On monogenity of certain number fields defined by trinomials

Abstract

Let $K=\mathbb{Q}(\theta)$ be a number field generated by a complex root $\theta$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \mathbb{Z}[x]$. There is an extensive literature on monogenity of number fields defined by trinomials. For example, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja provide some explicit conditions on $a$, $b$ and $n$ for $(1, \theta, \ldots, \theta^{n-1})$ to be a power integral basis in $K$. But, if $\theta$ does not generate a power integral basis of $\mathbb{Z}_K$, then Jhorar's and Khanduja's results cannot answer the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of non-monogenity of $K$. More precisely, when $\theta$ does not generate a power integral basis of $\mathbb{Z}_K$, 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.