The case of a global field "K" (such as a number field) is addressed by global class field theory. If "D" is a central simple algebra over "K" and "v" is a place of "K", then "D" ⊗ "K""v" is a central simple algebra over "K""v", the completion of "K" at "v". This defines a homomorphism from the Brauer group of "K" into the Brauer group of "K""v". A given central simple algebra "D" splits for all but finitely many "v", so that the image of "D" under almost all such homomorphisms is 0. The Brauer group Br("K") fits into an exact sequence constructed by Hasse: