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Albert–Brauer–Hasse–Noether theorem
In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field ''K'' which splits over every completion ''K''''v'' is a matrix algebra over ''K''. The theorem is an example of a local-global principle in algebraic number theory and
leads to a complete description of finite-dimensional division algebras over algebraic number fields in terms of their local invariants. It was proved independently by Helmut Hasse, Richard Brauer, and Emmy Noether and by Abraham Adrian Albert.
== Statement of the theorem ==
Let ''A'' be a central simple algebra of rank ''d'' over an algebraic number field ''K''. Suppose that for any valuation ''v'', ''A'' splits over the corresponding local field ''K''''v'':
:
Then ''A'' is isomorphic to the matrix algebra ''M''''d''(''K'').
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