On monogenity of certain number fields defined by trinomials

TL;DR

This paper investigates the monogenity of certain number fields generated by roots of specific trinomials, providing conditions under which these fields are not monogenic, especially when their rings of integers are not generated by a single element.

Contribution

It introduces new sufficient conditions for non-monogenity of number fields defined by trinomials, extending previous results to cases where the ring of integers is not generated by the root.

Findings

Identifies conditions for non-monogenity based on polynomial parameters

Provides explicit infinite families of non-monogenic fields for specific degrees

Uses Newton polygon techniques to analyze the monogenity problem

Abstract

Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Ga\'al studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of $\Z[\th]$. But if $ \Z[\th]$ 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 $\Z_K \neq \Z[\th]$, 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.