Modified cyclotomic polynomials

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{align*} \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{align*} where $\zeta_n=e^{\frac{2\pi i}{n}}$ is 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 $\mathbb{Q}$.