# The Stacks Project

## Tag 030M

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. $\square$

The code snippet corresponding to this tag is a part of the file fields.tex and is located in lines 3424–3435 (see updates for more information).

\begin{lemma}
\label{lemma-normal-case}
Let $E/F$ be a normal algebraic field extension. There exist subextensions
$E / E_{sep} /F$ and $E / E_{insep} / F$ such that
\begin{enumerate}
\item $F \subset E_{sep}$ is Galois and $E_{sep} \subset E$
is purely inseparable,
\item $F \subset E_{insep}$ is purely inseparable and $E_{insep} \subset E$
is Galois,
\item $E = E_{sep} \otimes_F E_{insep}$.
\end{enumerate}
\end{lemma}

\begin{proof}
We found the subfield $E_{sep}$ in Lemma \ref{lemma-separable-first}.
We set $E_{insep} = E^{\text{Aut}(E/F)}$. Details omitted.
\end{proof}

Comment #581 by Wei Xu on May 20, 2014 a 4:30 pm UTC

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 (site) on May 23, 2014 a 8:10 pm UTC

Thanks! Fixed here.

## Add a comment on tag 030M

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 lower-right corner).

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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?