On index divisors and monogenity of certain number fields defined by trinomials $x^7+ax+b$

TL;DR

This paper investigates the conditions under which certain number fields defined by seventh-degree trinomials are non-monogenic, extending previous results by characterizing when primes divide the index of the field.

Contribution

It provides a complete characterization of when a prime is a common index divisor for fields defined by $x^7+ax+b$, extending prior necessary conditions for non-monogenity.

Findings

Identifies conditions for primes to be common index divisors.

Extends previous non-monogenity criteria for these number fields.

Provides a characterization applicable to all primes for the given trinomials.

Abstract

For a number field $K$ defined by a trinomail $F(x) = x^7+ax+b \in \mathbb{Z}[x]$, Jakhar and Kumar gave some necessary conditions on $a$ and $b$, which guarantee the non-monogenity of $K$ \cite{A6}. In this paper, for every prime integer $p$, we characterize when $p$ is a common index divisor of $K$. In particular, if any one of these conditions holds, then $K$ is not monogenic. In such a way our proposed results extend those of Jakhar and Kumar.