On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$

Let $K $ be a septic number field generated by a root $\theta$ of an irreducible polynomial $ F(x)= x^7+ax^5+b \in\mathbb Z[x]$. In this paper, we explicitly characterize the index $i(K)$ of $K$. More precisely, for all $a$ and $b$, we show that $i(K) \in\{1, 2\}$. Our results answer completely to Problem 22 of W. Narkiewicz's book (2004) for these families of number fields. In particular, we provide sufficient conditions for which $K$ is not monogenic. We illustrate our results by some computational examples.