Counting finite-dimensional algebras over finite fields

TL;DR

This paper introduces an elementary counting method for finite-dimensional algebras over finite fields, demonstrated through the explicit example of 2-dimensional algebras over _{2}.

Contribution

The paper presents a new elementary approach for counting non-isomorphic algebras over finite fields, with detailed application to 2-dimensional cases.

Findings

Successfully counts 2-dimensional algebras over _{2}

Provides a general elementary method for algebra enumeration

Illustrates the method with explicit examples

Abstract

In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed dimension over a given finite field. We show how this method works for the explicit example of $2$-dimensional algebras over the field $\mathbb{F}_{2}$.