Arithmetic Aspects of Number Fields Generated by Polynomial Families

TL;DR

This paper investigates the arithmetic properties of number fields generated by specific polynomial families, deriving discriminant formulas, conditions for monogeneity, and analyzing Galois groups, with implications for algebraic number theory.

Contribution

It provides explicit discriminant formulas, necessary and sufficient conditions for monogeneity, and extends these results to polynomial compositions, advancing understanding of number field arithmetic.

Findings

Explicit discriminant formulas for the polynomial family

Conditions for monogeneity based on prime divisors of parameters

Identification of cases with full symmetric Galois group

Abstract

Let $f(x)=(x^{k}+c)^{m}-ax^{n}\in\mathbb{Z}[x]$ be an irreducible polynomial over $\mathbb{Q}$, where $k,m,n\in\mathbb{N}$ with $km>n$, and let $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $f(x)$. We investigate the arithmetic properties of the number fields that arise from this family. We first obtain an explicit formula for the discriminant of $f(x)$. Using this formula, we establish necessary and sufficient conditions for the monogeneity of $f(x)$, expressed in terms of the prime divisors of $a$ and $c$ and the parameters $k,m,n$. This yields infinite families of monogenic polynomials of arbitrary degree, including families with a non-square-free discriminant. Building on these results, we extend our algebraic characterization to composite polynomials, establishing some explicit conditions for the monogeneity of the composition of $f(x)$ with an arbitrary polynomial $g(x)$. From an analytic point of view, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. We further study non-monogeneity via the field index $i(K)$ and, for each prime $p$, provide sufficient conditions ensuring $\nu_p(i(K))=1$, yielding partial progress toward a problem of Narkiewicz. We also highlight a connection with a class of differential equations naturally associated with $f(x)$. As an application, we determine the conditions under which the splitting field of $f(x)$ has a full symmetric Galois group. Several explicit examples illustrate our results.