Formally real field

Remarks

Equivalently, for all $n \ge 1$ and $a_1, \dots, a_n \in F$, if $a_1^2 + \cdots + a_n^2 = 0$, then $a_1 = \cdots = a_n = 0$. By the Artin–Schreier theorem on orderings, $F$ is formally real if and only if $F$ carries a structure of ordered field. A formally real field has characteristic $0$, since in characteristic $p$ one has $-1 = 1^2 + \cdots + 1^2$ with $p - 1$ terms. Example: $\Q$ is formally real, because a sum of rational squares is at least $0$ in the usual order. Non-example: $\C$ is not formally real, because $-1 = i^2$; this is why $\C$ cannot be ordered.