Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. 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.

An ordered field necessarily has characteristic 0 since the elements 0 < 1 < 1 + 1 < 1 + 1 + 1 < ... necessarily are all distinct. Thus, an ordered field necessarily contains an infinite number of elements: a finite field cannot be ordered.

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. Any Dedekind-complete ordered field is isomorphic to the real 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 −1. Every ordered field is a formally real field.

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 maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.

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