Lemma 10.165.4. Let k be a field. Let K/k be a separable field extension. Then K is geometrically normal over k.
Proof. This is true because k^{perf} \otimes _ k K is a field. Namely, it is reduced by Lemma 10.43.6. By Lemma 10.45.4 (or by Definition 10.45.5) the field extension k^{perf}/k is purely inseparable. Hence by Lemma 10.46.10 the ring k^{perf} \otimes _ k K has a unique prime ideal. A reduced ring with a unique prime ideal is a field. \square
Comments (2)
Comment #8096 by Rubén Muñoz--Bertrand on
Comment #8209 by Stacks Project on