Brauer Group - Brauer Group and Class Field Theory

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 K_v is a non-archimedean local field, there is a canonical isomorphism inv_v: Br(K_v) → 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 ⊗ K_v is a central simple algebra over K_v, the local 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

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.