Brauer group of infinite extension of $\Bbb Q$ Let $K/\Bbb Q$ be an algebraic extension of infinite degree, do we have similar local-global exact sequence of brauer group in class field theory? In particular, can we compute $BrK$ or find conditions for the vanishing of brauer group? 
Let $K_n$ be an increasing chain of finite degree subfields of $K$ such that their union is $K$, what is the relationship of their brauer group and with that of $K$ ?
 A: There are many equivalent definitions of the Brauer group of a field, but the most convenient setting for answering your questions is Galois cohomology, see e.g. Serre's "Local Fields", chapter VII, end of §6. So:
The cohomological description of the Brauer group of any field $K$ is $Br(K) = H^2(K_s/K, K_s^∗)$ (where $K_s$ denotes a separable closure). If $L/K$ is any Galois extension (finite or not) with group $G$, there is a natural map $Br(K) \to Br(L)$ which is the cohomological restriction map $H^2(K_s/K, K_s^∗)\to H^2(K_s/L, K_s^∗)$, whose kernel and image are "known" thanks to the so called Hochschild-Serre spectral sequence: the kernel is the so called relative Brauer group $Br(L/K) = H^2(G,L^∗)$, and the image is the kernel of the cohomological transgression map $H^2(K_s/L, K_s^∗)^G \to H^3(G,L^∗)$. Note that if $L$ is the inductive limit of its subextensions $L_i$ of finite degrees, then $Br(L)$ is the inductive limit of the $Br(L_i)$'s w.r.t. restriction.
Coming to the specific case of number fields, i.e.  finite extensions $K/\mathbf Q$, the description of $Br(K)$ is the central part of Class Field Theory in its cohomological expression. Denote by $K_v$ the completion of $K$ w.r.t. to a place $v$. At archimedean places, we have $Br(\mathbf C)=0$ and $Br(\mathbf R)=\mathbf Z/2$ by direct computation. For the completion $K_v$ at a non archimedean place $v$, local CFT shows that $Br(K_v) \cong \mathbf Q/\mathbf Z$. An important point here is that for a finite extension $L_w/K_v$, the restriction map $Br(K_v) \to Br(L_w)$ corresponds to the multiplication by the degree of $L_w/K_v$ in $\mathbf Q/\mathbf Z$. The global-local principle states that $Br(K)$ injects canonically into the direct sum of all the $Br(K_v)$'s (archimedean or not), and the cokernel is $\cong \mathbf Q/\mathbf Z$ (actually it's the cokernel of the map "sum of the local coordinates") . This answers all your questions, I think.
