Polynomial equations over octonion algebras.

In this paper, we present a complete method for finding the roots of all polynomials of the form ϕ (z) = c n z n + c n − 1 z n − 1 + ⋯ + c 1 z + c 0 over a given octonion division algebra. When ϕ (z) is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of ϕ (z) , showing that the right eigenvalues form the conjugacy classes of the roots of ϕ (z) and the left eigenvalues form a larger set than the roots of ϕ (z). [ABSTRACT FROM AUTHOR]