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Concept
Ordered field
Summary
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field is isomorphic to the reals.
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. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is (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 first-order 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 higher-order, viewing positive cones as prepositive cones provides a larger context in which field orderings are partial orderings.
A field together with a (strict) total order on is an if the order satisfies the following properties for all
if then and
if and then
A or preordering of a field is a subset that has the following properties:
For and in both and are in
If then In particular,
The element is not in
A is a field equipped with a preordering Its non-zero 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 non-zero 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.
Official source
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