Higher level reduced Witt rings of skew fields
The purpose of this paper is to extend to skew fields \textit{E. Becker} and \textit{A. Rosenberg}'s theory of higher level reduced forms and reduced Witt rings [J. Algebra 92, 477--503 (1985; Zbl 0555.10009)]. We can then show that in the 2-primary case, skew fields give rise to spaces of signatures, \textit{C. Mulcahy}'s abstract setting for this theory [An abstract approach to higher level forms and rigidity. Commun. Algebra 16, No. 3, 577--612 (1988; Zbl 0644.10016); The representation theorem for spaces of signatures. Rocky Mt. J. Math. 19, No. 3, 807--810 (1989; Zbl 0705.11023)]. This allows us to extend many results on preordered fields to preordered skew fields, for example we can obtain a ``representation theorem'' for skew fields. The notions of a preorder (of level 1) and the reduced Witt ring associated to a preorder were extended to skew fields by \textit{A. Tschimmel-Faltings} [Über Anordnungsräume von Schiefkörpern (Ph.D. dissertation. Universität Münster, 1981)]. \textit{T. C. Craven} extended the notion of a higher level ordering to skew fields [J. Algebra 77, 74--96 (1982; Zbl 0493.10026)], and developed some of the valuation theory which we will need. The proofs given here follow those of the commutative case, and in some cases no extra work is needed. Occasionally we make use of the commutative case to prove facts ``one element at a time'' by reducing to a commutative subfield of our skew field.
scientific article; zbMATH DE number 4019150