# 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:[1][2]

- −1 is not a sum of squares in
*F*. In other words, the Stufe of*F*is infinite. (In particular, such a field must have characteristic 0, since in a field of characteristic*p*the element −1 is a sum of 1's.) This can be expressed in first-order logic by , , etc., one sentence for each number of variables. - There exists an element of
*F*that is not a sum of squares in*F*, and the characteristic of*F*is not 2. - If any sum of squares of elements of
*F*equals zero, then each of those elements must be zero.

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*.

## Real Closed Fields

A formally real field with no formally real proper algebraic extension is a real closed field.[3] 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,[3] and the non-negative elements are exactly the squares.

## Notes

- Rajwade, Theorem 15.1.
- Milnor and Husemoller (1973) p.60
- Rajwade (1993) p.216

## References

- Milnor, John; Husemoller, Dale (1973).
*Symmetric bilinear forms*. Springer. ISBN 3-540-06009-X. - Rajwade, A. R. (1993).
*Squares*. London Mathematical Society Lecture Note Series.**171**. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.