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Journal Articles Indagationes Mathematicae Year : 2021

Springer's odd degree extension theorem for quadratic forms over semilocal rings

Abstract

A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R-algebra of constant odd degree, which is étale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic.
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Dates and versions

hal-03124810 , version 1 (29-01-2021)
hal-03124810 , version 2 (18-06-2021)

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Philippe Gille, Erhard Neher. Springer's odd degree extension theorem for quadratic forms over semilocal rings. Indagationes Mathematicae, In press, ⟨10.1016/j.indag.2021.06.009⟩. ⟨hal-03124810v2⟩
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