A parametrization of nonassociative cyclic algebras of prime degree

TL;DR

This paper explicitly parametrizes isomorphism classes of nonassociative cyclic algebras of prime degree, including quaternion algebras, over various fields, providing a comprehensive classification and explicit descriptions especially over local nonarchimedean fields.

Contribution

It introduces a new explicit parametrization method for nonassociative cyclic algebras of prime degree, extending to quaternion algebras and local fields, and clarifies conditions for isomorphism.

Findings

Isomorphism classes are determined by the cyclic extension and Galois generator.

Provides explicit parametrization over local nonarchimedean fields.

Establishes uniqueness conditions for isomorphism classes.

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 when the base field contains a primitive $m$th 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.