Lemma 10.45.3. Let $K/k$ be a finitely generated field extension. There exists a diagram

where $k'/k$, $K'/K$ are finite purely inseparable field extensions such that $K'/k'$ is a separable field extension. In this situation we can assume that $K' = k'K$ is the compositum, and also that $K' = (k' \otimes _ k K)_{red}$.

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