Lemma 53.13.2. 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 the extension $k(X)/k(Y)$ of function fields is purely inseparable, then there exists a factorization

such that each $X_ i$ is a proper nonsingular curve and $X_ i \to X_{i + 1}$ is a degree $p$ morphism with $k(X_{i + 1}) \subset k(X_ i)$ inseparable.

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