Applications of Clifford algebras to involutions and quadratic forms

摘要

Let k be a field, char k not equal 2, F = k(x), D a biquaternion division algebra over k, and a an orthogonal involution on D with nontrivial discriminant. We show that there exists a quadratic form phi is an element of I-2 (F) such that dim phi = 8, [C(phi)] = [D], and phi does not decompose into a direct sum of two forms similar to two-fold Pfister forms. This implies in particular that the field extension F(D)IF is not excellent. Also we prove that if A is a central simple K-algebra of degree 8 with an orthogonal involution sigma, then a is hyperbolic if and only if sigma(K(A)) is hyperbolic. Finally, let sigma be a decomposable orthogonal involution on the algebra M-2m(K). In the case m <= 5 we give another proof of the fact that or is a Pfister involution. If m >= 2n(-2) - 2 and n >= 5, we show that q(sigma) is an element of I-n (K), where q(sigma) is a quadratic form corresponding to sigma. The last statement is founded on a deep result of Orlov et al. (2000) concerning generic splittings of quadratic forms.