Corollary 2

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

Algebra, VIII, p. 253, § 14. Central Simple Algebras

Let A be a central simple algebra of finite degree over K (for example, a field of finite degree over K with center K). There exists an integer $n\geqslant 1$ such that $[A : K] =n^2$ .

Let L be an extension of K and $n$ a strictly positive integer such that the L-algebras $A_{(L)}$ and $\mathbf{M}_n(L)$ are isomorphic. We have

[A : K] = [A_{(L)}: L] = [\mathbf{M}_n(L) : L] =n^2

In the notation of Corollary 2, the integer $n$ is called the reduced degree of A.

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.

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