Remark 5

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

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

Let A be a central simple algebra of finite degree over K whose reduced degree is a prime number $\ell$ . Then either A is a field, or A is isomorphic to $\mathbf{M}_{\ell}(K)$ . Indeed, A is isomorphic to an algebra of the form $\mathbf{M}_n$ (D), where D is a field with center K, and we have

\ell^2= [A : K] =n^2[D : K]

if A is not a field, then $n\not= 1$ , so $n=\ell$ and D = K.

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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