On the index of power compositional polynomials

Abstract

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