Remark 2

We can calculate a 2-cocycle $c$ associated with an algebra A split by a finite Galois extension L with group G as follows. First, there exists a K-homomorphism $\varphi : A\rightarrow \mathbf{M}_m$(L), where $[A : K] =m^2$. For $g\in G$, let $\varphi^g$ be the homomorphism from A to $\mathbf{M}_m(L)$ given by $a\mapsto \varphi (g^{-1}ag)$. By the Skolem–Noether theorem (VIII, p. 256, Theorem 3), for every $g\in G$, there exists an element $u_g$ of $\mathbf{G}\mathbf{L}_m(L)$ such that

for $a\in A$. We then set

We can also define an extension of G by $L^*$ using $\varphi :$ we consider the group $\Gamma \subset \mathbf{G}\mathbf{L}_m(L)$ consisting of the $\gamma$ for which there exists a $g\in G$ with

for every $a\in A$. The class of this extension is the inverse image by Ψ of the class of A in Br(L$/K$).