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⊆F and an octonion E-algebra R inside A, the set S of polynomials f(x)∈A[x] satisfying f(R)⊆R is an octonion (S∩F[x])-algebra, under the assumption that either 1/2∈R or char(F)≠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 1p(xp2−x)(xp−x) for prime p are integer-valued in the ring of polynomials A[x] over any real nonsplit Cayley-Dickson algebra A.