Theorem in number theory
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 Richard Brauer, Helmut Hasse, 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":
 formula_1
Then "A" is isomorphic to the matrix algebra "M""d"("K").
Applications.
Using the theory of Brauer group, one shows that two central simple algebras "A" and "B" over an algebraic number field "K" are isomorphic over "K" if and only if their completions "A""v" and "B""v" are isomorphic over the completion "K""v" for every "v".
Together with the Grunwald–Wang theorem, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is "cyclic", i.e. can be obtained by an explicit construction from a cyclic field extension "L"/"K" .
Notes.
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