On non-monogenic number fields defined by trinomials of type $x^n+a{x}^m+b$

Let $K=\text{Q}(\theta )$ be a number field generated by a complex root $\mathrm{th}$ 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 non-monogenity of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As application, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree $n=2^r\cdot 3^k$ with $r$ and $k$ are positive integers. We also give two infinite families of non-monogenic number fields defined by trinomials of degree $6$. Finally, we illustrate our results by giving some examples.