In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard orderings.
Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Every Dedekindcomplete ordered field is isomorphic to the reals. Squares are necessarily nonnegative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1 (which is negative in any ordered field). Finite fields cannot be ordered.
Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. This grew eventually into the Artin–Schreier theory of ordered fields and formally real fields.
There are two equivalent common definitions of an ordered field. The definition of total order appeared first historically and is a firstorder axiomatization of the ordering as a binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements. Although the latter is higherorder, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.
A field together with a total order on is an ordered field if the order satisfies the following properties for all
As usual, we write for and . The notations and stand for and , respectively. Elements with are called positive.
A prepositive cone or preordering of a field is a subset that has the following properties:^{[1]}
A preordered field is a field equipped with a preordering Its nonzero elements form a subgroup of the multiplicative group of
If in addition, the set is the union of and we call a positive cone of The nonzero elements of are called the positive elements of
An ordered field is a field together with a positive cone
The preorderings on are precisely the intersections of families of positive cones on The positive cones are the maximal preorderings.^{[1]}
Let be a field. There is a bijection between the field orderings of and the positive cones of
Given a field ordering ≤ as in the first definition, the set of elements such that forms a positive cone of Conversely, given a positive cone of as in the second definition, one can associate a total ordering on by setting to mean This total ordering satisfies the properties of the first definition.
Examples of ordered fields are:
The surreal numbers form a proper class rather than a set, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.
For every a, b, c, d in F:
Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is isomorphic to the rationals (as for any other field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves.
If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean. Otherwise, such field is a nonArchimedean ordered field and contains infinitesimals. For example, the real numbers form an Archimedean field, but hyperreal numbers form a nonArchimedean field, because it extends real numbers with elements greater than any standard natural number.^{[4]}
An ordered field F is isomorphic to the real number field R if and only if every nonempty subset of F with an upper bound in F has a least upper bound in F. This property implies that the field is Archimedean.
Vector spaces (particularly, nspaces) over an ordered field exhibit some special properties and have some specific structures, namely: orientation, convexity, and positivelydefinite inner product. See Real coordinate space#Geometric properties and uses for discussion of those properties of R^{n}, which can be generalized to vector spaces over other ordered fields.
Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares.^{[2]}^{[3]}
Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order need not be uniquely determined.) The proof uses Zorn's lemma.^{[5]}
Finite fields and more generally fields of positive characteristic cannot be turned into ordered fields, as shown above. The complex numbers also cannot be turned into an ordered field, as −1 is a square of the imaginary unit i. Also, the padic numbers cannot be ordered, since according to Hensel's lemma Q_{2} contains a square root of −7, thus 1^{2} + 1^{2} + 1^{2} + 2^{2} + √−7^{2} = 0, and Q_{p} (p > 2) contains a square root of 1 − p, thus (p − 1)⋅1^{2} + √1 − p^{2} = 0.^{[6]}
If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and × are continuous, so that F is a topological field.
The Harrison topology is a topology on the set of orderings X_{F} of a formally real field F. Each order can be regarded as a multiplicative group homomorphism from F^{∗} onto ±1. Giving ±1 the discrete topology and ±1^{F} the product topology induces the subspace topology on X_{F}. The Harrison sets form a subbasis for the Harrison topology. The product is a Boolean space (compact, Hausdorff and totally disconnected), and X_{F} is a closed subset, hence again Boolean.^{[7]}^{[8]}
A fan on F is a preordering T with the property that if S is a subgroup of index 2 in F^{∗} containing T − {0} and not containing −1 then S is an ordering (that is, S is closed under addition).^{[9]} A superordered field is a totally real field in which the set of sums of squares forms a fan.^{[10]}
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