Monogenic Fields from Polynomial Compositions with Applications

TL;DR

This paper characterizes when polynomial compositions generate monogenic number fields, providing conditions, asymptotic counts, and applications to polynomials with non-square-free discriminants.

Contribution

It establishes necessary and sufficient conditions for monogenicity of composed polynomials and derives asymptotic estimates for their counts.

Findings

Conditions for monogenicity of composed polynomials are established.

Asymptotic estimates for the number of monogenic polynomials are derived.

Construction of polynomials with non-square-free discriminants is demonstrated.

Abstract

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[\alpha]$ for some $\alpha \in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}_{K_i}:\mathbb{Z}[\alpha_i]]=1$, where $K_i=\mathbb{Q}(\alpha_i)$ and $\alpha_i$ is a root of the composed polynomial $f_i(x^k+b)$ for $i=1,2$. Here, $f_1(x)=x^n+c\sum_{j=1}^{n}(ax)^{n-j}\in\mathbb{Z}[x]$ and $f_2(x)=x^n+c\sum_{j=1}^{n}a^{j-1}x^{n-j}\in\mathbb{Z}[x]$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.