Note that while the adjective ‘ordered’ usually refers to a partial order, it is traditionally used more strictly when placed before ‘field’.
One often sees the definition using a total order 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.
Every complete ordered field is archimedean.
Suppose otherwise: let be given, and suppose is an upper bound of . Then is an upper bound of 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 is not a sum of squares.
Given an ordered field, any non-zero square is positive since either or is positive, and so is positive. Hence a sum of non-zero squares cannot be negative, and in particular cannot be equal to .
In the other direction, every real field 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 gives an ordering on .