Fields of u-invariant 2r + 1

Abstract

In this article we provide a uniform construction of fields with all known u-invariants. We also obtain the new values for the u-invariant: 2r + 1, for r > 3. The main tools here are the new discrete invariant of quadrics (so-called elementary discrete invariant), and the methods of [14] (which permit one to reduce the questions of rationality of elements of the Chow ring over the base field to that over bigger fields, the generic point of a quadric).