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 Kv 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 Kv:
- [math]\displaystyle{ A\otimes_K K_v \simeq M_d(K_v). }[/math]
Then A is isomorphic to the matrix algebra Md(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 Av and Bv are isomorphic over the completion Kv 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
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
- 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, Graduate Texts in Mathematics, 88, New York-Berlin: Springer-Verlag, ISBN 0-387-90693-2, https://archive.org/details/associativealgeb00pier_0
- Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series, 28, Oxford University Press, p. 276, ISBN 0-19-852673-3
- Roquette, Peter (2005), "The Brauer–Hasse–Noether theorem in historical perspective", Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften 15, http://www.rzuser.uni-heidelberg.de/~ci3/brhano.pdf, 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
- Albert, Nancy E. (2005), "A3 & His Algebra, iUniverse, ISBN:978-0-595-32817-8
Notes
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