Remark 2

Tag 008K, permanent. Cite it as [Bourbaki, Tag 008K].

Algebra, VIII, p. 125, § 7. Simple Rings

Let K be a commutative field. If A is a simple subalgebra of an algebra B of finite degree over K, then the left degree of B over A satisfies the relation $[B : A]_s[A : K] = [B : K]$ by Proposition 6 of VIII, p. 125. Likewise, we have $[B : A]_d[A : K] = [B : K]$ . The equality $[B : A]_s= [B : A]_d$ follows.

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

Nothing.

Cites

Proposition 6 of VIII, p. 125