Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras
Adam Chapman, Solomon Vishkautsan
TL;DR
This work investigates when roots of polynomials over Cayley–Dickson algebras guarantee a right factorization in $A[x]$, establishing this for linear and monic quadratic polynomials but showing failures in higher dimensions ($\dim A\ge 16$). It analyzes the companion polynomial $C_f(x)=f(x)\overline{f(x)}$ and clarifies its relationship to roots of $f$ across dimension thresholds, including counterexamples where $C_f$ has roots not matched by $f$. The paper also explores critical-point phenomena and demonstrates that Gauss–Lucas-type results do not universally extend to Cayley–Dickson algebras, providing explicit counterexamples. Finally, it develops a concrete left-eigenvalue framework for $2\times2$ octonion matrices, expressing eigenvalues in terms of roots of a quadratic $f(x)=b x^2+(a-d)x-c$ and showing that such matrices always have left eigenvalues, with extensions to associative division algebras. Together, these results deepen the understanding of polynomial factorization and spectral theory in nonassociative, noncommutative algebras with applications to octonion matrices.
Abstract
Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $α$ if and only $f(x)=g(x)\cdot (x-α)$ for some $g(x) \in A[x]$. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that it is when $f(x)$ is linear or monic quadratic, but it is false in general. Similar questions about the connections between $f$ and its companion $C_f(x)=f(x)\cdot \overline{f(x)}$ are studied. Finally, we compute the left eigenvalues of $2\times 2$ octonion matrices.