A parametrization of nonassociative cyclic algebras of prime degree

Nevins, Monica; Pumplün, Susanne

Abstract

We determine and explicitly parametrize the isomorphism classes of nonassociative quaternion algebras over a field of characteristic different from two, as well as the isomorphism classes of nonassociative cyclic algebras of odd prime degree m when the base field contains a primitive mth root of unity. In the course of doing so, we prove that any two such algebras can be isomorphic only if the cyclic field extension and the chosen generator of the Galois group are the same. As an application, we give a parametrization of nonassociative cyclic algebras of prime degree over a local nonarchimedean field F, which is entirely explicit under mild hypotheses on the residual characteristic. In particular, this gives a rich understanding of the important class of nonassociative quaternion algebras up to isomorphism over nonarchimedean local fields.

Citation

Nevins, M., & Pumplün, S. (2025). A parametrization of nonassociative cyclic algebras of prime degree. Journal of Algebra, 664(Part A), 631-654. https://doi.org/10.1016/j.jalgebra.2024.10.021

• Journal Article Type: Article
• Acceptance Date: Oct 21, 2024
• Online Publication Date: Oct 24, 2024
• Publication Date: Feb 15, 2025
• Deposit Date: Oct 21, 2024
• Publicly Available Date: Oct 25, 2025
• Journal: Journal of Algebra
• Print ISSN: 0021-8693
• Electronic ISSN: 1090-266X
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 664
• Issue: Part A
• Pages: 631-654
• DOI: https://doi.org/10.1016/j.jalgebra.2024.10.021
• Keywords: Nonassociative division algebras; nonassociative cyclic algebras
• Public URL: https://nottingham-repository.worktribe.com/output/40853446
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0021869324005660?via%3Dihub
• Additional Information: This article is maintained by: Elsevier; Article Title: A parametrization of nonassociative cyclic algebras of prime degree; Journal Title: Journal of Algebra; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.jalgebra.2024.10.021; Content Type: article; Copyright: © 2024 The Author(s). Published by Elsevier Inc.