Albert–Brauer–Hasse–Noether theorem

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In

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:

${\displaystyle A\otimes _{K}K_{v}\simeq M_{d}(K_{v}).}$

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

See also

• Class field theory
• Hasse norm theorem

References

• Zbl 0005.05003
• Brauer, R.; Hasse, H.; Noether, E. (1932), "Beweis eines Hauptsatzes in der Theorie der Algebren", J. reine angew. Math., 167: 399–404
• Fenster, D.D.; Schwermer, J. (2005), "Delicate collaboration: Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras", Archive for History of Exact Sciences, 59 (4): 349–379,
  doi:10.1007/s00407-004-0093-6
• Pierce, Richard (1982), Associative algebras,
  Zbl 0497.16001
• Zbl 1024.16008
• Zbl 1276.11001
• Albert, Nancy E. (2005), "A³ & His Algebra, iUniverse,
  ISBN 978-0-595-32817-8