Brauer group of global fields

The exact sequence splits (non-canonically) to prove that $Br(K)$ is a direct sum of a finite number of copies of $\Bbb Z/2\Bbb Z$ (coming from the real places) and countably many copies of $\Bbb Q/\Bbb Z$. So the $n$-torsion is always infinite (for $n\ge2$) and the group is $\ell$-divisible for all odd $\ell$. For $\ell=2$, $Br(K)$ is $2$-divisible iff $K$ is totally complex.