Monogenity of iterates of irreducible binomials

Abstract

One of the oldest problems of algebraic number theory is to find a method to determine if the ring of integers of a number field K is $\mathbb{Z}[\theta ]$ for some $\theta \in K$; a field for which the answer to this question is affirmative is referred to as a monogenic field. Suppose $f(x)=x^m-a\in \mathbb{Z}[x]$ is a monic irreducible polynomial and ${\alpha }_n\in \mathbb{C}$ is a root of $f^n(x)$, the n-fold composition of f. In this article, we prove a necessary and sufficient condition for $K_n=\mathbb{Q}({\alpha }_n)$ to be monogenic for every $n\in \mathbb{N}$ and $m\ge 2.$

Keywords:

• Irreducible polynomial
• iterates
• monogenic number field

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 11R04
• 37P05
• Secondary: 11R21

Acknowledgments

We thank the anonymous referee and the editor for their time and consideration.