On ordered division rings (Q2717736)

\textit{A. Prestel} [Lectures on formally real fields. Lect. Notes Math. 1093, Berlin: Springer Verlag (1984; Zbl 0548.12011)] introduced a notion of semiordering for fields such that the usual property, \(ab\) is positive if \(a,b\) are positive, is replaced by \(ab^2\) is positive if \(a\) is positive and \(b\neq 0\). The author studies these semiorderings in the case of division rings, and he shows that they behave just as in the commutative situation. A result is that a division ring admits a semiordering precisely when \(-1\) is not a sum of products of squares and, moreover, that every semiordered division ring is ordered. Also relations between valuations and semiorderings are proved.