Ordered field

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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

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

The relationship between the order and the associated positive cone is that

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

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.

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 $\mathbf{Q}(\sqrt 2)$ 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.