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 \[\left\{1,θ,θ^2,\ldots,θ^{2^{n-1}p^{m-1}(p-1)-1}\right\}\] is a basis for the ring of integers of $\mathbb{Q}(\theta )$, where $T(\theta )=0$.

10 pages