Proposition 3

Let A and B be central simple K-algebras of finite degree. The following properties are equivalent:

(i) We have [A] = [B] in the Brauer group Br(K).

(ii) There exists an integer $t\geqslant 1$ such that the K-algebra $A\otimes_KB^o$ is isomorphic to the matrix algebra $\mathbf{M}_t(K)$.

(iii) There exist strictly positive integers $r$ and $s$ such that the K-algebras $A\otimes_K\mathbf{M}_r(K)$ and $B\otimes_K\mathbf{M}_s(K)$ are isomorphic.

(iv) There exist a field D containing K and integers $m\geqslant 1$ and $n\geqslant 1$ such that A is isomorphic to $\mathbf{M}_m(D)$ and B to $\mathbf{M}_n(D)$.

(v) The K-algebras A and B are Morita equivalent.

Suppose that we have [A] = [B]. Since $[B^o]$ is the inverse of [B] in the Brauer group, we have $[K] = [B][B^o] = [A][B^o] = [A\otimes_KB^o]$. By Lemma 1, there exists an integer $t\geqslant 1$ such that the algebras $A\otimes_KB^o$ and $\mathbf{M}_t(K)$ are isomorphic. So (i) implies (ii).

Suppose that (ii) holds. Since $B^o\otimes_KB$ is isomorphic to a matrix algebra $\mathbf{M}_s(K)$ with $s\geqslant 1$ (VIII, p. 252, Theorem 1), the algebra $A\otimes_KB^o\otimes_KB$ is isomorphic to $A\otimes_K\mathbf{M}_s$(K), on the one hand, and to $\mathbf{M}_t(K)\otimes_KB$, on the other. So property (ii) implies property (iii).

Suppose that (iii) holds. By Wedderburn’s theorem (VIII, p. 120, Theorem 1), there exist integers $m\geqslant 1$ and $n\geqslant 1$ and fields D and $D'$ of finite degree over K with center K such that A is isomorphic to $\mathbf{M}_m(D)$ and B to $\mathbf{M}_n(D')$. Then the algebra $A\otimes_K\mathbf{M}_r(K)$ is isomorphic to $\mathbf{M}_{mr}$(D), and the algebra $B\otimes_K\mathbf{M}_{r'}(K)$ is isomorphic to $\mathbf{M}_{nr'}(D')$ (Lemma 2). By Corollary 2 of VIII, p. 121, the K-algebras D and $D'$ are isomorphic. So (iii) implies (iv).

Suppose that (iv) holds. The algebras A and D are Morita equivalent, as are the algebras B and D (VIII, p. 114, Example 2). Therefore, A and B are Morita equivalent, so (iv) implies (v).

Finally, the implication (v)$\Rightarrow$(i) is the definition of the set Br(K).

When the equivalent properties of the proposition hold, we say that the algebras A and B are similar.