The Stacks project

Lemma 9.16.6. Let $L/K$ be a finite extension. Let $M/L$ be the normal closure of $L$ over $K$. Then there is a surjective map

\[ L \otimes _ K L \otimes _ K \ldots \otimes _ K L \longrightarrow M \]

of $K$-algebras where the number of tensors can be taken $[L : K]_ s \leq [L : K]$.

Proof. Choose an algebraic closure $\overline{K}$ of $K$. Set $n = [L : K]_ s = |\mathop{\mathrm{Mor}}\nolimits _ K(L, \overline{K})|$ with equality by Lemma 9.14.8. Say $\mathop{\mathrm{Mor}}\nolimits _ K(L, \overline{K}) = \{ \sigma _1, \ldots , \sigma _ n\} $. Let $M' \subset \overline{K}$ be the $K$-subalgebra generated by $\sigma _ i(L)$, $i = 1, \ldots , n$. Then $M'$ is a field since any $K$-subalgebra of $\overline{K}$ is a field. Any $K$-algebra map from $M'$ to $\overline{K}$ permutes the $\sigma _ i$ so sends $M'$ into and onto $M'$. By construction the field $M'$ is generated by conjugates of elements of $\sigma _1(L)$. Having said this it follows from Lemma 9.15.5 that $M'$ is normal over $K$ and that it is the smallest normal subextension of $\overline{K}$ containing $\sigma _1(L)$. By uniqueness of normal closure we have $M \cong M'$. Finally, there is a surjective map

\[ L \otimes _ K L \otimes _ K \ldots \otimes _ K L \longrightarrow M', \quad \lambda _1 \otimes \ldots \otimes \lambda _ n \longmapsto \sigma _1(\lambda _1) \ldots \sigma _ n(\lambda _ n) \]

and note that $n \leq [L : K]$ by definition. $\square$

Comments (2)

Comment #9016 by Zhenhua Wu on

The part ''It follows from Lemma 09HQ that is normal over and that it is the smallest normal subextension of containing needs clarification. To be specific: 1)why is an field; 2)how do we use lemma 09HQ to show is normal; 3) how do we show that it is the smallest normal subextension of containing .

Comment #9195 by on

OK, yes this was a bit terse. I added a few sentences. Since this will be used only much later in the project, I think it can be a little bit less detailed than the rest of the chapter. Changes are here.

There are also:

  • 2 comment(s) on Section 9.16: Splitting fields

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