Abstract
In this paper, we present a complete method for finding the roots of all polynomials of the form [Formula: see text] over a given octonion division algebra. When [Formula: see text] is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of [Formula: see text], showing that the right eigenvalues form the conjugacy classes of the roots of [Formula: see text] and the left eigenvalues form a larger set than the roots of [Formula: see text].
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