= Solution
Assume now that $g$ is irreducible and separable. Its splitting field $L/K$ is <Finite Galois extension>[Galois], and its Galois group acts transitively on the roots $y_i$ by the <Galois group of a polynomial>. Part (c) extends every such automorphism uniquely to $M$. If $x_i^p=y_i$, the extension sends $x_i$ to the unique $p$th root above the image of $y_i$. The <transitivity lifted through unique pth roots> therefore shows that
$$
\operatorname{Aut}(M/K)
$$
acts transitively on the roots of $f$.
Every automorphism of $M/K$ preserves $L$, because $L$ is the splitting field of $g$ over $K$. Transitivity therefore implies that either all roots $x_i$ lie in $L$ or none does. Since $x_i^p\in L$, the <minimal polynomial of a purely inseparable element> has degree either one or $p$. Consequently,
$$
\boxed{\text{either every root lies in }L,\text{ or every root has degree }p\text{ over }L.}
$$
Let $h\in K[T]$ be a monic irreducible factor of $f$. If $h$ is inseparable, part (a) gives
$$
h(T)=q(T^p)
$$
for a nonconstant monic $q\in K[T]$. Since $q(T^p)$ divides $g(T^p)$, polynomial division in $K[U]$ and substitution $U=T^p$ show that $q$ divides $g$. Irreducibility of $g$ forces $q=g$, hence $h=f$. Therefore
$$
\boxed{h=f\quad\text{or}\quad h\text{ is separable},}
$$
which is the <irreducible factors after Frobenius substitution> dichotomy.
If every coefficient of $g$ is a $p$th power, say
$$
g(T)=\sum_i a_i^pT^i,
$$
then
$$
f(T)=g(T^p)
=\left(\sum_i a_iT^i\right)^p,
$$
so $f$ is reducible.
Conversely, suppose $f$ is reducible and factor it into distinct monic irreducibles:
$$
f=\prod_i h_i^{e_i}.
$$
Every $h_i$ is then a proper factor, hence separable by the preceding dichotomy. Since $f'=0$, unique factorization and $h_i'\ne0$ force $p\mid e_i$ for every $i$. Thus $f=q^p$ for some monic $q\in K[T]$. The <Frobenius endomorphism> raises coefficients to their $p$th powers, and the coefficient of $T^{ip}$ in $f=g(T^p)$ is the coefficient of $T^i$ in $g$. Hence every coefficient of $g$ is a $p$th power in $K$. We have proved the <reducibility criterion after Frobenius substitution>:
$$
\boxed{g(T^p)\text{ is reducible over }K
\iff\text{ every coefficient of }g\text{ is a }p\text{th power in }K.}
$$
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