The Irreducibility and Monogenicity of Power-Compositional Trinomials

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 ${\mathcal F}(x):=x^m+Ax^{m-1}+B$\mathrm{.}Inthisarticle,wedeterminesetsofconditionson$m$,$A$,and$B$,suchthatthepower-compositionaltrinomial${\mathcal F}(x^{p^n})$ismonogenicforallintegers$n\ge 0$andagivenprime$p$. Furthermore, we prove the actual existence of infinite families of such trinomials $\mathcal{F}(x)$.