Proposition 2

The set Br(K) is the set of invertible elements of $\mathscr{M}_K$, where the inverse of an element [A] of Br(K) is $[A^o]$. Consequently, the law of composition on the monoid $\mathscr{M}_K$ endows Br(K) with the structure of an abelian group.

Let A be a central simple algebra of finite degree over K. The algebra $A^o$ is central simple and of finite degree over K (VIII, p. 251). The algebra $A\otimes_KA^o$ is isomorphic to a matrix algebra $\mathbf{M}_n$(K), where $n^2=$ dim(A) (VIII, p. 252, Theorem 1). We therefore have $[A][A^o] = [A\otimes_KA^o] = [K]$ (Lemma 1), which proves that [A] is invertible, with inverse $[A^o]$.

Conversely, let A be a K-algebra of finite degree. If [A] is invertible in $\mathscr{M}_K$, then there exists a K-algebra B of finite degree such that [A][B] = [K]; by formula (2) and Lemma 1, this means that the K-algebra $A\otimes_KB$ is isomorphic to a matrix algebra $\mathbf{M}_n(K)$ with $n\geqslant 1$. By Remark 1 of VIII, p. 251, the algebra A is then central simple.