On characterization of prime divisors of the index of a quadrinomial

TL;DR

This paper characterizes prime divisors of the discriminant of certain quadrinomials that do not divide the index, providing criteria for the monogenity of related number fields.

Contribution

It offers a complete characterization of prime divisors of the discriminant not dividing the index for quadrinomials, and establishes conditions for field monogenity.

Findings

Identifies all prime divisors of the discriminant not dividing the index.

Provides necessary and sufficient conditions for monogenity of specific number fields.

Enhances understanding of the relationship between discriminant divisors and field structure.

Abstract

Let $\theta$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $\theta$ over the rationals. Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $f(x).$ As a fascinating corollary, we deduce necessary and sufficient conditions for the monogenity of the field $K=\mathbb{Q}(\theta),$ where $\theta$ is associated with certain quadrinomials.