Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Nonexcellence

A field extension L/F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L/F is called universally excellent if L · E/E is excellent for any field extension E/F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi-Brauer varieties and its relationship with the relative Galois cohomology group H3(L/F) for a generic (common) splitting field L of the corresponding central simple F-algebras.