The index of certain nonic number fields defined by $x^9+ax^2+b$

In this paper, for any nonic number field $K$ defined by a monic irreducible trinomial $F(x)=x^9+ax^2+b \in \mathbb{Z}[x]$, we calculate $\nu_p(i(K))$ for every rational prime $p$. In particular, we characterize the index $i(K)$ of this family of number fields. As an application of our results, if $i(K)\neq1$, then $K$ is not monogenic. We illustrate our results by some computational examples.