Octonion Polynomials with Values in a Subalgebra

Abstract

In this paper, we prove that given an octonion algebra $A$ over a field $F$, a subring $E \subseteq F$ and an octonion $E$-algebra $R$ inside $A$, the set $S$ of polynomials $f(x) \in A[x]$ satisfying $f(R) \subseteq R$ is an octonion $(S\cap F[x])$-algebra, under the assumption that either $\frac{1}{2} \in R$ or $\operatorname{char}(F) \neq 0$, and $R$ contains the standard generators of $A$ and their inverses. The project was inspired by a question raised by Werner on whether integer-valued octonion polynomials over the reals form a nonassociative ring. We also prove that the polynomials $\frac{1}{p}(x^{p^2}-x)(x^p-x)$ for prime $p$ are integer-valued in the ring of polynomials $A[x]$ over any real nonsplit Cayley-Dickson algebra $A$.