On common index divisors and monogenity of certain number fields defined by $x^5+ax+b$

Abstract

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 $F(x)=x^5+ax+b\in \mathbb{Z}[x]$. 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 $K$.

Submitted. arXiv admin note: text overlap with arXiv:2111.05899, arXiv:2106.01252, arXiv:2109.08765