On indices of quintic number fields defined by x 5 + ax + b

Abstract

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.