On characterization of Monogenic number fields associated with certain quadrinomials and its applications

TL;DR

This paper characterizes prime divisors of discriminants of specific quadrinomials, establishes conditions for the associated number fields to be monogenic, and explores solutions to related differential equations.

Contribution

It provides a complete characterization of prime divisors affecting monogenicity in number fields generated by certain quadrinomials and links these properties to differential equations.

Findings

Prime divisors of discriminants not dividing the index are characterized.

Necessary and sufficient conditions for monogenicity are established.

Solutions to differential equations related to the polynomial are investigated.

Abstract

Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $\theta$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ 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 $\theta.$ As an interesting result, we establish necessary and sufficient conditions for the field $K=\mathbb{Q}(\theta)$ to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x).$