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

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