Real closed field
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.
A real closed field is a field F in which any of the following equivalent conditions are true:
- 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[clarification needed] is true in F if and only if it is true in the reals.
- 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 with coefficients in F has at least one root in F.
- 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.
- F is not algebraically closed but its algebraic closure is a finite extension.
- F is not algebraically closed but the field extension 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.)
- 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.
- F is a real closed ring.
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 (note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃z y = x + z2). For example, the real closure of the rational numbers is the field of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926.
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.
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 real closed ring. For example the real closure of the field is the ring (the two copies correspond to the two orderings of ). Whereas the real closure of the ordered subfield of is again the field .
Model theory: 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 ordered field axioms
- an axiom asserting that every positive number has a square root, and
- an axiom scheme asserting that all polynomials of odd degree have at least one root,
one obtains a first-order theory. Alfred Tarski (1951) proved that the theory of real closed fields in the first order language of 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 theoretic consequences hereof: The theory of real closed fields is complete, o-minimal and decidable.
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 of the real field axioms, and thus is also decidable.
The decision procedures are not necessarily practical. The 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.
The algorithm Tarski proposed for quantifier elimination has NONELEMENTARY complexity, meaning that no tower 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) doubly exponential: there exists a family Φn of formulas with n quantifiers, of length O(n) and constant degree such that any quantifier-free formula equivalent to Φn must involve polynomials of degree and length , using the Ω asymptotic notation. Ben-Or, Kozen, and Reif (1986) proved that the theory of real closed fields is decidable in exponential space, and therefore in doubly exponential time.
Basu and Roy (1996) proved that there exists a well-behaved algorithm to decide the truth of a formula ∃x1,…,∃xk P1(x1,…,xk)⋈0∧…∧Ps(x1,…,xk)⋈0 where ⋈ is <, > or =, with complexity in arithmetic operations sk+1dO(k). In fact, the existential theory of the reals can be decided in PSPACE.
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 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 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 .
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 ηα 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 ηα if and only if they are -saturated, and moreover two ηα real closed fields both of cardinality 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 , where M is a maximal ideal not leading to a field order-isomorphic to . This is the most commonly used 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 then we have a unique ηβ field of size ηβ.)
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 of formal power series on the Sierpiński group.
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
- the Puiseux series with real coefficients
- Rajwade (1993) pp. 222–223
- Efrat (2006) p. 177
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