Brauer group

《Local Fields》 · 第 156 页 · Chapter X，Section 4「Change of Base Field」

In particular, if $k'=k$, and if $K$ and $K'$ are isomorphic, then there is a canonical isomorphism of $H^q(K/k)$ with $H^q(K'/k)$; this applies notably when $K$ is the separable closure $k_s$ of $k$, in which case the groups $H^q(k_s/k)$ will also be denoted $H^q(\ /k)$. […] The group $H^2(\ /k)$ is called the Brauer group of the field $k$, and will also be denoted $B_k$. By definition, it is the direct limit of the $H^2(K/k)$ as $K$ runs through the set of finite Galois extensions of $k$.