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Irreducibility of Hurwitz polynomials over the ring of integers

오동렬|서예림
강원경기수학회 (The Kangweon-Kyungki Mathematical Society)|2019.06|한국수학논문집 (Korean Journal of Mathematics)|제27권 제2호|pp. 465-474

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키워드

atomic.Hurwitz polynomial ringirreducible Hurwitz polynomialprimitive polynomial
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초록

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.

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.

참고문헌 (7)

  1. A. Benhissi, "Ideal structure of Hurwitz series rings", Contrib. Alg. Geom, 48, pp.251~256 (1997)
  2. A. Benhissi, "Basic properties of Hurwitz series rings", Ric. Mat, 61, pp.255~273 (2012)
  3. W.F. Keigher, "Adjunctions and comonads in differential algebra", Pacific J. Math, 59, pp.99~112 (1975)
  4. W. F. Keigher, "On the ring of Hurwitz series", Comm. Algebra, 25, pp.1845~1859 (1997)
  5. 임정욱, "Composite Hurwitz Rings Satisfying the Ascending Chain Condition on Principal Ideals", Kyungpook Mathematical Journal, 56(4), pp.1115~1123 (2016), 경북대학교 자연과학대학 수학과
  6. J.W. Lim, "Chain conditions on composite Hurwitz series rings", Open Math, 15, pp.1161~1170 (2017)
  7. Z. Liu, "Hermite and PS-rings of Hurwitz series", Comm. Algebra, 28, pp.299~305 (2000)

관련 논문AI추천

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