On index divisors and monogenity of certain septic number fields defined by x⁷ + ax³ + b

In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial F(x)=x7+ax3+b∈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)≠1, then K is not mongenic. We illustrate our results by some computational examples.