The Index of Composition of polynomials and its applications

The index of a given monic irreducible polynomial $f(x)\in\mathbb Z[x]$ having a root $θ$ is the index of $\mathbb Z[θ]$ in the ring of algebraic integers of $\mathbb Q(θ).$ Further, $f(x)$ is monogenic if its index is $1$. In this article, we prove that the index of $f(x)$ always divides the index