Korean J. Math. Vol. 30 No. 3 (2022) pp.425-431
DOI: https://doi.org/10.11568/kjm.2022.30.3.425

Factorization in the ring $h(\mathbb{Z}, \mathbb{Q})$ of composite Hurwitz polynomials

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Published: Sep 30, 2022
Keywords:
Composite Hurwitz polynomial ring, irreducible composite Hurwitz polynomial, quasi-primary, atomicComposite Hurwitz polynomial ring, atomic
Mathematics Subject Classification:
13A05, 13A15, 13F15, 13F20

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Dong Yeol Oh
Department of Mathematics Education, Chosun University, Gwangju 61452, Republic of Korea
Ill Mok Oh
Department of Mathematics Education, Chosun University, Gwangju 61452, Republic of Korea

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 $*$-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 $\leq 3$ in $h(\mathbb{Z}, \mathbb{Q})$ to be irreducible.



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