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.