The Index of Composition of polynomials and its applications

Source abstract

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 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 $\langle I_n\rangle$ 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.