  Let $K=\Q(\theta)$ be a number field generated by a complex root $\th$ of a
monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. 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 $n=2^r\cdot3^k$
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.
