Corollary 1

Let A be a central simple K-algebra of finite degree, and let L be an extension of K of finite degree. Suppose $[A : K] = [L : K]^2$. Then L is a splitting field for A if and only if A contains a subalgebra isomorphic to L.

Suppose that there exists a morphism $\varphi$ from L to A. Let M be a maximal semisimple commutative subalgebra containing $\varphi (L)$. By Proposition 3 of VIII, p. 262, we have $[A : K] = [M : K]^2$, hence [M : K] = [L : K] and $M =\varphi (L)$. By Proposition 5, the field L splits A. Conversely, suppose that the field L splits A; then it is isomorphic to a maximal commutative subalgebra of a central simple K-algebra B similar to A (Proposition 5). We have $[B : K] = [L : K]^2$ (VIII, p. 262, Proposition 3), hence [B : K] = [A : K]. Consequently, B is isomorphic to A (VIII, p. 280, Corollary). Corollary 1 follows.