On the Index of Certain Nonic Number Fields Defined by \(x^9+ax^5+b\)

In this paper, for any nonic number field \(K\) generated by a root \(\alpha\) of a monic irreducible trinomial \(F(x)=x^9+ax^5+b \in \mathbb{Z}[x]\) 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 [23] for 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.