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}.
Proof. By Lemma 10.42.4 we can find such a diagram with K'/k' separably generated. By Lemma 10.44.3 this implies that K' is separable over k'. The compositum k'K is a subextension of K'/k' and hence k' \subset k'K is separable by Lemma 10.42.2. The ring (k' \otimes _ k K)_{red} is a domain as for some n \gg 0 the map x \mapsto x^{p^ n} maps it into K. Hence it is a field by Lemma 10.36.19. Thus (k' \otimes _ k K)_{red} \to K' maps it isomorphically onto k'K. \square
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