Quadratic modules and the orthogonal group over polynomial rings

The following conjecture of Parimala is proved: Any quadratic space over a polynomial ring with coefficients from an algebraically closed field of characteristic different from 2 is extended from the coefficient field. In the case of an arbitrary field of characteristic different from 2, an analogous result is obtained for quadratic spaces whose Witt index is at least 2. Also proved are general cancellation theorems for quadratic modules and a stabilization theorem for the orthogonal group over arbitrary polynomial rings.