in ''F'' such that ''a'' = ''b''<sup>2</sup> or ''a'' = −''b''<sup>2</sup>.
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| #''F'' is not [[algebraically closed]] but its algebraic closure is a [[finite extension]].
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| #''F'' is not algebraically closed but the [[field extension]] <math>F(\sqrt{-1})</math> is algebraically closed.
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| #There is an ordering on ''F'' which does not extend to an ordering on any proper [[algebraic extension]] of ''F''.
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| #''F'' is a formally real field such that no proper algebraic extension of ''F'' is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.)
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| #There is an ordering on ''F'' making it an ordered field such that, in this ordering, the [[intermediate value theorem]] holds for all polynomials over ''F''.
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| #''F'' is a [[real closed ring]].
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| If ''F'' is an ordered field (not just orderable, but a definite ordering ''P'' is fixed as part of the structure), the '''Artin–Schreier theorem''' states that ''F'' has an algebraic extension, called the '''real closure''' ''K'' of ''F'', such that ''K'' is a real closed field whose ordering is an extension of the given ordering ''P'' on F, and is unique up to a unique isomorphism of fields<ref>Rajwade (1993) pp.222–223</ref> (note that every [[ring homomorphism]] between real closed fields automatically is [[order isomorphism|order preserving]], because ''x'' ≤ ''y'' if and only if ∃''z'' ''y'' = ''x''+''z''<sup>2</sup>). For example, the real closure of the rational numbers is the field <math>\mathbb{R}_{alg}</math> of real [[algebraic number]]s. The theorem is named for [[Emil Artin]] and [[Otto Schreier]], who proved it in 1926.
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| If (''F'',''P'') is an ordered field, and ''E'' is a [[Galois extension]] of ''F'', then by [[Zorn's Lemma]] there is a maximal ordered field extension (''M'',''Q'') with ''M'' a subfield of ''E'' containing ''F'' and the order on ''M'' extending ''P'': ''M'' is the '''relative real closure''' of (''F'',''P'') in ''E''. We call (''F'',''P'') '''real closed relative to''' ''E'' if ''M'' is just ''F''. When ''E'' is the [[algebraic closure]] of ''F'' we recover the definitions above.<ref name=Efr177>Efrat (2006) p.177</ref>
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| If ''F'' is a field (so this time, no order is fixed, and it is even not necessary to assume that ''F'' is orderable) then ''F'' still has a real closure, which in general is not a field anymore, but a
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| [[real closed ring]]. For example the real closure of the field <math>\mathbb{Q}(\sqrt 2)</math> is the ring <math>\mathbb{R}_{alg}\times \mathbb{R}_{alg}</math> (the two copies correspond to the two orderings of <math>\mathbb{Q}(\sqrt 2)</math>). Whereas the real closure of the '''ordered''' subfield <math>\mathbb{Q}(\sqrt 2)</math>
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| of <math>\mathbb{R}</math> is again the field <math>\mathbb{R}_{alg}</math>.
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| ==Model theory: decidability and quantifier elimination==
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| The theory of real closed fields was invented by algebraists, but taken up with enthusiasm by logicians. By adding to the [[ordered field]] axioms
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| *an axiom asserting that every positive number has a square root, and
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| *an axiom scheme asserting that all [[polynomial]]s of odd degree have at least one [[root of a function|root]],
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| one obtains a [[first-order theory]]. Tarski (1951) proved that the theory of real closed fields in the first order language of [[Partially ordered_ring|partially ordered rings]] (consisting of the [[binary predicate]] symbols "=" and "≤", the operations of addition, subtraction and multiplication and the constant symbols 0,1) admits [[elimination of quantifiers]]. The most important [[model theory|model theoretic]] consequences hereof: The theory of real closed fields is [[Complete_theory|complete]], [[o-minimal]] and [[decidability (logic)|decidable]].
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| Decidability means that there exists at least one [[decision procedure]], i.e., a well-defined algorithm for determining whether a sentence in the first order language of real closed fields is true. [[Euclidean geometry]] (without the ability to measure angles) is also a [[model theory|model]] of the real field axioms, and thus is also decidable.
