Real closed field

In mathematics, a real closed field is a field F in which any of the following equivalent conditions are true:

There is an ordering 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.
F is not algebraically closed but its algebraic closure is a finite extension.
F is not algebraically closed but the field extension $F({\sqrt {-1}})$ is algebraically closed.
There is an ordering on F which does not extend to an ordering on any proper algebraic extension of F.
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.)

The proof that these properties are all equivalent is not easy.

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.

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 $F({\sqrt {-1}})$ 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.

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.

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.

Tarski's algorithm for quantifier elimination has non-elementary complexity, meaning that no tower $2^{2^{\cdot ^{\cdot ^{\cdot ^{n}}}}}$ 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 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 $2^{2^{\Omega }(n)}$ and length $2^{2^{\Omega }(n)}$ , using the Ω asymptotic notation.

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.

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 confinality 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 $\aleph _{0}$ .

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:

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 $\aleph _{\alpha }$ , 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 $\aleph _{\alpha }$ -saturated, and moreover two η_α real closed fields both of cardinality $\aleph _{\alpha }$ are order isomorphic.

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 ${\mathbb {R}}^{\mathbb {N}}/{\mathbf {M} }$ , where M is a maximal ideal not leading to a field order-isomorphic to ${\mathbb {R}}$ . 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 $\aleph _{\beta }$ then we have a unique η_β field of size η_β.)

Moreover, we do not need ultrapowers to construct Ϝ, we can do so much more constructively as the subfield of ${\mathbb {R}}((G))$ 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 $\aleph _{1}$ , Κ has cardinality $\aleph _{2}$ , 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 $\aleph _{2}$ instead of $\aleph _{1}$ , cofinality $\aleph _{1}$ instead of $\aleph _{0}$ , and weight $\aleph _{1}$ instead of $\aleph _{0}$ , 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

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