Factorization in the ring h(Z,Q) of composite Hurwitz polynomials

Abstract

Let $\mathbb{Z}$ and $\mathbb{Q}$ be the ring of integers and the field of rational numbers, respectively. Let $h(\mathbb{Z},\mathbb{Q})$ be the ring of composite Hurwitz polynomials. In this paper, we study the factorization of composite Hurwitz polynomials in $h(\mathbb{Z},\mathbb{Q})$. We show that every nonzero nonunit element of $h(\mathbb{Z},\mathbb{Q})$ is a finite $\ast$-product of quasi-primary elements and irreducible elements of $h(\mathbb{Z},\mathbb{Q})$. By using a relation between usual polynomials in $\mathbb{Q}[x]$ and composite Hurwitz polynomials in $h(\mathbb{Z},\mathbb{Q})$, we also give a necessary and sufficient condition for composite Hurwitz polynomials of degree $\le 3$ in $h(\mathbb{Z},\mathbb{Q})$ to be irreducible.