The Irreducibility and Monogenicity of Power-Compositional Trinomials

Abstract

A polynomial $f(x)\in \mathbb{Z}[x]$ of degree $N$ is called \emph{monogenic} if $f(x)$ is irreducible over $\mathbb{Q}$ and $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{N-1}\}$ is a basis for the ring of integers of $\mathbb{Q}(\theta )$, where $f(\theta )=0$. Define $\mathcal{F}(x):=x^m+A{x}^{m-1}+B$. In this article, we determine sets of conditions on $m$, $A$, and $B$, such that the power-compositional trinomial $\mathcal{F}(x^{p^n})$ is monogenic for all integers $n\ge 0$ and a given prime $p$. Furthermore, we prove the actual existence of infinite families of such trinomials $\mathcal{F}(x)$.