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.

The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as first-order sentences in the language of fields, and are equivalent to the above definition.

A formally real field F is a field that satisfies in addition 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 prepositive cones 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 P ⊂ F. One uses this positive cone to define an ordering: a ≤ b if and only if b − a belongs to P.

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.

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