Generic algebras with involution of degree 8m

Abstract

The centers of the generic central simple algebras with involution are interesting objects in the theory of central simple algebras. These fields also arise as invariant fields for linear actions of projective orthogonal or symplectic groups. In this paper, we prove that when the characteristic is not 2, these fields are retract rational, in the case the degree is 8 m and m is odd. We achieve this by proving the equivalent lifting property for the class of central simple algebras of degree 8 m with involution. A companion paper [D.J. Saltman, Invariant fields of symplectic and orthogonal groups, preprint] deals with the case of m , 2 m , and 4 m where stronger rationality results are proven.