Weakly *-ordered *-fields
Let \(D\) be a division ring with involution *, center \(F\) and \([D : F] < \infty \). Since furthermore \(D\) is required to have a weak *-ordering, only the following two cases occur: (1) \((D,*)\) is a standard quaternion-algebra or (2) \([D : F]\) is odd. In \(\S 2\) the author looks at the more general case of Baer orderings of \(D\) and gives valuation theoretical conditions under which in case (2) a Baer ordering of \((F,*)\) extends to a Baer ordering of \((D,*) \). \(\S 3\) is devoted to weak preorderings and to generalizing results known for ordinary preorderings. One main result is Theorem 3.4, which gives a homeomorphism between the space of all weak *-orderings of \((D,*)\) and the space of all orderings of \(F' = \text{Fix} (*)\) that contain \(J (\{1\}) \cap F'\), where \(J (\{1\})\) is the smallest Baer preordering of \((D,*)\). The other important result is Proposition 3.9 which states that a weak preordering \(T\) of \((D,*)\) is indeed the intersection of all weak *-orderings of \((D,*)\) that contain \(T\). In \(\S 4\) the author introduces compatibility between *-valuations and weak preorderings and generalizes results known for compatibility in the commutative case. Finally, \(\S 5\) deals with \(T\)-forms, a generalization of abstract quadratic forms, and contains as a main result a representation criterion (Theorem 5.5).
scientific article; zbMATH DE number 762334
scientific article