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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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참고문헌 (7)
참고문헌 신청
A. Benhissi / 1997 / Ideal structure of Hurwitz series rings / Contrib. Alg. Geom 48:251~256
A. Benhissi / 2012 / Basic properties of Hurwitz series rings / Ric. Mat 61:255~273
W.F. Keigher / 1975 / Adjunctions and comonads in differential algebra / Pacific J. Math 59:99~112
W. F. Keigher / 1997 / On the ring of Hurwitz series / Comm. Algebra 25:1845~1859
임정욱 / 2016 / Composite Hurwitz Rings Satisfying the Ascending Chain Condition on Principal Ideals / Kyungpook Mathematical Journal 56(4):1115~1123
A. Benhissi / 2012 / Basic properties of Hurwitz series rings / Ric. Mat 61:255~273
W.F. Keigher / 1975 / Adjunctions and comonads in differential algebra / Pacific J. Math 59:99~112
W. F. Keigher / 1997 / On the ring of Hurwitz series / Comm. Algebra 25:1845~1859
임정욱 / 2016 / Composite Hurwitz Rings Satisfying the Ascending Chain Condition on Principal Ideals / Kyungpook Mathematical Journal 56(4):1115~1123
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