On the indices of certain number fields defined by polynomials $x^{6}+ax+b$

Abstract

Let $K$ be a sextic number field defined by a trinomial $F(x)=x^{6}+a x+b \in \mathbb{Z}[x]$. In this paper, for any prime integer $p$, we compute the $p$-adic valuation of the field index $i(K)$. In what follows, we explicitly compute the full index $i(K)$. In particular, if $i(K)$ is nontrivial, then $K$ is not monogenic. The study of the monogenicity of $K$ can be performed in some cases, when $i(K)=1$.