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 $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Define ${\mathcal F}(x):=x^m+Ax^{m-1}+B$. 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 ${\mathcal F}(x)$.

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