Inmathematics, areal closed field is afield${\displaystyle F}$ that has the samefirst-order properties as the field ofreal numbers. (First-order properties are those properties that can be expressed with thelogic symbols${\displaystyle \mathrm{\forall },\mathrm{\exists },\vee ,\wedge ,\mathrm{\neg },\to }$ and the arithmetic symbols${\displaystyle 0,1,+,-,\times ,÷,=}$, where the domain of allquantifiers is the set${\displaystyle F}$; it is hence not allowed to quantify over natural numbers, subsets of${\displaystyle F}$, sequences in${\displaystyle F}$, functions${\displaystyle F\to F}$ etc.) Some examples of real closed fields are the field of real numbers itself, the field of realalgebraic numbers, and fields ofhyperreal numbers that include infinitesimals. In algebra, most theorems that involve the real numbers remain true when formulated for arbitrary real closed fields.

A real closed field is a fieldF in which any of the following equivalent conditions is true:

1. F iselementarily equivalent to the field of real numbers. In other words, it has the same first-order properties as the reals: anysentence in the first-order language offields is true inFif and only if it is true in the reals.
2. There is atotal order onF turningF into anordered field such that, in this ordering, every positive element ofF has asquare root inF and anypolynomial ofodddegree withcoefficients inF has at least oneroot inF.
3. There is a total order onF turningF into an ordered field such that, in this ordering, theintermediate value theorem holds for all polynomials with coefficients inF.
4. F is aformally real field (meaning there exists a total order onF turningF into an ordered field) such that every polynomial of odd degree with coefficients inF has at least one root inF, and for every elementa ofF there isb inF such thata =b² ora = −b².
5. F is notalgebraically closed, but itsalgebraic closure is afinite extension ofF.
6. F is not algebraically closed but thefield extension${\displaystyle F(\sqrt{-1})}$ is algebraically closed.
7. There is an ordering onF that does not extend to an ordering on any properalgebraic extension ofF.
8. F is a formally real field such that no proper algebraic extension ofF is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.)
9. F is aweakly o-minimal ordered field.^[1]

The following fields are real closed, which can be shown by verifying property 2 above:

• the field of realalgebraic numbers;
• the field ofcomputable real numbers;
• the field ofdefinable real numbers;
• the field ofreal numbers;
• the field ofPuiseux series with real coefficients;
• theLevi-Civita field;
• thehyperreal number fields;
• thesuperreal number fields;
• the field ofsurreal numbers (this is aproper class, not aset).

IfF is an ordered field, theArtin–Schreier theorem states thatF has an algebraic extension, called thereal closureK ofF, such thatK is a real closed field whose ordering is an extension of the given ordering onF, and is uniqueup to a uniqueisomorphism of fields identical onF^[2] (note that everyring homomorphism between real closed fields automatically isorder preserving, becausex ≤ y if and only if ∃z :y = x + z²). For example, the real closure of the ordered field ofrational numbers is the field${\displaystyle {\mathbb{R}}_{\mathrm{a}\mathrm{l}\mathrm{g}}}$ of real algebraic numbers. Thetheorem is named forEmil Artin andOtto Schreier, whoproved it in 1926.

If (F,P) is an ordered field, andE is aGalois extension ofF, then byZorn's lemma there is a maximal ordered field extension (M,Q) withM asubfield ofE containingF and the order onM extendingP. ThisM, together with its orderingQ, is called therelative real closure of (F,P) inE. We call (F,P) real closed relative toE ifM is justF. WhenE is the algebraic closure ofF the relative real closure ofF inE is actually thereal closure ofF described earlier.^[3]

IfF is a field (not ordered or even orderable) thenF still has a real closure, which may not be a field anymore, but just areal closed ring. For example, the real closure of the field${\displaystyle \mathbb{Q}(\sqrt{2})}$ is thering${\displaystyle {\mathbb{R}}_{\mathrm{a}\mathrm{l}\mathrm{g}}\times {\mathbb{R}}_{\mathrm{a}\mathrm{l}\mathrm{g}}}$ (the two copies correspond to the two orderings of${\displaystyle \mathbb{Q}(\sqrt{2})}$). On the other hand, if${\displaystyle \mathbb{Q}(\sqrt{2})}$ is considered as an ordered subfield of${\displaystyle \mathbb{R}}$, its real closure is again the field${\displaystyle {\mathbb{R}}_{\mathrm{a}\mathrm{l}\mathrm{g}}}$.

