  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 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.
