Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-16T13:24:19.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Spaces of Orderings IV

Published online by Cambridge University Press:  20 November 2018

Murray Marshall
Murray Marshall*
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
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.

A major goal of this paper is to give a proof of the following isotropy criterion: Let X = (X,G) be a space of orderings in the terminology of [9] or [10], and let f be a form defined over G.Then f is anisotropic over X if and only if f is anisotropic over some finite subspace of X.This is the content of Theorem 1.4, and generalizes [1, Corollary 3.4]. Moreover, in view of the known structure of finite spaces (see [9]), this has, essentially, the strength of [2, Satz 3.9] or [12, Theorem 8.12]. The technique used to prove this criterion is roughly patterned on that of [6], and yields some interesting by-products: An interesting invariant of a space of orderings called the chain length is introduced (Definition 1.1) and spaces of orderings with finite chain length are classified (Theorem 1.6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 603 - 627
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Becker, E. and Brôcker, L., On the description of the reduced Witt ring, J. Alg. 52 (1978), 328–346.Google Scholar
2. Brôcker, L., Zur théorie der quadratischen formen ilber formal reellen Kôrpern, Math. Ann. 210 (1974), 223, 256.Google Scholar
3. Brôcker, L., Characterization of fans and hereditarily pythagorian fields, Math. Z. 152 (1976), 149–163.Google Scholar
4. Brôcker, L., tJber die anzahl der anordnungen eines kommutativen kôrpers, Archiv der Math. 29 (1977), 458–464.Google Scholar
5. Brown, R., The reduced Witt ring of a formally real field, Trans. Amer. Math. Soc. 230 (1977), 257–292.Google Scholar
6. Brown, R. and Marshall, M., The reduced theory of quadratic forms, to appear, Rocky Mtn. J. of Math.Google Scholar
7. Craven, T., Characterizing reduced Witt rings of fields, J. of Alg. 58 (1978), 68–77.Google Scholar
8. Kleinstein, J. and Rosenberg, A., Succinct and representational Witt rings, to appear, Can. J. Math.Google Scholar
9. Marshall, M., Classification of finite space of orderings, Can. J. Math. 31 (1979), 320–330.Google Scholar
10. Marshall, M., Quotients and inverse limits of spaces of orderings, Can. J. Math. 31 (1979), 604–616.Google Scholar
11. Marshall, M., The Witt ring of a space of orderings, Trans. Amer. Math. Soc. 258 (1980), 505–521.Google Scholar
12. Prestel, A., Lectures on formally real fields, Lecture Notes, IMPA, Rio de Janeiro (1976).Google Scholar