On the index divisors and monogenity of certain nonic number fields

In this paper, for any nonic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^9+ax+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz (\cite{Nar}) for this family of number fields. In particular if $i(K)\neq 1$, then $K$ is not mongenic. We illustrate our results by some computational examples.