Quadratic spaces and quaternion algebras

It is shown that the existence of a nontrivial quaternion algebra over a field F of characteristic not equal to 2 implies the existence of quadratic F[X₁...,X_n]-spaces (n2) which are not extensions of F.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 110–120, 1978.