On common index divisor of the number fields defined by $x^7+ax+b$

TL;DR

This paper investigates the conditions under which a prime divides the index of number fields generated by roots of specific seventh-degree polynomials, providing criteria for non-monogenic fields based on coefficients.

Contribution

It offers necessary and sufficient conditions involving polynomial coefficients for prime divisors of the index, and identifies cases leading to non-monogenic fields.

Findings

Identifies conditions on coefficients for primes dividing the index.

Provides sufficient conditions for the field to be non-monogenic.

Includes illustrative examples demonstrating the criteria.

Abstract

Let $f(x)=x^7+ax+b$ be an irreducible polynomial having integer coefficients and $K=\mathbb{Q}(\theta)$ be an algebraic number field generated by a root $\theta$ of $f(x)$. In the present paper, for every rational prime $p$, our objective is to determine the necessary and sufficient conditions involving only $a,~b$ so that $p$ is a divisor of the index of the field $K$. In particular, we provide sufficient conditions on $a$ and $b$, for which $K$ is non-monogenic. In a special case, we show that if either $8$ divides both $a\pm1$, $b$ or $32$ divides both $a+4$, $b$, then $K$ is non-monogenic. We illustrate our results through examples.