Lemma 53.13.4. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If $X$ is smooth and $k(Y) \subset k(X)$ is purely inseparable, then there is a unique $n \geq 0$ and a unique isomorphism $Y = X^{(p^ n)}$ such that $f$ is the $n$-fold relative Frobenius of $X/k$.

**Proof.**
The $n$-fold relative Frobenius of $X/k$ is defined in Varieties, Remark 33.36.11. The lemma follows by combining Lemmas 53.13.3 and 53.13.2.
$\square$

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