ON NONMONOGENIC NUMBER FIELDS DEFINED BY TRINOMIALS OF TYPE xn+axm+b

Abstract

Let $K=\mathbb{Q}(𝜃)$ be a number field generated by a complex root $𝜃$ of a monic irreducible trinomial $F(x)=x^n+a{x}^m+b\in \mathbb{Z}[x]$. In this paper, we deal with the problem of the nonmonogenity of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As an application, we show that there are infinite families of nonmonogenic number fields defined by trinomials of degrees $n=s\cdot 2^r\cdot 3^k$, where $s$, $r$, and $k$ are positive integers. Finally, we illustrate our results by giving some examples.