The Stacks project

Proof. Let $\sigma $ be a generator of $\text{Gal}(L/K)$. Consider $\sigma : L \to L$ as a $K$-linear operator. Observe that $\sigma ^ p - 1 = 0$ as a linear operator. Applying linear independence of characters (Lemma 9.13.1), there cannot be a polynomial of degree $< p$ annihilating $\sigma $. We conclude that the minimal polynomial of $\sigma $ is $x^ p - 1 = (x - 1)^ p$. This implies that there exists $w \in L$ such that $(\sigma - 1)^{p - 1}(w) = y$ is nonzero. Then $\sigma (y) = y$, i.e., $y \in K$. Thus $z = y^{-1}(\sigma - 1)^{p - 2}(w)$ satisfies $\sigma (z) = z + 1$. Since $z \not\in K$ we have $L = K[z]$. Moreover since $\sigma (z^ p - z) = (z + 1)^ p - (z + 1) = z^ p - z$ we see that $z^ p - z \in K$ and the proof is complete. $\square$