Formally real field

In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions.
The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as (infinitely many) first-order sentences in the language of fields and are equivalent to the above definition.
A formally real field "F" is a field that also satisfies one of the following equivalent properties:
It is easy to see that these three properties are equivalent. It is also easy to see that a field that admits an ordering must satisfy these three properties.
A proof that if "F" satisfies these three properties, then "F" admits an ordering uses the notion of and positive cones. Suppose -1 is not a sum of squares, then a Zorn's Lemma argument shows that the prepositive cone of sums of squares can be extended to a positive cone . One uses this positive cone to define an ordering: if and only if belongs to "P".
Real closed fields.
A formally real field with no formally real proper algebraic extension is a real closed field. If "K" is formally real and Ω is an algebraically closed field containing "K", then there is a real closed subfield of Ω containing "K". A real closed field can be ordered in a unique way, and the non-negative elements are exactly the squares.