Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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MODIFIED CYCLOTOMIC POLYNOMIALS

  • Ae-Kyoung, Cha (Department of Mathematics Education Korea National University of Education) ;
  • Miyeon, Kwon (Department of Mathematics University of Wisconsin-Platteville) ;
  • Ki-Suk, Lee (Department of Mathematics Education Korea National University of Education) ;
  • Seong-Mo, Yang (Department of Mathematics Education Korea National University of Education)
  • Received : 2021.12.01
  • Accepted : 2022.05.02
  • Published : 2022.11.30
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Abstract

Let H be a subgroup of $\mathbb{Z}^*_n$ (the multiplicative group of integers modulo n) and h1, h2, …, hl distinct representatives of the cosets of H in $\mathbb{Z}^*_n$. We now define a polynomial Jn,H(x) to be $$J_{n,H}(x)=\prod^l_{j=1} \left( x-\sum_{h{\in}H} {\zeta}^{h_jh}_n\right)$$, where ${\zeta}_n=e^{\frac{2{\pi}i}{n}}$ is the nth primitive root of unity. Polynomials of such form generalize the nth cyclotomic polynomial $\Phi_n(x)={\prod}_{k{\in}\mathbb{Z}^*_n}(x-{\zeta}^k_n)$ as Jn,{1}(x) = Φn(x). While the nth cyclotomic polynomial Φn(x) is irreducible over ℚ, Jn,H(x) is not necessarily irreducible. In this paper, we determine the subgroups H for which Jn,H(x) is irreducible over ℚ.

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Acknowledgement

This work was financially supported by the 2021 Sabbatical Leave Research Grant funded by Korea National University of Education.

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