Lemma 9.27.3 (030M)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 9: Fields
Section 9.27: Linearly disjoint extensions
Lemma 9.27.3 (cite)

previous definition

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

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

Comments (2)

Comment #581 by Wei Xu on May 20, 2014 at 16:30

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 $\mathbf{Q}$ .

Comment #595 by Johan on May 23, 2014 at 20:10

Thanks! Fixed here.

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