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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.