On the index of power compositional polynomials

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