Narrow Results

We provide an analogue of Wedderburn’s factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree $n$ with $n$ conjugacy classes of roots, counting multiplicities.

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