Let $K=\mathbb{Q}(\alpha )$ be a number field generated by a complex root α of a monic irreducible trinomial $F(x)=x^5+a{x}^2+b\in \mathbb{Z}[x].$ In this paper, for every prime integer p, we give necessary and sufficient conditions on a and b so that p is a common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic.

In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial $F(x)=x^7+a{x}^3+b\in \mathbb{Z}[x]$, we describe all prime power divisors of the index of K answering Problem 22 of Narkiewicz [ 26 Narkiewicz, W. (2004). Elementary and Analytic Theory of Algebraic Numbers, 3rd ed. Berlin: Springer. [Crossref] , [Google Scholar] ]. In particular, if $i(K)\ne 1$, then K is not mongenic. We illustrate our results by some computational examples.