Nonassociative differential extensions of characteristic p

TL;DR

This paper introduces and studies nonassociative differential extensions over fields of characteristic p, providing criteria for these algebras to be division and generalizing classical associative results.

Contribution

It develops a framework for nonassociative differential extensions, extending classical associative algebra results, and constructs new families of nonassociative division algebras.

Findings

Criteria for nonassociative algebras to be division

Construction of nonassociative division algebra families

Generalization of associative cyclic extensions

Abstract

Let F be a field of characteristic p. We define and investigate nonassociative differential extensions of F and of a central simple division algebra over F and give a criterium for these algebras to be division. As special cases, we obtain classical results for associative algebras by Amitsur and Jacobson. We construct families of nonassociative division algebras which can be viewed as generalizations of associative cyclic extensions of a purely inseparable field extension of exponent one or a central division algebra. Division algebras which are nonassociative cyclic extensions of a purely inseparable field extension of exponent one are particularly easy to obtain.