On monogenity of certain number fields defined by trinomials

Abstract

Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x)=x^n+ax+b\in \mathbb{Z}[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of $\mathbb{Z}[þ]$. But if $\mathbb{Z}[þ]$ is not integrally closed, then Jhorar and Khanduja's results cannot answer on the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of $K$. More precisely, when ${\mathbb{Z}}_K\ne \mathbb{Z}[þ]$, we give sufficient conditions on $n$, $a$ and $b$ for $K$ to be not monogenic. For $n\in \{5,6,3^r,2^k\cdot 3^r,2^s\cdot 3^k+1\}$, we give explicitly some infinite families of these number fields that are not monogenic. Finally, we illustrate our results by some computational examples.

submitted on August 2, 2021