Monogenic cyclotomic compositions

Abstract

Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=\Phi_{p^m}\left(\Phi_{2^n}(x)\right)$, where $\Phi_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\mathbb Q$ and that \[\left\{1,\theta,\theta^2,\ldots,\theta^{2^{n-1}p^{m-1}(p-1)-1}\right\}\] is a basis for the ring of integers of $\mathbb Q(\theta)$, where $T(\theta)=0$.