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 ≠2 is isotropic if it is so after an odd degree field extension. In this paper we generalize Springer’s theorem as follows. Let R be 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 qS is isotropic. We show that then already q is isotropic.