On polynomial equations over split octonions

TL;DR

This paper solves polynomial equations over split octonions with scalar coefficients and determines the n-th roots of an octonion, advancing understanding of octonionic algebraic structures.

Contribution

It provides a complete solution to polynomial equations with scalar coefficients over split octonions and computes octonion roots, which was previously unresolved.

Findings

All polynomial equations with scalar coefficients are solvable over split octonions.

Explicit formulas for n-th roots of octonions are derived.

The results deepen the algebraic understanding of split octonions.

Abstract

Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.