A field $F$ is called formally real if $-1$ can not be expressed as a sum of squares (of elements of $F$ ).

Given a field $F$ , let $S_F$ be the set of all sums of squares in $F$ . The following are equivalent conditions that $F$ is formally real:

1. $-1\notin S_F$
2. $S_F\not= F$ and $\operatorname{char}(F)\ne 2$
3. $\sum {a_i}^2=0$ implies each $a_i=0$ , where $a_i\in F$
4. $F$ can be ordered (There is a total order $<$ which makes $F$ into an ordered field)

Some Examples:

• $\mathbb{R}$ and $\mathbb{Q}$ are both formally real fields.
• If $F$ is formally real, so is $F(\alpha)$ , where $\alpha$ is a root of an irreducible polynomial of odd degree in $F[x]$ . As an example, $\mathbb{Q}(\sqrt[3]{2}\omega)$ is formally real, where $\omega\not= 1$ is a third root of unity.
• $\mathbb{C}$ is not formally real since $-1=i^2$ .
• Any field of characteristic non-zero is not formally real; it is not even orderable.