  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.
