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Let $K = \mathbb{Q} (\th) $ be a number field generated by a complex root$\th$ of a monic irreducible trinomial of type $ F(x)= x^{2^r}+ax^m+b \in\Z[x]$ and $\Z_K$ its ring of integers. In this paper, we study the monogenityof $K$. More precisely, when $m=1$, we provide some explicit conditions on $a$,$b$, and $r$ for which $K$ is not monogenic. We also construct a family ofirreducible trinomials which are not monogenic, but their roots generatemonogenic number fields. To illustrate our results, we give some computationalexamples.

On monogenity of certain number fields defined by $x^{2^r}+ax^m+b$