Clifford algebras of $p$-central sets

TL;DR

The paper introduces a generalized Clifford algebra related to p-central sets, explores its structure using central simple algebra theory, and applies graph theory to solve specific cubic equations in algebraic integers.

Contribution

It generalizes the concept of Clifford algebras for p-central sets and develops a graph-theoretic method for analyzing their generators when p=3.

Findings

Established a structure theory for the generalized Clifford algebra

Proposed a graph-theoretic approach for p=3 case

Derived solutions to a specific cubic equation in algebraic integers

Abstract

A generalization of the term "generalized Clifford algebras" (as appears in papers on advances in applied Clifford algebras) is introduced. This algebra is studied by means of structure theory of central simple algebras. A graph theoretical approach is proposed for studying the generating set of this algebra in case where the prime number under discussion is three. Finally, it is shown how to obtain solutions in to the equation $\alpha Y^3=\alpha X_1^3+\beta X_2^3+\alpha^2 \beta^2 X_3^3$ in $\mathbb{Z}[\rho]$ where $\rho$ is the primitive third root of unity.