Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online July 12, 2022

Copyright © The Korean Mathematical Society.

Modified Cyclotomic Polynomials

Ae-Kyoung Cha, Miyeon Kwon, Ki-Suk Lee, and Seong-Mo Yang

Korea National University of Education, University of Wisconsin-Platteville

Abstract

Let $H$ be a subgroup of $\mathbb{Z}_n^\ast$ (the multiplicative
group of integers modulo $n$) and $h_1,h_2,\ldots,h_l$ distinct
representatives of the cosets of $H$ in $\mathbb{Z}_n^\ast$. We
now define a polynomial $J_{n,H}(x)$ to be
\begin{eqnarray*}
\begin{split}
J_{n,H}(x)=\prod\limits_{j=1}^{l} \bigg( x-\sum\limits_{h \in
H}\zeta_n^{h_jh} \bigg),
\end{split}
\end{eqnarray*}
where $\zeta_n=e^{\frac{2\pi i}{n}}$ be the $n$th primitive root
of unity. Polynomials of such form generalize the $n$th cyclotomic
polynomial $\Phi_n(x)=\prod_{k \in
\mathbb{Z}_n^\ast}(x-\zeta_n^k)$ as $J_{n,\{1\}}(x)=\Phi_n(x)$.
While the $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible
over $\mathbb{Q}$, $J_{n,H}(x)$ is not necessarily irreducible. In
this paper, we determine the subgroups $H$ for which $J_{n,H}(x)$
is irreducible over $\mathbb{Q}$.

Keywords: Cyclotomic polynomials; irreducible polynomials; Galois group

MSC numbers: 12E05

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