In mathematics, a real closed field is a field F in which any of the following equivalent conditions are true: For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

1. There is a total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and any polynomial of odd degree with coefficients in F has at least one root in F.
2. 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=b² or a=-b².
3. F is not algebraically closed but its algebraic closure is a finite extension.
4. F is not algebraically closed but the field extension is algebraically closed.
5. There is an ordering on F which does not extend to an ordering on any proper algebraic extension of F.
6. 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.)
7. 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.

The proof that these properties are all equivalent is not easy. In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics, an ordered field is a field together with an ordering of its elements. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... This article is about the term degree as used in mathematics. ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, a formally real field in field theory is a field that shares certain algebraic properties with the real number field. ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ... In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... In mathematics, a formally real field in field theory is a field that shares certain algebraic properties with the real number field. ... In Mathematical analysis, the intermediate value theorem is either of two theorems of which an account is given below. ...

If F is an ordered field (not just orderable, but a definite ordering 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 on F, and is unique up to order isomorphism. For example, the real closure of the rational numbers are the real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926. In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ... Otto Schreier (born March 3, 1901 in Vienna, Austria; died June 2, 1929 in Hamburg, Germany) was an Austrian mathematician who made major contributions in combinatorial group theory. ... Year 1926 (MCMXXVI) was a common year starting on Friday (link will display the full calendar) of the Gregorian calendar. ...

Model theory

Two real closed fields, isomorphic as fields, are necessarily isomorphic as ordered fields; any field isomorphism of real closed fields is isotonic, or order-preserving, as the ordering of a real closed field is definable by a first-order formula from its field operations: x ≤ y if and only if ∃z y = x+z². For any field F such that is an algebraically closed field, there is a unique ordering which makes F a real closed field (and it is given by the formula above).

Decidability and quantifier elimination

The theory of real closed fields was invented by algebraists but taken up with enthusiasm by logicians. By adding to the finite list of ordered field axioms an axiom saying that square roots of positive numbers exist, as well as an axiom scheme saying there exists a root for any polynomial of odd order, one obtains a first-order theory. Tarski's theorem tells us that the theory of real closed fields, including a "<" predicate symbol, admits elimination of quantifiers, which in turn entails it is a complete and decidable theory. Quantifier elimination is a technique in logic, model theory, and theoretical computer science. ... A logical system or theory is decidable if the set of all well-formed formulas valid in the system is decidable. ...

The latter means that we can always tell by a decision procedure whether some sentence in the first-order language with relation symbols for inequality and equality, and functions for addition and multiplication, is true. Euclidean geometry (without the ability to measure angles) can be axiomatized using the real field axioms, and thus is decidable. In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i. ...

This decision procedure, however, is not necessarily practical. The algorithmic complexities of the currently known decision procedures are very high and practical execution times can be prohibitive except for very simple, small problems. Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. ...

Tarski's algorithm for quantifier elimination has non-elementary complexity, meaning that no tower can bound the execution time of the algorithm if n is the size of the problem. Davenport and Heinz proved in 1988 that quantifier elimination is in fact (at least) doubly exponential: there exists a family Φ_n of formulas with n quantifiers, of length O(n) and constant degree such that any quantifier-free formula equivalent to Φ_n must involve polynomials of degree and length , using the Ω asymptotic notation. In computational complexity theory, the complexity class ELEMENTARY is the union of the classes in the exponential hierarchy. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

Basu and Roy (1996) proved that there exists a well-behaved algorithm to decide the truth of a formula ∃x₁,…,∃x_k P₁(x₁,…,x_k)⋈0∧…∧P_s(x₁,…,x_k)⋈0 where ⋈ is <, > or =, with complexity in arithmetic operations s^k+1d^O(k).

Order properties

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. A non-archimedean field is, of course, a field that is not archimedean, and there are real closed non-archimedean fields; for example any field of hyperreal numbers is real closed and non-archimedean. An archimedean field is an ordered field with the archimedean property. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, an Archimedean field is an ordered field with the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ...

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 . In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a &#8804; b. ...

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

• The cardinality of F.

• The cofinality of F.

To this we may add

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

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 generalized continuum hypothesis. There are also particular properties which may or may not hold: In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

• 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.

• An ordered field F has the η_α property for the ordinal number α if for any two subsets L and U of F of cardinality less than , 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_α if and only if they are -saturated, and moreover two η_α real closed fields both of cardinality are order isomorphic.

In mathematical logic, and in particular model theory, a saturated model M is one which realizes as many complete types as may be reasonably expected given its size. ...

The generalized continuum hypothesis

The characteristics of real closed fields become much simpler if we are willing to assume the generalized continuum hypothesis. If the continuum hypothesis holds, all real closed fields with cardinality the continuum and having the η₁ property are order isomorphic. This unique field Ϝ can be defined by means of an ultrapower, as , where M is a maximal ideal not leading to a field order-isomorphic to . This is the most commonly used hyperreal number field in nonstandard 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 then we have a unique η_β field of size η_β.) In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... An ultrapower is an important special case of the ultraproduct construction. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of...

Moreover, we do not need ultrapowers to construct Ϝ, we can do so much more constructively as the subfield of of formal power series on the Sierpinski group with a countable number of nonzero terms.

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 , Κ has cardinality , 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 instead of , cofinality instead of , and weight instead of , and with the η₁ property in place of the η₀ property (which merely means between any two real numbers we can find another.)

Examples of real closed fields

• the real algebraic numbers
• the computable numbers
• the definable numbers
• the real numbers
• superreal numbers
• hyperreal numbers

In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn&#8722;1 + ··· + an &#8722;1x + an = 0 where n is a positive integer called... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ... A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The superreal numbers compose a more inclusive category than hyperreal number. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ...

References

• Chang, Chen Chung and Keisler, H. Jerome: Model Theory, North-Holland, 1989.
• H. Garth Dales and W. Hugh Woodin: Super-Real Fields, Clarendon Press, 1996.
• Computational Real Algebraic Geometry, Bhubaneswar Mishra, Handbook of Discrete and Computational Geometry, CRC Press, 1997 (Postscript version); also in 2004 edition, p. 743, ISBN 1-58488-301-4
• Saugata Basu, Richard Pollack and Marie-Françoise Roy, Algorithms in real algebraic geometry, Springer, Algorithms and computation in mathematics, 2003, ISBN 3540330984 (online version)
• Bob F. Caviness, Jeremy R. Johnson, editors, Quantifier elimination and cylindrical algebraic decomposition, Springer, 1998, ISBN 3211827943