Iterates of Quadratics and Monogenicity

TL;DR

This paper studies the properties of iterated quadratic polynomials, specifically focusing on conditions for monogenicity and prime splitting in their generated number field extensions, providing new criteria and families with consistent monogenicity.

Contribution

It establishes necessary and sufficient conditions for monogenicity of iterated quadratic polynomials and constructs families where this property holds for all iterations.

Findings

Derived criteria for monogenicity of iterated quadratic polynomials.

Constructed families with monogenic iterates for all n.

Analyzed prime splitting in related number field extensions.

Abstract

We investigate monogenicity and prime splitting in extensions generated by roots of iterated quadratic polynomials. Let $f(x)\in\mathbb{Z}[x]$ be an irreducible, monic, quadratic polynomial, and write $f^n(x)$ for the $n^{\text{th}}$ iterate. We obtain necessary and sufficient conditions for $f^n(x)$ to be monogenic for each $n$. We use this to construct multiple families where $f^n(x)$ is monogenic for every $n>0$.