Reducibility criterion after Frobenius substitution
= Reducibility criterion after Frobenius substitution
Under the same hypotheses, $g(T^p)$ is reducible exactly when every coefficient of $g$ is a $p$th power in $K$. If $g=\sum a_i^pT^i$, then
$$
g(T^p)=\left(\sum a_iT^i\right)^p.
$$
Conversely, if $g(T^p)$ is reducible, all its irreducible factors are separable. Its zero derivative forces every factor multiplicity to be divisible by $p$, so it is a $p$th power in $K[T]$ and all its coefficients are $p$th powers.