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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 ) =  x5  +  ax3  +  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.

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