Ordered field

An ordered field, in mathematics, is a field (F,+,*) together with a total order <= on F which 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

It follows from these axioms that for every a, we have either -a <= 0 <= a or a <= 0 <= -a. We are allowed to "add inequalities" (if a <= b and c <= d, then a + b <= c + d) and "multiply inequalities with positive elements" (if a <= b and 0 <= c, then ac <= bc). Also, squares are positive: 0 <= a² for all a in F; in particular 0 < 1. Furthermore, one can deduce that 0 < 1 + 1 + ... + 1 for any number of summands; this implies that the field F has characteristic 0.

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
• the surreal numbers; in fact every ordered field can be embedded into the surreal numbers

Finite fields and the complex numbers cannot be turned into ordered fields.