Irreducible factors after Frobenius substitution

Let $g\in K[T]$ be irreducible and separable in characteristic $p$, and put $f(T)=g(T^p)$. Every monic irreducible factor $h$ of $f$ is either $f$ itself or separable. Indeed, if $h$ were inseparable, then $h(T)=q(T^p)$; divisibility of $q(T^p)$ into $g(T^p)$ forces $q\mid g$, hence $q=g$ and $h=f$.

Under the same hypotheses, $g(T^p)$ is reducible exactly when every coefficient of $g$ is a $p$th power in $K$. If $g=\sum a_i^p{T}^i$, thenConversely, 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.