Editing Ordered field (section)

{{Short description|Algebraic object with an ordered structure}}
In [[mathematics]], an '''ordered field''' is a [[field (mathematics)|field]] together with a [[total order]]ing of its elements that is compatible with the field operations.  Basic examples of ordered fields are the [[Rational number|rational numbers]] and the [[real numbers]], both with their standard orderings.

Every [[Field extension|subfield]] of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is [[Isomorphism|isomorphic]] to the [[rational number]]s. Every [[Dedekind-complete]] ordered field is isomorphic to the reals. [[square (algebra)|Squares]] are necessarily non-negative in an ordered field. This implies that the [[complex number]]s cannot be ordered since the square of the [[imaginary unit]] ''i'' is {{num|−1}} (which is negative in any ordered field).  [[Finite field]]s 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 (mathematician)|Hans Hahn]]. This grew eventually into the [[Artin–Schreier theorem|Artin–Schreier theory]] of ordered fields and [[formally real field]]s.