Ordered fieldmathematics, 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
- 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 ≤ a2 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.
Examples of ordered fields are:surreal numbers form a proper class rather than a set, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal 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 Q2 contains a square root of -7 and Qp (p > 2) contains a square root of 1-p.