Brauer Group and Class Field Theory
The notion of Brauer group plays an important role in the modern formulation of the class field theory. If Kv is a non-archimedean local field, there is a canonical isomorphism invv: Br(Kv) → Q/Z constructed in local class field theory. An element of the Brauer group of order n can be represented by a cyclic division algebra of dimension n2.
The case of a global field K is addressed by the global class field theory. If D is a central simple algebra over K and v is a valuation then D ⊗ Kv is a central simple algebra over Kv, the local completion of K at v. This defines a homomorphism from the Brauer group of K into the Brauer group of Kv. 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
where S is the set of all valuations of K and the right arrow is the direct sum of the local invariants and the Brauer group of the real numbers is identified with (1/2)Z/Z. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from the global class field theory. The group Q/Z on the right may be interpreted as the "Brauer group" of the class formation of idele classes associated to K.
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