Thelanguage of real closed fields${\displaystyle {\mathcal{L}}_{\text{rcf}}}$ includes symbols for the operations of addition and multiplication, the constants 0 and 1, and the order relation≤ (as well as equality, if this is not considered a logical symbol). In this language, the (first-order) theory of real closed fields,${\displaystyle {\mathcal{T}}_{\text{rcf}}}$, consists of all sentences that follow from the following axioms:

• theaxioms ofordered fields;
• the axiom asserting that every positive number has a square root;
• for every odd number${\displaystyle d}$, the axiom asserting that all polynomials of degree${\displaystyle d}$ have at least one root.

All of these axioms can be expressed infirst-order logic (i.e.quantification ranges only over elements of the field). Note that${\displaystyle {\mathcal{T}}_{\text{rcf}}}$ is just the set of all first-order sentences that are true about the field of real numbers.

Tarski showed that${\displaystyle {\mathcal{T}}_{\text{rcf}}}$ iscomplete, meaning that any${\displaystyle {\mathcal{L}}_{\text{rcf}}}$-sentence can be proven either true or false from the above axioms. Furthermore,${\displaystyle {\mathcal{T}}_{\text{rcf}}}$ isdecidable, meaning that there is analgorithm to determine the truth or falsity of any such sentence. This was done by showingquantifier elimination: there is an algorithm that, given any${\displaystyle {\mathcal{L}}_{\text{rcf}}}$-formula, which may containfree variables, produces an equivalent quantifier-free formula in the same free variables, whereequivalent means that the two formulas are true for exactly the same values of the variables. Tarski's proof uses a generalization ofSturm's theorem. Since the truth of quantifier-free formulas without free variables can be easily checked, this yields the desired decision procedure. These results were obtainedc. 1930 and published in 1948.[4]

TheTarski–Seidenberg theorem extends this result to the followingprojection theorem. IfR is a real closed field, a formula withn free variables defines a subset ofRⁿ, the set of the points that satisfy the formula. Such a subset is called asemialgebraic set. Given a subset ofk variables, theprojection fromRⁿ toR^k is thefunction that maps everyn-tuple to thek-tuple of the components corresponding to the subset of variables. The projection theorem asserts that a projection of a semialgebraic set is a semialgebraic set, and that there is an algorithm that, given a quantifier-free formula defining a semialgebraic set, produces a quantifier-free formula for its projection.

In fact, the projection theorem is equivalent to quantifier elimination, as the projection of a semialgebraic set defined by the formulap(x,y) is defined by

${\displaystyle (\mathrm{\exists }x)P(x,y),}$

wherex andy represent respectively the set of eliminated variables, and the set of kept variables.

The decidability of a first-order theory of the real numbers depends dramatically on the primitive operations and functions that are considered (here addition and multiplication). Adding other functions symbols, for example, thesine or theexponential function, can provide undecidable theories; seeRichardson's theorem andDecidability of first-order theories of the real numbers.

Furthermore, the completeness and decidability of the first-order theory of the real numbers (using addition and multiplication) contrasts sharply withGödel's andTuring's results about the incompleteness and undecidability of the first-order theory of the natural numbers (using addition and multiplication). There is no contradiction, since the statement "x is an integer" cannot be formulated as a first-order formula in the language${\displaystyle {\mathcal{L}}_{\text{rcf}}}$.

Tarski's original algorithm forquantifier elimination hasnonelementarycomputational complexity, meaning that no tower

${\displaystyle 2^{2^{{\cdot }^{{\cdot }^{{\cdot }^n}}}}}$

can bound the execution time of the algorithm ifn is the size of the input formula. Thecylindrical algebraic decomposition, introduced byGeorge E. Collins, provides a much more practicable algorithm of complexity

${\displaystyle d^{2^{O(n)}}}$

wheren is the total number of variables (free and bound),d is the product of the degrees of the polynomials occurring in the formula, andO(n) isbig O notation.

