On the index divisors and monogenity of number fields defined by x ⁵ + ax ³ + b

Abstract

The main goal of this paper is to provide a complete answer to the Problem 22 of Narkiewicz[24] for any quintic number field K generated by a complex root α of a monic irreducible trinomial F(x) = x ⁵ + ax ³ + b ∈ ℤ [x]. Namely we calculate the index of the field K. In particular, if the index is not trivial, then K is not mongenic. Finally, we illustrate our results by some computational examples.