  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 monogenity
of $K$. 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 of
irreducible trinomials which are not monogenic, but their roots generate
monogenic number fields. To illustrate our results, we give some computational
examples.
