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 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:

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

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 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 .

See also

• Class field theory
• Hasse norm theorem

References

• Albert, A.A.; Hasse, H. (1932), "A determination of all normal division algebras over an algebraic number field", Trans. Amer. Math. Soc., 34 (3): 722–726, doi:10.1090/s0002-9947-1932-1501659-x, 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, doi:10.1515/crll.1932.167.399
• 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, Graduate Texts in Mathematics, vol. 88, New York-Berlin: Springer-Verlag, ISBN 0-387-90693-2, Zbl 0497.16001
• Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series, vol. 28, Oxford University Press, p. 276, ISBN 0-19-852673-3, Zbl 1024.16008
• Roquette, Peter (2005), "The Brauer–Hasse–Noether theorem in historical perspective" (PDF), Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, 15, CiteSeerX 10.1.1.72.4101, MR 2222818, Zbl 1060.01009, retrieved 2009-07-05 Revised version — Roquette, Peter (2013), Contributions to the history of number theory in the 20th century, Heritage of European Mathematics, Zürich: European Mathematical Society, pp. 1–76, ISBN 978-3-03719-113-2, Zbl 1276.11001
• Albert, Nancy E. (2005), "A³ & His Algebra, iUniverse, ISBN 978-0-595-32817-8

Notes