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

TL;DR

This paper explicitly calculates the field index of quintic number fields generated by roots of specific trinomials, providing conditions for non-monogenity and addressing a classical problem in algebraic number theory.

Contribution

It offers a complete explicit computation of the field index for quintic fields defined by $x^5+ax+b$, solving a problem posed by Narkiewicz.

Findings

Explicit formulas for the prime divisors of the field index

Conditions guaranteeing non-monogenity of the fields

Complete evaluation of the index for all primes

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