The goal of this paper is to calculate explicitly the field index of any quintic number field $K$ generated by a complex root $\al$ of a monic irreducible trinomial $F(x) = x^5+ax+b \in \Z[x]$. In such a way we provide a complete answer to the Problem 22 of Narkiewicz \cite{ WN}. 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 $K$.
Categories: math.NT