On index divisors and monogenity of certain sextic number fields defined by $x^6+a{x}^5+b$

The main goal of this paper is to provide a complete answer to the Problem 22 of Narkiewicz \cite{Nar} for any sextic number field $K$ generated by a complex root $\alpha$ of a monic irreducible trinomial $F(x)=x^6+a{x}^5+b\in \mathbb{Z}[x]$. Namely we calculate the index of the field $K$. In particular, if $i(K)\mathrm{\ne }1$, then $K$ is not mongenic. Finally, we illustrate our results by some computational examples.