A formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4. As an application, an alternative proof of Shapiro's "Pfister factor conjecture" is given for tensor products of at most five quaternion algebras. 2004 Elsevier Inc. All rights reserved.Throughout this paper, the characteristic of the base field F is supposed to be different from 2. Recall from [4, (2.5)] that a symplectic (respectively orthogonal) involution on a central simple F -algebra A is a map A → A which after scalar extension to a splitting field K can be identified with the adjoint involution of a nonsingular alternating (respectively symmetric) bilinear form over a K-vector space. The discriminant of an orthogonal involution ρ on a central simple F -algebra of even degree was defined by Jacobson, Tits, and Knus-Parimala-Sridharan, see [4, §7]. It is an element of the Galois cohomology group H 1 (F, µ 2 ) F × /F ×2 which we denote by disc ρ.For symplectic involutions σ, σ 0 on a central simple F -algebra A of degree divisible by 4, a relative discriminant ∆ σ 0 (σ ) ∈ H 3
In this note we show that the hermitian level of a quaternion division algebra with involution of second kind, is always a power of 2, when it is ®nite. This result holds for a ®eld with trivial or non-trivial involution, and quaternion division algebras with involution of ®rst kind [6,5,9] .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.