# Ordered field

 related topics {math, number, function}

In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations. An example of an ordered field is the field of real numbers. This concept was introduced by Emil Artin in 1927.

## Contents

### Definition

There are two equivalent definitions, depending on which properties one takes as the definition for an ordered field.

### Def 1: A total order on F

A field (F,+,*) together with a total order ≤ on F is an ordered field if the order satisfies the following properties:

• if ab then a + cb + c
• if 0 ≤ a and 0 ≤ b then 0 ≤ a b

It follows from these axioms that for every a, b, c, d in F:

• Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
• We are allowed to "add inequalities": If ab and cd, then a + cb + d
• We are allowed to "multiply inequalities with positive elements": If ab and 0 ≤ c, then acbc.

### Def 2: An ordering on F

An ordering of a field F is a subset PF that has the following properties:

• F is the disjoint union of P, −P, and the element 0. That is, for each xF, exactly one of the following conditions is true: x = 0, xP or −xP.
• For x and y in P, both x+y and xy are in P.

The elements of the subset P are called the positive elements of F.

We next define x < y to mean that y − xP (so that y − x > 0 in a sense). This relation satisfies the expected properties:

• If x < y and y < z, then x < z. (transitivity)
• If x < y and z > 0, then xz < yz.
• If x < y and x,y > 0, then 1/y < 1/x

The statement xy will mean that either x < y or x = y.

### Properties of ordered fields

• 1 is positive. (Proof: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) = 1 is positive, which is a contradiction)
• An ordered field has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the field had characteristic p > 0, then −1 would be the sum of p − 1 ones, but −1 is not positive). In particular, finite fields cannot be ordered.
• Squares are non-negative. 0 ≤ a² for all a in F. (Follows by a similar argument to 1 > 0)