In mathematics, a formally real field in field theory is a field that shares certain algebraic properties with the real number field. A formally real field F may be characterized in any of the following equivalent ways:
- −1 is not a sum of squares in F.
- There exists an element of F which is not a sum of squares in F.
- If any sum of squares of elements of F equals zero, each element must equal zero.
- F can be totally ordered in such a way as to become an ordered field.
A formally real field with no formally real algebraic extension is a real closed field.
