Ordered field

In mathematics, an ordered field is a field which has a linear order structure which is compatible with the field operations.

Formally, F is an ordered field if there is a linear order ≤ on F which satisfies
 * $$x \le y \Rightarrow x+z \le y+z ; \,$$
 * If $$0 \le z \,$$ then $$x \le y \Rightarrow x.z \le y.z ; \,$$
 * For each element $$x \le 0 \,$$ or $$x \ge 0 \,$$;
 * If $$x \le 0\,$$ and $$x \ge 0\,$$ then $$x = 0. \,$$

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
 * $$C \cup (-C) = F ;\,$$
 * $$C \cap (-C) = \{0\} .\,$$


 * $$x \ge y \Leftrightarrow x-y \in C .\,$$

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 can be ordered if and only if -1 is not a sum of squares in F.

Examples

 * 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 $$\mathbf{Q}(\sqrt 2)$$ has two possible structures as ordered field, corresponding to the embeddings into R in which $$\sqrt 2$$ takes on the two possible real values.
 * No finite field can be ordered.