Davenport and Heintz (1988) proved that thisworst-case complexity is nearly optimal for quantifier elimination by producing a familyΦ_n of formulas of lengthO(n), withn quantifiers, and involving polynomials of constant degree, such that any quantifier-free formula equivalent toΦ_n must involve polynomials of degree${\displaystyle 2^{2^{\mathrm{\Omega }(n)}}}$ and length${\displaystyle 2^{2^{\mathrm{\Omega }(n)}},}$ where${\displaystyle \mathrm{\Omega }(n)}$ isbig Omega notation. This shows that both the time complexity and the space complexity of quantifier elimination are intrinsicallydouble exponential.

For the decision problem, Ben-Or,Kozen, andReif (1986) claimed to have proved that the theory of real closed fields is decidable inexponential space, and therefore in double exponential time, but their argument (in the case of more than one variable) is generally held as flawed; see Renegar (1992) for a discussion.

For purely existential formulas, that is for formulas of the form

∃x₁, ..., ∃x_kP₁(x₁, ...,x_k) ⋈ 0 ∧ ... ∧P_s(x₁, ...,x_k) ⋈ 0,

where⋈ stands for either<, > or =, the complexity is lower. Basu andRoy (1996) provided a well-behaved algorithm to decide the truth of such an existential formula with complexity ofs^k+1d^O(k) arithmetic operations andpolynomial space.

Any real closed field can be turned into anordered field in just one way: the positive elements are precisely the squares of non-zero elements.

A crucially important property of the real numbers is that it is anArchimedean field, meaning it has the Archimedean property that for any real number, there is aninteger larger than it inabsolute value. Note that this statement is not expressible in the first-order language of ordered fields, since it is not possible to quantify over integers in that language.

There are real-closed fields that arenon-Archimedean; for example, any field ofhyperreal numbers is real-closed and non-Archimedean. These fields contain infinitely large (larger than any integer) and infinitesimal (positive but smaller than any positive rational) elements.

The Archimedean property is related to the concept ofcofinality. A setX contained in an ordered setF is cofinal inF if for everyy inF there is anx inX such thaty <x. In other words,X is an unbounded sequence inF. The cofinality ofF is thecardinality 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${\displaystyle {\mathrm{\aleph }}_0}$.

We have therefore the following invariants defining the nature of a real closed fieldF:

• The cardinality ofF.
• The cofinality ofF.

To this we may add

• The weight ofF, which is the minimum size of a dense subset ofF.

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 thegeneralized continuum hypothesis. There are also particular properties that may or may not hold:

• A fieldF iscomplete if there is no ordered fieldK properly containingF such thatF is dense inK. If the cofinality ofF isκ, this is equivalent to sayingCauchy sequences indexed byκ areconvergent inF.
• An ordered fieldF has theeta set property η_α, for theordinal numberα, if for any two subsetsL andU ofF of cardinality less than${\displaystyle {\mathrm{\aleph }}_{\alpha }}$ such that every element ofL is less than every element ofU, there is an elementx inF withx larger than every element ofL and smaller than every element ofU. This is closely related to themodel-theoretic property of being asaturated model; any two real closed fields are η_α if and only if they are${\displaystyle {\mathrm{\aleph }}_{\alpha }}$-saturated, and moreover two η_α real closed fields both of cardinality${\displaystyle {\mathrm{\aleph }}_{\alpha }}$ areorder isomorphic.

The characteristics of real closed fields become much simpler if we are willing to assume thegeneralized continuum hypothesis. If the continuum hypothesis holds, all real closed fields withcardinality of the continuum and having theη₁ property are order isomorphic. This unique fieldϜ can be defined by means of anultrapower, as${\displaystyle {\mathbb{R}}^{\mathbb{N}}/\mathbf{M}}$, whereM is a maximal ideal not leading to a field order-isomorphic to${\displaystyle \mathbb{R}}$. This is the most commonly usedhyperreal number field innonstandard 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${\displaystyle {\mathrm{\aleph }}_{\beta }}$ then we have a uniqueη_β field of size${\displaystyle {\mathrm{\aleph }}_{\beta }}$.)

Moreover, we do not need ultrapowers to constructϜ, we can do so much more constructively as the subfield of series with acountable number of nonzero terms of the field${\displaystyle \mathbb{R}[[G]]}$ offormal power series on atotally orderedabeliandivisible groupG that is anη₁ group of cardinality${\displaystyle {\mathrm{\aleph }}_1}$ (Alling 1962).

