A Generic Characterization of Direct Summands for Orthogonal Involutions

Abstract

The “transcendental methods” in the algebraic theory of quadratic forms are based on two major results, proved in the 1960s by Cassels and Pfister, and known as the representation and the subform theorems. A generalization of the representation theorem was proven by Jean–Pierre Tignol in 1996, in the setting of central simple algebras with involution. This article studies the subform question for orthogonal involutions. A generic characterization of direct summands is given; an analogue of the subform theorem is proven for division algebras and algebras of index at most 2.

Key Words:

• Algebras with involution
• Quadratic forms
• Representation theorem
• Subform theorem

2000 Mathematics Subject Classification:

• Primary 16W10
• Secondary 11E39, 11E81, 16K20

ACKNOWLEDGMENTS

I would like to thank Philippe Gille, Bruno Kahn, R. Parimala, and Jean–Pierre Tignol for useful discussions on this question while this work was in progress.

Notes

Communicated by M. Vigue.