On the index of power compositional polynomials

TL;DR

This paper characterizes when power compositional polynomials are monogenic, linking their algebraic properties to the monogenicity of the original polynomial, with applications to specific polynomial families.

Contribution

It provides necessary and sufficient conditions for monogenicity of power compositional polynomials and relates it to the original polynomial's properties.

Findings

Conditions for monogenicity of $f(x^k)$ based on $f(x)$

Characterization of monogenicity for polynomials of the form $x^d + A h(x)$

Infinite families of monogenic polynomials provided as examples

Abstract

The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $\theta$ is the index $[\mathbb{Z}_K:\mathbb{Z}[\theta]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(\theta)$. If $[\mathbb{Z}_K:\mathbb{Z}[\theta]]=1$, 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 $d>1, \operatorname{deg} h(x)<d$ and $|h(0)|=1$, we prove that for each positive integer $k$ with $\operatorname{rad}(k)\mid \operatorname{rad}(A)$, 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.