Higher level reduced Witt rings of skew fields
type: Item
instance of: scholarly article
title: Higher level reduced Witt rings of skew fields
zbMATH Open document ID: 0627.10014
DOI: 10.1007/BF01162872
author: Victoria Powers
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173942
review text: 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.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 12E15
zbMATH DE Number: 4019150
zbMATH Keywords: 2-primary
zbMATH Keywords: skew fields
zbMATH Keywords: spaces of signatures
zbMATH Keywords: preordered skew fields
zbMATH Keywords: higher level reduced Witt rings
MaRDI profile type: Publication
cites work: Partial orders on a field and valuation rings
cites work: Local global theorems for diagonal forms of higher degree.
cites work: Q3918202
cites work: Reduced forms and reduced Witt rings of higher level
cites work: Q4126487
cites work: Witt rings and orderings of skew fields
cites work: Classification of Finite Spaces of Orderings
cites work: Valuations of skew fields and projective Hjelmslev spaces
cites work: An abstract approach to higher level forms and rigidity
cites work: The representation theorem for spaces of signatures
cites work: Characterizing reduced Witt rings of higher level
cites work: Lectures on formally real fields
cites work: Q5796078
description: scientific article; zbMATH DE number 4019150