General Theory
For an arbitrary field K, the Brauer group may be expressed in terms of Galois cohomology as follows:
Here, Ks is the separable closure of K, which coincides with the algebraic closure when K is a perfect field.
A generalisation of the Brauer group to the case of commutative rings by M. Auslander and O. Goldman, and more generally to schemes, was introduced by Alexander Grothendieck. In their approach, central simple algebras over a field are replaced with Azumaya algebras.
Read more about this topic: Brauer Group
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