On polynomial equations over split-octonions: the arbitrary field case

TL;DR

This paper solves polynomial equations over split-octonions in arbitrary fields and determines roots of octonions, advancing algebraic understanding of these non-associative structures.

Contribution

It provides a complete solution to polynomial equations over split-octonions over arbitrary fields and characterizes roots of octonions.

Findings

Solved all polynomial equations with scalar coefficients over split-octonions

Determined square and cubic roots of octonions

Extended algebraic methods to non-associative structures

Abstract

Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.