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| The decision procedures are not necessarily ''practical''. The [[algorithmic complexity|algorithmic complexities]] of all known decision procedures for real closed fields are very high, so that practical execution times can be prohibitively high except for very simple problems. | |
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| The algorithm Tarski proposed for [[quantifier elimination]] has [[NONELEMENTARY]] complexity, meaning that no tower <math>2^{2^{\cdot^{\cdot^{\cdot^n}}}}</math> can bound the execution time of the algorithm if ''n'' is the size of the problem. Davenport and Heintz (1988) proved that quantifier elimination is in fact (at least) [[Double exponential function|doubly exponential]]: there exists a family Φ<sub>n</sub> of formulas with ''n'' quantifiers, of length ''O''(''n'') and constant degree such that any quantifier-free formula equivalent to Φ<sub>n</sub> must involve polynomials of degree <math>2^{2^{\Omega(n)}}</math> and length <math>2^{2^{\Omega(n)}}</math>, using [[Big O notation#Related asymptotic notations: O.2C o.2C .CE.A9.2C .CF.89.2C .CE.98.2C .C3.95|the Ω asymptotic notation]]. Ben-Or, Kozen, and Reif (1986) proved that the theory of real closed fields is decidable in [[EXPSPACE|exponential space]], and therefore in doubly exponential time.
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| Basu and [[Marie-Françoise Roy|Roy]] (1996) proved that there exists a well-behaved algorithm to decide the truth of a formula ∃x<sub>1</sub>,…,∃x<sub>k</sub> P<sub>1</sub>(x<sub>1</sub>,…,x<sub>k</sub>)⋈0∧…∧P<sub>s</sub>(x<sub>1</sub>,…,x<sub>k</sub>)⋈0 where ⋈ is <, > or =, with complexity in arithmetic operations s<sup>k+1</sup>d<sup>O(k)</sup>. In fact, the [[existential theory of the reals]] can be decided in [[PSPACE]].
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| == Order properties ==
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| A crucially important property of the real numbers is that it is an [[Archimedean field]], meaning it has the Archimedean property that for any real number, there is an integer larger than it in [[absolute value]]. An equivalent statement is that for any real number, there are integers both larger and smaller. Such real closed fields that are not Archimedean, are [[non-Archimedean ordered field]]s. For example, any field of [[hyperreal numbers]] is real closed and non-Archimedean.
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| The Archimedean property is related to the concept of [[cofinality]]. A set X contained in an ordered set F is cofinal in F if for every y in F there is an x in X such that y < x. In other words, X is an unbounded sequence in F. The cofinality of F is the size of the smallest cofinal set, which is to say, the size of the smallest cardinality giving an unbounded sequence. For example natural numbers are cofinal in the reals, and the cofinality of the reals is therefore <math>\aleph_0</math>.
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| We have therefore the following invariants defining the nature of a real closed field F:
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| * The cardinality of F.
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| * The cofinality of F.
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| To this we may add
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| * The weight of F, which is the minimum size of a dense subset of F.
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| These three cardinal numbers tell us much about the order properties of any real closed field, though it may be difficult to discover what they are, especially if we are not willing to invoke [[continuum hypothesis|generalized continuum hypothesis]]. There are also particular properties which may or may not hold:
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| * A field F is '''complete''' if there is no ordered field K properly containing F such that F is dense in K. If the cofinality of K is κ, this is equivalent to saying Cauchy sequences indexed by κ are convergent in F.
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| * An ordered field F has the η<sub>α</sub> property for the ordinal number α if for any two subsets L and U of F of cardinality less than <math>\aleph_\alpha</math>, at least one of which is nonempty, and such that every element of L is less than every element of U, there is an element x in F with x larger than every element of L and smaller than every element of U. This is closely related to the model-theoretic property of being a [[saturated model]]; any two real closed fields are η<sub>α</sub> if and only if they are <math>\aleph_\alpha</math>-saturated, and moreover two η<sub>α</sub> real closed fields both of cardinality <math>\aleph_\alpha</math> are order isomorphic.
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| == The generalized continuum hypothesis ==
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| The characteristics of real closed fields become much simpler if we are willing to assume the [[continuum hypothesis|generalized continuum hypothesis]]. If the continuum hypothesis holds, all real closed fields with cardinality the continuum and having the η<sub>1</sub> property are order isomorphic. This unique field Ϝ can be defined by means of an [[ultraproduct|ultrapower]], as <math>\Bbb{R}^{\Bbb{N}}/{\mathbf M}</math>, where '''M''' is a maximal ideal not leading to a field order-isomorphic to <math>\Bbb{R}</math>. This is the most commonly used [[hyperreal numbers|hyperreal number field]] in [[non-standard analysis]], and its uniqueness is equivalent to the continuum hypothesis. (Even without the continuum hypothesis we have that if the cardinality of the continuum is
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| <math>\aleph_\beta</math> then we have a unique η<sub>β</sub> field of size η<sub>β</sub>.) | |
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| Moreover, we do not need ultrapowers to construct Ϝ, we can do so much more constructively as the subfield of series with a countable number of nonzero terms of the field <math>\Bbb{R}((G))</math> of [[formal power series]] on the [[Sierpiński set|Sierpiński group]].
