The Index of Composition of polynomials and its applications

Abstract

The index of a given monic irreducible polynomial $f(x)\in \mathbb{Z}[x]$ having a root $\theta$ is the index of $\mathbb{Z}[\theta ]$ in the ring of algebraic integers of $\mathbb{Q}(\theta ).$ 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 of $f(g(x)),$ where $f(x),g(x)\in \mathbb{Z}[x]$. Additionally, we give the precise power of the index of $f(x)$ dividing the index of $f(g(x))$. Also, we provide a necessary condition for the monogenity of $f(g(x))$. As an application of our results, we prove that the $n$-fold composition $f^n(x)=(f\circ f\cdots \circ f)(x)$ of a polynomial $f(x)$ is monogenic for all $n\in \mathbb{N}$ if and only if the sequence $⟨I_n⟩$ is convergent, where $I_n$ is the index of $f^n(x)$. Moreover, we show that for a given $n\in \mathbb{N}$, if $I_n$ is $m^{\text{th}}$ power free integer, then the $n$-tower $\mathbb{Q}\subseteq K_1\subseteq \cdots \subseteq K_{n-1}\subseteq K_n,$ defined by $n$ iterates of a polynomial $f(x)\in \mathbb{Z}[x]$ contains at least $\max \{\lfloor n-{\log }_{°f}m\rfloor ,0\}$ monogenic number fields.