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

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

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

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

Proof. We found the subfield $E_{sep}$ in Lemma 9.14.6. We set $E_{insep} = E^{\text{Aut}(E/F)}$. Details omitted. $\square$

Comment #581 by Wei Xu on

Line 2511, a typo: "There exist subextensions $E / E_{sep} /F'$" should be "There exist subextensions $E / E_{sep} /F$".

In the "Waring part", Line 2482 -- Line 2491, the field of rational numbers should all be denoted $\mathbf{Q}$.

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