nLab
ordered field

Ordered fields

Idea

An ordered field is a field equipped with a compatible linear order.

Note that while the adjective ‘ordered’ usually refers to a partial order, it is traditionally used more strictly when placed before ‘field’.

Definition

An ordered field is a field KK equipped with a linear order <\lt such that:

  • 1>01 \gt 0,
  • If a,b>0a, b \gt 0, then so are a+ba + b and aba b.

One often sees the definition using a total order \leq instead of the linear order <\lt. This makes no difference in classical mathematics, but we need to use linear orders in constructive mathematics if we wish to have the usual examples.

Examples

The field \mathbb{R} of real numbers is the? Dedekind-complete ordered field.

The field \mathbb{Q} of rational numbers is a subfield? of \mathbb{R} that is too small to be complete.

The field of surreal numbers is a field extension of \mathbb{R} that is too large to be complete.

Properties

Every ordered field must have characteristic 00, since we can prove by induction that n>0n \gt 0 for every positive natural number nn.

The archimedean ordered fields are precisely the subfield?s of the field of real numbers.

Proposition

Every complete ordered field is archimedean.

Proof

Suppose otherwise: let a,b>0a, b \gt 0 be given, and suppose bb is an upper bound of a,2a,3a,a, 2a, 3a, \ldots. Then bab - a is an upper bound of 0,a,2a,0, a, 2a, \ldots and consequently there can be no least upper bound of the sequence, contradicting Dedekind completeness.

The following is a result in classical mathematics.

Proposition

A field admits an order (“is orderable”) if and only if it is a real field, i.e., if the element 1-1 is not a sum of squares.

Proof

Given an ordered field, any non-zero square is positive since either α-\alpha or α\alpha is positive, and so (α) 2=α 2(-\alpha)^2 = \alpha^2 is positive. Hence a sum of non-zero squares cannot be negative, and in particular cannot be equal to 1-1.

In the other direction, every real field FF may be embedded in a real closed field (this requires Zorn's lemma), and a real closed field admits a unique ordering. The restriction of this ordering to the embedded field FF gives an ordering on FF.

Revised on August 11, 2011 19:22:38 by Toby Bartels (71.31.222.2)