Involutions and stable subalgebras

TL;DR

This paper studies étale subalgebras within central simple algebras with involution, providing a unified approach across characteristics and offering a conceptual proof of a key theorem about division algebras of degree eight.

Contribution

It introduces a unified method for analyzing involutions and symmetric subalgebras, leading to a new conceptual proof of Rowen's theorem on division algebras.

Findings

Unified treatment of involutions across all characteristics

Conceptual proof of Rowen's theorem for degree eight division algebras

Identification of maximal subfields as triquadratic extensions

Abstract

Given a central simple algebra with involution over an arbitrary field, \'etale subalgebras contained in the space of symmetric elements are investigated. The method emphasizes the similarities between the various types of involutions and privileges a unified treatment for all characteristics whenever possible. As a consequence a conceptual proof of a theorem of Rowen is obtained, which asserts that every division algebra of exponent two and degree eight contains a maximal subfield that is a triquadratic extension of the centre.