Communications in Mathematics |

Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ 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.

Source: arXiv.org:2311.16704

Volume: Volume 33 (2025), Issue 3 (Special issue: European Non-Associative Algebra Seminar)

Published on: February 27, 2024

Accepted on: February 9, 2024

Submitted on: November 29, 2023

Keywords: Mathematics - Rings and Algebras,Mathematics - Number Theory,primary 17A75, secondary 17A45, 17A35, 17D05, 37P35, 37C25

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