Polynomial Equations over Octonion Algebras

Abstract

In this paper we present a complete method for finding the roots of all polynomials of the form $\ddot{I}•(z)=c_n{z}^n+c_{n-1}{z}^{n-1}+\cdots +c_{1}z+c_0$ over a given octonion division algebra. When $\ddot{I}•(z)$ is monic we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of $\ddot{I}•(z)$, showing that the right eigenvalues form the conjugacy classes of the roots of $\ddot{I}•(z)$ and the left eigenvalues form a larger set than the roots of $\ddot{I}•(z)$.