Abstract:

A field extension L / F is called excellent if, for any quadratic form over F, the anisotropic part (L)an of 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.

Keywords: quadratic form over a field; excellent field extension; central simple algebra; Severi–Brauer variety; Chow group; Galois cohomology

Language: English

Document Type: Regular paper

Affiliations: 1: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, 198904, Russia. e-mail: oleg@izh.usr.pu.ru 2: Westfälische Wilhelms-Universität, Mathematisches Institut, Einsteinstraße 62, D-48149 Münster, Germany. e-mail: karpenk@math.uni-muenster.de