Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-08T19:28:46.022Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Spaces of Signatures

Published online by Cambridge University Press:  20 November 2018

Victoria Powers
Victoria Powers*
Affiliation:
Emory University, Atlanta, Georgia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 5 , 01 October 1989 , pp. 808 - 829
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Becker, E. and Rosenberg, A., Reduced forms and reduced Witt rings of higher level, J. Algebra 92 (1985), 477–503.Google Scholar
2. Craven, T., Witt rings and orderings of skew fields, J. Algebra 77 (1982), 74–96.Google Scholar
3. Kalhoff, F., Spaces of orderings and Witt rings of planar ternary rings, J. Pure Appl. Math., to appear.Google Scholar
4. Marshall, M., Classification of finite spaces of orderings, Can. J. Math 31 (1979), 320–330.Google Scholar
5. Marshall, M. and Mulcahy, C., The Witt ring of a space of signatures. Preprint.Google Scholar
6. Marshall, M. and Walter, L., Signatures of higher level on rings with many units. Preprint.Google Scholar
7. Mulcahy, C., Spaces of signatures Ph.D. thesis, Cornell University (1985).Google Scholar
8. Mulcahy, C., An abstract approach to higher level forms and rigidity, Comm. Algebra 16 (1988), 577–612.Google Scholar
9. Mulcahy, C., The representation theorem for spaces of signatures, J. Algebra 3 (1988), 105–122.Google Scholar
10. Powers, V., Characterizing reduced Witt rings of higher level, Pac. J. Math. 728 (1987), 333–347.Google Scholar
11. Powers, V., Higher level reduced Witt rings of skew fields, Math. Zeit. 198 (1988), 545–554.Google Scholar