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Spaces of Orderings IV

Published online by Cambridge University Press:  20 November 2018

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

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