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The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

Author: Stephen Scully
Journal: Proc. Amer. Math. Soc. 146 (2018), 1-13
MSC (2010): Primary 11E81; Secondary 11E08
Published electronically: October 5, 2017
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Abstract: Let $ R$ be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over $ R$ remain anisotropic after base change to any odd-degree finite étale extension of $ R$. This generalization of the classical Artin-Springer theorem (concerning the situation where $ R$ is a field) was previously established in the case where all residue fields of $ R$ are infinite by I. Panin and U. Rehmann. The more general result presented here permits one to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic $ \neq 2$ to the case where the ring has at least one residue field which is finite.

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Stephen Scully
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton Alberta, Canada T6G 2G1

Keywords: Quadratic forms, semi-local rings, Artin-Springer theorem
Received by editor(s): February 26, 2016
Published electronically: October 5, 2017
Additional Notes: The author was supported by a PIMS postdoctoral fellowship held at the University of Alberta.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society

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