# 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 3423–3434 (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).