On index divisors and non-monogenity of certain quintic number fields defined by x⁵ + ax^m + bx + c

Abstract

In this paper, for any nonic number field K generated by a root α of a monic irreducible trinomial F(x) = x 9 + ax + b ∈ 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 ([19]) for this family of number fields. In particular if i(K) 1, then K is not mongenic. We illustrate our results by some computational examples.