Editing Ordered field (section)

==Examples of ordered fields==
Examples of ordered fields are:
* the field <math>\Q</math> of [[rational number]]s with its standard ordering (which is also its only ordering);
* the field <math>\R</math> of [[real number]]s with its standard ordering (which is also its only ordering);
* any subfield of an ordered field, such as the real [[algebraic numbers]] or the [[computable number]]s, becomes an ordered field by restricting the ordering to the subfield;
* the field <math>\mathbb{Q}(x)</math> of [[rational functions]] <math>p(x)/q(x)</math>, where <math>p(x)</math> and <math>q(x)</math> are [[polynomial]]s with rational coefficients and <math>q(x) \ne 0</math>, can be made into an ordered field by fixing a real [[transcendental number]] <math>\alpha</math> and defining <math>p(x)/q(x) > 0</math> if and only if <math>p(\alpha)/q(\alpha) > 0</math>.  This is equivalent to embedding <math>\mathbb{Q}(x)</math> into <math>\mathbb{R}</math> via <math>x\mapsto \alpha</math> and restricting the ordering of <math>\mathbb{R}</math> to an ordering of the image of <math>\mathbb{Q}(x)</math>. In this fashion, we get many different orderings of <math>\mathbb{Q}(x)</math>.
* the field <math>\mathbb{R}(x)</math> of [[rational functions]] <math>p(x)/q(x)</math>, where <math>p(x)</math> and <math>q(x)</math> are [[polynomial]]s with real coefficients and <math>q(x) \ne 0</math>, can be made into an ordered field by defining <math>p(x)/q(x) > 0</math> to mean that <math>p_n/q_m > 0</math>, where <math>p_n \neq 0</math> and <math>q_m \neq 0</math> are the leading coefficients of <math>p(x) = p_n x^n + \dots + p_0</math> and <math>q(x) = q_m x^m + \dots + q_0</math>, respectively. Equivalently: for rational functions <math>f(x), g(x)\in \mathbb{R}(x)</math> we have <math>f(x) < g(x)</math> if and only if <math>f(t) < g(t)</math> for all sufficiently large <math>t\in\mathbb{R}</math>. In this ordered field  the polynomial <math>p(x)=x</math> is greater than any constant polynomial and the ordered field is not [[Archimedean field|Archimedean]].
* The field <math>\mathbb{R}((x))</math> of [[formal Laurent series]] with real coefficients, where ''x'' is taken to be infinitesimal and positive
* the [[transseries]]
* [[real closed field]]s
* the [[superreal number]]s
* the [[hyperreal number]]s

The [[surreal numbers]] form a [[class (set theory)|proper class]] rather than a [[Set (mathematics)|set]], but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.