Springer’s Odd Degree Extension Theorem for quadratic forms over semilocal rings

A fundamental result of Springer says that a quadratic form over field characteristic ≠2 is isotropic if it so after an odd degree extension. In this paper we generalize Springer’s theorem as follows. Let R be arbitrary semilocal ring, let S finite R-algebra constant degree, which étale or generated by one element, and q nonsingular R-quadratic whose base ring extension qS isotropic. We show then already