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 1s.) This can be expressed in first-order logic by ${\displaystyle \forall x_{1}(-1\neq x_{1}^{2})}$, ${\displaystyle \forall x_{1}x_{2}(-1\neq x_{1}^{2}+x_{2}^{2})}$, etc., with 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

1. Rajwade, Theorem 15.1.
2. Milnor and Husemoller (1973) p.60
3. 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. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.