On indices of quintic number fields defined by <i>x</i> ⁵ + <i>ax</i> + <i>b</i>

Abstract The goal of this paper is to calculate explicitly the field index of any quintic number field K generated by a complex root α of a monic irreducible trinomial F ( x ) = x 5 + ax + b ∈ ℤ[ x ]. In such a way we provide a complete answer to the Problem 22 of Narkiewicz [36] for this class of number fields. Namely, for every prime integer p , we evaluate the highest power of p dividing i ( K ). In particular, we give sufficient conditions on a and b , which guarantee the non-monogenity of K .