Remark 1

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

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

Let A and B be K-algebras. If $A\otimes_KB$ is a central simple algebra, then so are A and B (III, §4, No. 4, p. 468, Corollary of Proposition 6 and VIII, p. 221, Proposition 6). Conversely, if A and B are central simple algebras and if one of them has finite degree over K, then $A\otimes_KB$ is a central simple algebra (III, §4, No. 4, p. 468, Corollary of Proposition 6 and VIII, p. 222, Corollary 2).

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
Proposition 2, § 15

Cites

III, §4, No. 4, p. 468, Corollary of Proposition 6 not in this corpus
VIII, p. 221, Proposition 6
VIII, p. 222, Corollary 2