ON the index divisors of certain number fields

TL;DR

This paper investigates the conditions under which certain degree 6 number fields generated by roots of specific quadrinomials are non-monogenic, providing explicit criteria and prime divisibility results, and addressing a classical problem in algebraic number theory.

Contribution

It offers explicit criteria based on polynomial coefficients for non-monogenicity of these fields and determines the highest powers of primes dividing their indices, partially solving Narkiewicz's Problem 22.

Findings

Identifies conditions for non-monogenicity based on polynomial coefficients.

Determines the highest power of primes dividing the index of the field.

Provides examples illustrating the theoretical results.

Abstract

Let $K=\Q(\theta)$ be an algebraic number field with $\theta$ a root of an irreducible quadrinomial $f(x) = x^6+ax^m+bx+c\in\Z[x] $ with $m\in\{2,3,4,5\}$. In the present paper, we give some explicit conditions involving only $a,~b,~c$ and $m$ for which $K$ is non-monogenic. In each case, we provide the highest power of a rational prime $p$ dividing index of the field $K$. In particular, we provide a partial answer to the Problem $22$ of Narkiewicz \cite{Nar} for these number fields. Finally, we illustrate our results through examples.