Corollary 1
Tag 06M1, permanent. Cite it as [Bourbaki, Tag 06M1].
Algebra, chapter V, § 9. Conjugate elements and quasi-Galois extensions
— If E is an extension of finite degree of K, then the quasi-Galois extension N of K generated by E is of finite degree over K. For we have where A is finite, hence the set B of conjugates of elements of A is finite, whence the corollary follows by Th. 2 (V, p. 18).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.