Factorization in the Composition Algebras

Abstract

Let $\mathcal{O}$ be a maximal arithmetic in one of the four (non-split) composition algebras over ℝ, and let [ρ] = mn be the norm of an element ρ in $\mathcal{O}$. Rehm [14] describes an algorithm for finding all factorizations of ρ as ρ = αβ, where [α] = m, [β] = n and (m,n) = 1. Here, we extend the algorithm to general ρ, m, and n, providing precise geometrical configurations for the sets of left- and right-hand divisors.