Lemma 9.21.5. Let $L/K$ be a finite separable extension of fields. Let $M$ be the normal closure of $L$ over $K$ (Definition 9.16.4). Then $M/K$ is Galois.

**Proof.**
The subextension $M/M_{sep}/K$ of Lemma 9.14.6 is normal by Lemma 9.15.4. Since $L/K$ is separable we have $L \subset M_{sep}$. By minimality $M = M_{sep}$ and the proof is done.
$\square$

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