Ϝ however is not a complete field; if we take its completion, we end up with a fieldΚ of larger cardinality.Ϝ has the cardinality of the continuum, which by hypothesis is${\displaystyle {\mathrm{\aleph }}_1}$,Κ has cardinality${\displaystyle {\mathrm{\aleph }}_2}$, and containsϜ as a dense subfield. It is not an ultrapower but itis 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${\displaystyle {\mathrm{\aleph }}_2}$ instead of${\displaystyle {\mathrm{\aleph }}_1}$, cofinality${\displaystyle {\mathrm{\aleph }}_1}$ instead of${\displaystyle {\mathrm{\aleph }}_0}$, and weight${\displaystyle {\mathrm{\aleph }}_1}$ instead of${\displaystyle {\mathrm{\aleph }}_0}$, and with theη₁ property in place of theη₀ property (which merely means between any two real numbers we can find another).

Tarski's axioms are an axiom system for the first-order ("elementary") portion ofEuclidean geometry. Using those axioms, one can show that the points on a line form a real closed field R, and one can introduce coordinates so that the Euclidean plane is identified with R². Employing the decidability of the theory of real closed fields, Tarski then proved that the elementary theory of Euclidean geometry is complete and decidable.^[4]

• Euclidean ordered field
• p-adically closed field
• real-closed ring

• Alling, Norman L. (1962), "On the existence of real-closed fields that are η_α-sets of power ℵ_α.",Trans. Amer. Math. Soc.,103:341–352,doi:10.1090/S0002-9947-1962-0146089-X,MR 0146089
• Basu, Saugata,Richard Pollack, andMarie-Françoise Roy (2003) "Algorithms in real algebraic geometry" inAlgorithms and computation in mathematics. Springer.ISBN 3-540-33098-4 (online versionArchived 2022-11-16 at theWayback Machine)
• Michael Ben-Or, Dexter Kozen, and John Reif,The complexity of elementary algebra and geometry, Journal of Computer and Systems Sciences 32 (1986), no. 2, pp. 251–264.
• Caviness, B F, and Jeremy R. Johnson, eds. (1998)Quantifier elimination and cylindrical algebraic decomposition. Springer.ISBN 3-211-82794-3
• Chen Chung Chang andHoward Jerome Keisler (1989)Model Theory. North-Holland.
• Dales, H. G., andW. Hugh Woodin (1996)Super-Real Fields. Oxford Univ. Press.
• Davenport, James H.;Heintz, Joos (1988)."Real quantifier elimination is doubly exponential".J. Symb. Comput.5 (1–2):29–35.doi:10.1016/s0747-7171(88)80004-x.Zbl 0663.03015.
• Efrat, Ido (2006).Valuations, orderings, and MilnorK-theory. Mathematical Surveys and Monographs. Vol. 124. Providence, RI:American Mathematical Society.ISBN 0-8218-4041-X.Zbl 1103.12002.
• Macpherson, D., Marker, D. and Steinhorn, C.,Weakly o-minimal structures and real closed fields, Trans. of the American Math. Soc., Vol. 352, No. 12, 1998.
• Mishra, Bhubaneswar (1997) "Computational Real Algebraic Geometry," inHandbook of Discrete and Computational Geometry. CRC Press. 2004 edition, p. 743.ISBN 1-58488-301-4
• Rajwade, A. R. (1993).Squares. London Mathematical Society Lecture Note Series. Vol. 171.Cambridge University Press.ISBN 0-521-42668-5.Zbl 0785.11022.
• Renegar, James (1992)."On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals".Journal of Symbolic Computation.13 (3):255–299.doi:10.1016/S0747-7171(10)80003-3.
• Passmore, Grant (2011).Combined Decision Procedures for Nonlinear Arithmetics, Real and Complex(PDF) (PhD).University of Edinburgh.
• Alfred Tarski (1951)A Decision Method for Elementary Algebra and Geometry. Univ. of California Press.
• Erdös, P.; Gillman, L.; Henriksen, M. (1955),"An isomorphism theorem for real-closed fields",Ann. of Math., 2,61 (3):542–554,doi:10.2307/1969812,JSTOR 1969812,MR 0069161

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