  Let $F$ be a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$
and $P(T) \in O_F[T]$ a monic separable polynomial such that $P(0) \not=0$ and
$P(1) \not=0$. We give precise sufficient conditions on a given positive
integer $k$ for the following condition to hold: there exist infinitely many
non-zero prime ideals $\mathcal{P}$ of $O_F$ such that the reduction modulo
$\mathcal{P}$ of $P(T)$ has a root in the residue field $O_F/\mathcal{P}$, but
the reduction modulo $\mathcal{P}$ of $P(T^k)$ has no root in
$O_F/\mathcal{P}$. This makes a result from a previous paper (motivated by a
problem in field arithmetic) asserting that there exist (infinitely many) such
integers $k$ more precise.
