ON the index divisors of certain number fields

Let $K=\text{Q}(\theta )$ be an algebraic number field with $\theta$ a root of an irreducible quadrinomial $f(x)=x^6+a{x}^m+bx+c\in \mathbb{Z}[x]$ 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.