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
Comments (0)
There are also: