## Tag `030M`

Chapter 9: Fields > Section 9.27: Linearly disjoint extensions

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. $\square$

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}
```

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