An efficient method for solving equations in generalized quaternion and octonion algebras

TL;DR

This paper introduces formulas for transforming bases between different generalized quaternion and octonion algebras, facilitating solutions to algebraic equations and extending De Moivre's formula in these contexts.

Contribution

It provides explicit basis transformation formulas for generalized quaternion and octonion algebras, enabling new methods for solving algebraic equations within these structures.

Findings

Derived basis transformation formulas for generalized quaternion algebras.

Extended De Moivre's formula to generalized quaternion and octonion algebras.

Applied results to solve algebraic equations in these algebras.

Abstract

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of these results to the algebraic equations and De Moivre s formula in generalized quaternion algebras and in generalized octonion division algebras.