Exercise 4, § 9

Let $K$ be a field, $f$ an irreducible polynomial in $K[X]$, separable over $K$ and of degree $n$ and let $\alpha_i \ (1 \leq i \leq n)$ be its roots in an algebraic closure $\Omega$ of $K$. Let $g$ be a polynomial in $K[X]$, and $h$ an irreducible factor in $K[X]$ of the polynomial $f(g(X))$. Show that the degree of $h$ is a multiple $rn$ of $n$ and that $h$ has exactly $r$ roots in common with each of the polynomials $g(X) - \alpha_i \ (1 \leq i \leq n)$ of $\Omega[X]$ (consider the conjugates over $K$ of a root of $h$).