Proposition 5

Let A be a central simple K-algebra of finite degree, and let L be an extension of K of finite degree. The following properties are equivalent:

(i) The extension L is a splitting field for A.

(ii) There exists a central simple K-algebra of finite degree similar to A containing a maximal commutative subalgebra isomorphic to L.

Let us prove that (ii) implies (i); it suffices to consider the case when L is a maximal commutative subalgebra of A. Let $\psi : A\otimes_KA^o\rightarrow$ End$_K(A)$ be the canonical isomorphism that sends $a\otimes a'$ to the K-linear mapping $x\mapsto axa'$ (VIII, p. 252, Theorem 1). We view A as a right vector space over L; then $\psi$ sends $A\otimes_KL$ into the subspace End$_L(A)$ of End$_K(A)$ and induces, by restriction, an injective homomorphism of L-algebras $\psi ': A\otimes_KL\rightarrow$ End$_L(A)$. Set $n= [L : K]$. By VIII, p. 262, Proposition 3, we have $[A : L] =n$, and therefore $[A\otimes_KL : L] = [A : K] =n^2$ and [End$_L(A) : L] =n^2$. Consequently, $\psi '$ is an isomorphism, which proves (i).

The converse follows from Lemma 3 below.