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

Dong Yeol Oh, Ye Lim Seo

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.

Keywords


Hurwitz polynomial ring, irreducible Hurwitz polynomial, primitive polynomial, atomic.

Subject classification

13A05, 13A15, 13F15, 13F20

Sponsor(s)



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References


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