On common index divisors and monogenity of of the nonic number field defined by a trinomial $x^9+ax+b$

Let $K$ be a nonic number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x)=x^9+ax+b\in \mathbb{Z}[x]$, where $ab\mathrm{\ne }0$. Let $i(K)$ be the index of $K$. A rational prime $p$ dividing $i(K)$ is called a prime common index divisor of $K$. In this paper, for every rational prime $p$, we give necessary and sufficient conditions depending only $a$ and $b$ for which $p$ is a common index divisor of $K$. As application of our results we identify infinite parametric families of non-monogenic nonic numbers fields defined by such trinomials. At the end, some numerical examples illustrating our theoretical results are given.