The discriminant Pfister form of an algebra with involution of capacity four

TL;DR

This paper introduces a unified approach using a discriminant Pfister form to characterize the decomposability of algebras with involution of specific degrees, providing new insights especially in characteristic 2.

Contribution

It develops a discriminant Pfister form that unifies decomposability criteria for various involutions on central simple algebras, including new results in characteristic 2.

Findings

Unified criteria for decomposability of algebras with involution

New result for symplectic involutions in characteristic 2

Characterization of algebra decomposability via Pfister forms

Abstract

To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with involution. In this way we obtain a unified approach to known decomposability criteria for several cases, and a new result for symplectic involutions on degree $8$ algebras in characteristic 2.