Pure Imaginary Roots of Quaternion Standard Polynomials

TL;DR

This paper introduces a new method for solving quaternion equations, providing explicit solutions for cubic cases with pure imaginary roots and analyzing the roots of two-sided quaternion polynomials.

Contribution

It presents a novel approach to solving quaternion equations, including explicit solutions for cubic equations with pure imaginary roots and insights into root counts.

Findings

Explicit solutions for cubic quaternion equations with pure imaginary roots

Reobtained formulas for quadratic quaternion equations

Analysis of pure imaginary roots in two-sided quaternion polynomials

Abstract

In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic quaternion equation, as long as the equation has at least one pure imaginary root. We also discuss the number of essential pure imaginary roots of a two-sided quaternion polynomial.