The Galois group of $f(x)$ over $K$ acts on the roots of $f(x)$ in a splitting field $L/K$. Since the characteristic of $K$ is $p > 0$, the order of the Galois group divides $n!$.
We hope that this article has been helpful in providing solutions to Chapter 14 of Dummit and Foote and in introducing readers to the fascinating world of Galois Theory.
Let $r_1, r_2, \ldots, r_n$ be the roots of $f(x)$ in a splitting field $L/K$. Since $f(x)$ is separable, the roots $r_i$ are distinct. Let $\sigma \in \text{Gal}(L/K)$ be an automorphism of $L$ that fixes $K$. Then $\sigma(r_i)$ is also a root of $f(x)$ for each $i$. Since $\sigma$ is a bijection on the roots of $f(x)$, the Galois group of $f(x)$ over $K$ acts transitively on the roots.