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

TL;DR

This paper extends Springer's theorem from fields to semilocal rings, showing that isotropy of quadratic forms over certain odd degree extensions implies isotropy over the base ring.

Contribution

The paper generalizes Springer's theorem to semilocal rings with specific finite algebra extensions, broadening its applicability.

Findings

Quadratic form isotropy over semilocal rings is preserved under certain odd degree extensions.

Extension conditions include étale or single-generator algebra structures.

The generalization applies to nonsingular 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 {\'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.