A Study of monogenity of Binomial Composition

TL;DR

This paper investigates the monogenicity of number fields generated by roots of specific binomial compositions, characterizing primes affecting their integer bases and identifying conditions for monogenic pairs of binomials.

Contribution

It provides a complete characterization of primes dividing the index in fields generated by binomial compositions and identifies classes of binomials with monogenic properties.

Findings

Characterization of primes dividing the index in binomial composition fields

Conditions under which both binomials and their compositions are monogenic

Explicit examples of monogenic binomial pairs

Abstract

Let $\theta$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, \theta, \theta^2, \ldots, \theta^{n-1} \}$ is a basis for the ring $\Z_K$ of integers of $K = \Q(\theta)$. In this article, we study about the monogenity of number fields generated by a root of composition of two binomials. We characterise all the primes dividing the index of the subgroup $\Z[\theta]$ in $\Z_K$ where $K = \Q(\theta)$ with $\theta$ having minimal polynomial $F(x) = (x^m-b)^n - a \in \Z[x]$, $m\geq 1$ and $n \geq 2$. As an application, we provide a class of pairs of binomials $f(x)=x^n-a$ and $g(x)=x^m-b$ having the property that both $f(x)$ and $f(g(x))$ are monogenic.