# 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

- 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.

## Artin-Schreier theorem

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.

## 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 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.