Lemma 9.27.3. Let $E/F$ be a normal algebraic field extension. There exist subextensions $E / E_{sep} /F$ and $E / E_{insep} / F$ such that

$F \subset E_{sep}$ is Galois and $E_{sep} \subset E$ is purely inseparable,

$F \subset E_{insep}$ is purely inseparable and $E_{insep} \subset E$ is Galois,

$E = E_{sep} \otimes _ F E_{insep}$.

## Comments (2)

Comment #581 by Wei Xu on

Comment #595 by Johan on