nLab
ordered field

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 $K$ equipped with a linear order $<$ such that:

• $1>0$,
• If $a,b>0$, then so are $a+b$ and $ab$.

One often sees the definition using a total order $\le$ instead of the linear order $<$. 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 $ℝ$ of real numbers is the Dedekind-complete ordered field.

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

The field of surreal numbers is a field extension of $ℝ$ that is too large to be complete.

Properties

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

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 $a,b>0$ be given, and suppose $b$ is an upper bound of $a,2a,3a,\dots$. Then $b-a$ is an upper bound of $0,a,2a,\dots$ 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 $-1$ is not a sum of squares.

Proof

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

In the other direction, every real field $F$ 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 $F$ gives an ordering on $F$.