The Stacks project

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. $\square$


Comments (2)

Comment #581 by Wei Xu on

Line 2511, a typo: "There exist subextensions " should be "There exist subextensions ".

In the "Waring part", Line 2482 -- Line 2491, the field of rational numbers should all be denoted .


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 030M. Beware of the difference between the letter 'O' and the digit '0'.