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| Ϝ however is not a complete field; if we take its completion, we end up with a field Κ of larger cardinality. Ϝ has the cardinality of the continuum which by hypothesis is <math>\aleph_1</math>, Κ has cardinality <math>\aleph_2</math>, and contains Ϝ as a dense subfield. It is not an ultrapower but it ''is'' a hyperreal field, and hence a suitable field for the usages of nonstandard analysis. It can be seen to be the higher-dimensional analogue of the real numbers; with cardinality <math>\aleph_2</math> instead of <math>\aleph_1</math>, cofinality <math>\aleph_1</math> instead of <math>\aleph_0</math>, and weight <math>\aleph_1</math> instead of <math>\aleph_0</math>, and with the η<sub>1</sub> property in place of the η<sub>0</sub> property (which merely means between any two real numbers we can find another).
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| == Examples of real closed fields ==
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| * the real [[algebraic numbers]]
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| * the [[computable number]]s
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| * the [[definable number]]s
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| * the [[real number]]s
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| * [[superreal number]]s
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| * [[hyperreal number]]s
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| * the [[Puiseux series]] with real coefficients
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * Basu, Saugata, [[Richard M. Pollack|Richard Pollack]], and [[Marie-Françoise Roy]] (2003) "Algorithms in real algebraic geometry" in ''Algorithms and computation in mathematics''. Springer. ISBN 3-540-33098-4 ([http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html online version])
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| * Michael Ben-Or, Dexter Kozen, and John Reif, ''[http://www.cs.duke.edu/~reif/paper/benor/realclosed.pdf The complexity of elementary algebra and geometry]'', Journal of Computer and Systems Sciences 32 (1986), no. 2, pp. 251–264.
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| * Caviness, B F, and Jeremy R. Johnson, eds. (1998) ''Quantifier elimination and cylindrical algebraic decomposition''. Springer. ISBN 3-211-82794-3
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| * [[Chen Chung Chang]] and [[Howard Jerome Keisler]] (1989) ''Model Theory''. North-Holland.
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| * Dales, H. G., and [[W. Hugh Woodin]] (1996) ''Super-Real Fields''. Oxford Univ. Press.
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| * {{cite journal | last1=Davenport | first1=James H. | author1-link=James H. Davenport | last2=Heintz | first2=Joos | title=Real quantifier elimination is doubly exponential | journal=J. Symb. Comput. | volume=5 | number=1-2 | pages= 29–35 | year=1988 | zbl=0663.03015 }}
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| * {{cite book | last=Efrat | first=Ido | title=Valuations, orderings, and Milnor ''K''-theory | series=Mathematical Surveys and Monographs | volume=124 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2006 | isbn=0-8218-4041-X | zbl=1103.12002 }}
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| * Mishra, Bhubaneswar (1997) "[http://www.cs.nyu.edu/mishra/PUBLICATIONS/97.real-alg.ps Computational Real Algebraic Geometry,]" in ''Handbook of Discrete and Computational Geometry''. CRC Press. 2004 edition, p. 743. ISBN 1-58488-301-4
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| * {{cite book | title=Squares | volume=171 | series=London Mathematical Society Lecture Note Series | first=A. R. | last=Rajwade | publisher=[[Cambridge University Press]] | year=1993 | isbn=0-521-42668-5 | zbl=0785.11022 }}
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| *[[Alfred Tarski]] (1951) ''A Decision Method for Elementary Algebra and Geometry''. Univ. of California Press.
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| *[Paul Erdos] (1955) P. Erdös, L. Gillman, and M. Henriksen, An isomorphism theorem for real-closed fields, Ann. of Math. (2) 61 (1955), 542–554. MR 0069161 (16,993e)
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| ==External links==
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| * [http://www.maths.manchester.ac.uk/raag/ ''Real Algebraic and Analytic Geometry Preprint Server'']
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| * [http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/ ''Model Theory preprint server'']
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| [[Category:Field theory]]
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| [[Category:Real closed field|*]]
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| [[Category:Real algebraic geometry]]
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