The Stacks project

Lemma 58.32.2. Let $k$ be a field of characteristic $p > 0$. Let $X \to \mathbf{A}^ n_ k$ be an étale morphism with $X$ affine. Then there exists a finite étale morphism $X \to \mathbf{A}^ n_ k$.

Proof. Write $X = \mathop{\mathrm{Spec}}(C)$. Set $A = 0$ and denote $I = k[x_1, \ldots , x_ n]$. By assumption there exists some étale $k$-algebra map $\varphi : k[x_1, \ldots , x_ n] \to C$. Denote $\tau : C/\varphi (I)C \to 0$ the unique surjection. We may choose $\varphi $ and $\tau $ such that $N(C, \varphi , \tau )$ is minimal. By Lemma 58.32.1 we get $N(C, \varphi , \tau ) = 0$. Hence $\varphi $ is finite étale. $\square$

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