Corollary 1 (*« Artin-Schreier Theorem »*)

— A necessary and sufficient condition for there to exist an ordering on a commutative field E which makes it an ordered field, is that the relation $x_1^2 + \ldots + x_n^2 = 0$ imply $x_1 = \ldots = x_n = 0$.

The necessity is obvious. Conversely, the stated condition implies that E is of characteristic zero, hence an extension of Q; then condition $(OE)$ is satisfied, and Th. 1 shows that there exists on E the structure of an ordered extension of Q, that is an ordered field structure.

There does not exist any ordered field structure on a field E in which $-1$ is a square, in particular on an algebraically closed field.