Exercise 2, § 5

Let $K$ be a field of characteristic $p > 0$ and $f$ an irreducible monic polynomial of $K[X]$. Show that in $K[X]$ the polynomial $f(X^p)$ is either irreducible or the $p$-th power of an irreducible polynomial, depending on whether or not there exists a coefficient of $f$ not belonging to $K^p$ (decompose $f(X^p)$ into factors of the first degree in $\Omega[X]$, where $\Omega$ is an algebraically closed extension of $K$).