On three-dimensional associative algebras

TL;DR

This paper classifies three-dimensional associative algebras over various fields, providing canonical forms and comparing with existing classifications over complex numbers and nilpotent cases.

Contribution

It offers a comprehensive list of three-dimensional associative algebras with canonical representatives over fields of characteristic not two or three, extending previous classifications.

Findings

List of associative algebras with canonical forms

Comparison with complex number classifications

Comments on nilpotent cases

Abstract

This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of characteristic different from two and three. We also compare our lists with the most recent classifications over the complex numbers and with the nilpotent case over arbitrary base fields in dimension three, adding some comments.