On the index of power compositional polynomials

Abstract

The index of a monic irreducible polynomial $f(x)\in \mathbb{Z}[x]$ having a root $\hat{I}¸$ is the index $[{\mathbb{Z}}_K:\mathbb{Z}[\hat{I}¸]]$, where ${\mathbb{Z}}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(\hat{I}¸)$. If $[{\mathbb{Z}}_K:\mathbb{Z}[\hat{I}¸]]=1$, then $f(x)$ is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial $f(x^k)$ belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our results, for a polynomial $f(x)=x^d+A\cdot h(x)\in \mathbb{Z}[x],$ with $d>1,\mathrm{deg}h(x)<d$ and $|h(0)|=1$, we prove that for each positive integer $k$ with $\mathrm{rad}(k)\mid \mathrm{rad}(A)$, the power compositional polynomial $f(x^k)$ is monogenic if and only if $f(x)$ is monogenic, provided that $f(x^k)$ is irreducible. At the end of the paper, we give infinite families of polynomials as examples.