About: Real closed field

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In mathematics, a real closed field is a 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 numbers.

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rdf:type	motorsport season
rdfs:label	Real closed field (en) حقل مغلق حقيقي (ar) Reell abgeschlossener Körper (de) Cuerpo real cerrado (es) Corps réel clos (fr) 실폐체 (ko) 実閉体 (ja) Ciało (formalnie) rzeczywiste (pl) Corpo real fechado (pt) 實閉域 (zh)
rdfs:comment	في الرياضيات، حقل مغلق حقيقي (بالإنجليزية: Real closed field)‏ هو حقل F، له نفس الخصائص من المستوى الأول، التي يملكها حقل الأعداد الحقيقية. (ar) En mathématiques, un corps réel clos est un corps totalement ordonnable dont aucune extension algébrique propre n'est totalement ordonnable. (fr) In mathematics, a real closed field is a 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 numbers. (en) 체론과 모형 이론에서 실폐체(實閉體, 영어: real closed field)는 실수체와 기본 동치인 체이다. (ko) 数学における実閉体（じつへいたい、英: real closed field）は実数体と一階の性質が同じである体を言う。実数体、実代数的数体、超実数体などがその例を与える。 (ja) Corpo real fechado, em álgebra abstrata, é um tipo de corpo que tem, em comum com os reais, a propriedade de que menos um não é o quadrado de algum elemento, nem a soma de quadrados, e, além disto, é um corpo maximal no sentido de que a única forma de aumentar este corpo e continuar mantendo esta propriedade é através de elementos . Formalmente: Um corpo formalmente real é um corpo que satisfaz: * é um corpo * Um corpo real fechado R é um corpo formalmente real tal que, se E é uma extensão algébrica de R, e E é um corpo formalmente real, então E = R. (pt) 在數學中，實閉域或實封閉域是一類有序域，使得其中每個正元素皆可表為平方，且任何奇數次多項式都有根。以下將給出幾種等價的定義。 (zh) Die reell abgeschlossenen Körper sind in der Algebra Körper, die mit dem Körper der reellen Zahlen einige wesentliche Eigenschaften gemeinsam haben: Zum Beispiel haben Polynome mit ungeradem Grad dort stets eine Nullstelle und diese Körper lassen sich mit einer durch die Körperstruktur eindeutig bestimmten Ordnungsrelation ausstatten, mit der sie zu geordneten Körpern werden. (de) Ciało (formalnie) rzeczywiste – ciało w którym zachodzi czyli, jeśli suma kwadratów elementów z ciała wynosi zero, to każdy z tych elementów musi być równy zero. Powyższy warunek jest równoważny każdej z dwóch poniższych własności: * nie jest sumą kwadratów w * ciało może być liniowo uporządkowane. Ciało, które nie jest formalnie rzeczywiste, nazywamy nierzeczywistym. Ciało rzeczywiście domknięte to ciało spełniające którykolwiek z następujących równoważnych warunków: Artin wykorzystał te wyniki do rozwiązania 17. problemu Hilberta. (pl)
dcterms:subject	Real algebraic geometry Real closed field Field (mathematics)
Wikipage page ID	1066093 (xsd:integer)
Wikipage revision ID	1121898288 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Cardinality of the continuum Puiseux series Quantifier elimination Semialgebraic set Totally ordered group Algebraic closure Algebraic extension Algorithm Real algebraic geometry Howard Jerome Keisler Richard M. Pollack Cylindrical algebraic decomposition University of Edinburgh Decidability (logic) Decidability of first-order theories of the real numbers Degree of a polynomial Dexter Kozen EXPSPACE Elementarily equivalent Order isomorphism Worst-case complexity Countably infinite Mathematics Chen Chung Chang Non-Archimedean ordered field Emil Artin Function (mathematics) George E. Collins Model-theoretic Continuum hypothesis Theorem Ordered field Sine Complete theory Computable number Computational complexity Zorn's Lemma Real closed field Archimedean field Parity (mathematics) Surreal number Total order W. Hugh Woodin Divisible group Galois extension Eta set Absolute value Alfred Tarski Algebraic number American Mathematical Society Exponential function Field (mathematics) Finite extension First-order logic Formal power series Nonstandard analysis PSPACE Cardinality Cauchy sequence Formal language Formally real field Double exponential time Isomorphism John Reif Free variable Well-formed formula Mathematical proof Quantifier (logic) Real closed ring Ring (mathematics) Hyperreal number Superreal number Abelian group Field (mathematics)

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