In mathematics, a real closed field is an ordered field F in which any of the following equivalent conditions are true:
- Every non-negative 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.
- The field extension is algebraically closed.
- F has no proper algebraic extension to an ordered field.
F is a formally real field such that any algebraic extension of F is not formally real.
If F is any ordered field, one of the theorems called the Artin-Schreier theorem states that F has an algebraic extension, the real closure K of F, such that K is real closed and whose ordering is an extension of the ordering on F, which 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.
The theory of real closed fields was invented by algebraists but taken up with enthusiasm by logicians. If you add to the finite list of field axioms an axiom saying that square roots of positive numbers exist, an axiom scheme saying there exists a root for any polynomial of odd order, and another axiom scheme saying if a sum of squares is zero then each number being squared is zero 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 decidable theory ; 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. The theory of real closed fields is therefore complete.
Two real closed fields, isomorphic as fields, are necessarily isomorphic as ordered fields; in fact, the ordering of a real closed field is definable by a first-order formula from its field operations: x ≤ y iff ∃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).
Considered simply as a field with the cardinality of the continuum, there is up to isomorphism only one field which is not algebraically closed but which becomes so by adjoining the square root of minus one. From the above, we see that it would be a mistake to simply identify this field with the real numbers, whose order properties we need to take into account when doing, for instance, real analysis. In fact, other real closed fields with the cardinality of the continuum exist. If h is a field isomorphism between F and K, then it is an isomorphism of ordered fields if and only if h and h-1 are isotonic maps. This means that if a < b in F, then h(a) < h(b) in K, and if c < d in K, then h-1(c) < h-1 in F; to characterize what is unique about the real numbers means we must characterize its order properties, which make it unique as a real closed field, and hence as an ordered field.
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 is an integer 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 .
We have therefore the following invariants defining the nature of a real closed field 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 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 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 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
- 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.
Last updated: 05-13-2005 07:56:04