Remark 2

Tag 00GG, permanent. Cite it as [Bourbaki, Tag 00GG].

Algebra, VIII, p. 251, § 14. Central Simple Algebras

Let A be a K-algebra and L an extension of the field K. If the L-algebra $A_{(L)}$ is central simple, then the K-algebra A is central simple. Conversely, if one of the degrees [A : K] and [L : K] is finite and if A is a central simple K-algebra, then the L-algebra $A_{(L)}$ is central simple. This follows from Corollary 2 of VIII, p. 222.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Theorem 1, § 14
§ 15. Brauer Groups

Cites

Corollary 2 of VIII, p. 222