Formally, F is an ordered field if there is a linear order ≤ on F which satisfies
- If then
- For each element or ;
- If and then
Alternatively, the order may be defined in terms of a positive cone, a subset C of F which is closed under addition and multiplication, contains the 0 and 1 elements, and which has the properties that
The relationship between the order and the associated positive cone is that
It is possible for a field to have more than one linear order compatible with the field operations, but in any case the squares must lie in the positive cone.
A field F is formally real if -1 is not a sum of squares in F. The Artin-Schreier theorem states that a field F can be ordered if and only if it is formally real.
- The rational numbers form an ordered field in a unique way.
- The real numbers form an ordered field in a unique way: the squares form the positive cone.
- The complex numbers cannot be given an ordered field structure since both 1 and -1 are squares.
- The quadratic field has two possible structures as ordered field, corresponding to the embeddings into R in which takes on the two possible real values.
- No finite field can be ordered.