# Ordered field

In mathematics, an**ordered field**is a field (

*F*,+,*) together with a total order ≤ on

*F*that is compatible with the algebraic operations in the following sense:

- if
*a*≤*b*then*a*+*c*≤*b*+*c* - if 0 ≤
*a*and 0 ≤*b*then 0 ≤*a b*

*a*,

*b*,

*c*,

*d*in

*F*:

- Either −
*a*≤ 0 ≤*a*or*a*≤ 0 ≤ −*a*. - We are allowed to "add inequalities": If
*a*≤*b*and*c*≤*d*, then*a*+*c*≤*b*+*d* - We are allowed to "multiply inequalities with positive elements": If
*a*≤*b*and 0 ≤*c*, then*ac*≤*bc*. - Squares are non-negative: 0 ≤
*a*^{2}for all*a*in*F*; in particular 0 < 1. - One can deduce that 0 < 1 + 1 + ... + 1 for any number of summands; this implies that the field
*F*has characteristic 0.

*Archimedean*. For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.

If *F* is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and * are continuous.

Examples of ordered fields are:

- the rational numbers
- the real numbers
- the hyperreal numbers

Finite fields cannot be turned into ordered fields, because they do not have characteristic 0. The complex numbers also cannot be turned into an ordered field, as they contain a square root of `-`1, which no ordered field can do. Also, the p-adic numbers cannot be ordered, since **Q**_{2} contains a square root of -7 and **Q**_{p} (p > 2) contains a square root of 1-p.