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arXiv · 2101.12553

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

Philippe Gille·Erhard Neher
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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 {\'e}tale 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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Philippe Gille, Erhard Neher. 2021-01-29. Springer's odd degree extension theorem for quadratic forms over semilocal rings. https://arxiv.org/abs/2101.12553

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