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

• $x\leq y\Rightarrow x+z\leq y+z;\,$
• If $0\leq z\,$ then $x\leq y\Rightarrow x.z\leq y.z;\,$
• For each element $x\leq 0\,$ or $x\geq 0\,$;
• If $x\leq 0\,$ and $x\geq 0\,$ 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

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

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.