Characterizing reduced Witt rings of higher level

Abstract

Mulcahy's Spaces of Signatures (SOS) is an abstract setting for the reduced Witt rings of higher level of Becker and Rosenberg just as Marshall's Spaces of Orderings is an abstract setting for the ordinary reduced Witt ring. Finitely constructive SOS's are those built up in a finite number of steps from the smallest SOS using 2 operations. We show that finitely constructive SOS's are precisely those that arise from preordered fields (subject to a certain finiteness condition). This allows us to give an inductive construction for the reduced Witt rings of higher level for certain preordered fields, which generalizes a result of Craven for the ordinary reduced Witt ring. We also obtain a generalization of Brcker's results on die possible number of orderings of a field.