Korean J. Math. Vol. 27 No. 2 (2019) pp.465-474
DOI: https://doi.org/10.11568/kjm.2019.27.2.465

Irreducibility of Hurwitz polynomials over the ring of integers

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Published: Jun 30, 2019
Keywords:
Hurwitz polynomial ring, irreducible Hurwitz polynomial, primitive polynomial, 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
Ye Lim Seo
Department of Mathematics Education Chosun University, Gwangju 61452, Republic of Korea

Abstract

Let $\mathbb{Z}$ be the ring of integers and $\mathbb{Z}[X]$ (resp., $h(\mathbb{Z})$) be the ring of polynomials (resp., Hurwitz polynomials) over $\mathbb{Z}$. In this paper, we study the irreducibility of Hurwitz polynomials in $h(\mathbb{Z})$. We give a sufficient condition for Hurwitz polynomials in $h(\mathbb{Z})$ to be irreducible, and we then show that $h(\mathbb{Z})$ is not isomorphic to $\mathbb{Z}[X]$. By using a relation between usual polynomials in $\mathbb{Z}[X]$ and Hurwitz polynomials in $h(\mathbb{Z})$, we give a necessary and sufficient condition for Hurwitz polynomials over $\mathbb{Z}$ to be irreducible under additional conditions on the coefficients of Hurwitz polynomials.


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