Abstract
In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed, finite dimension over a given finite field. We show how this method works in the case of 2-dimensional algebras over the field {mathbb {F}}_{2}.
Highlights
Classifying finite-dimensional algebras over a given field is usually a very hard problem
The first general result was a classification by Hendersson and Searle of 2-dimensional algebras over the base field R, which appeared in 1992 ([1])
This was generalised in 2000 by Petersson ([3]), who managed to give a full classification of 2-dimensional algebras over an arbitrary base field
Summary
Classifying finite-dimensional algebras over a given field is usually a very hard problem. The first general result was a classification by Hendersson and Searle of 2-dimensional algebras over the base field R, which appeared in 1992 ([1]). This was generalised in 2000 by Petersson ([3]), who managed to give a full classification of 2-dimensional algebras over an arbitrary base field. Our aim in this paper is to give perhaps not a classification but at least a way to compute the exact number of non-isomorphic n-dimensional algebras over a fixed finite field by elementary means. In the first three sections, we give a proof based on concrete calculations, while Sect.
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