ordered field

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’.

An **ordered field** is a field $K$ equipped with a linear order $\lt$ such that:

- $1 \gt 0$,
- If $a, b \gt 0$, then so are $a + b$ and $a b$.

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.

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.

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

The archimedean ordered fields are precisely the subfields of the field of real numbers.

Every complete ordered field is archimedean.

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

The following is a result in classical mathematics.

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.

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 $F$.

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