Albert–Brauer–Hasse–Noether theorem

Albert–Brauer–Hasse–Noether theorem This article includes a list of references, lecture connexe ou liens externes, mais ses sources restent floues car il manque de citations en ligne. Merci d'aider à améliorer cet article en introduisant des citations plus précises. (Avril 2016) (Découvrez comment et quand supprimer ce modèle de message) 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.

Contenu 1 Énoncé du théorème 2 Applications 3 Voir également 4 Références 5 Notes 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: {style d'affichage Aotimes _{K}K_{v}simeq M_{ré}(K_{v}).} 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, c'est à dire. 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, est ce que je: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 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, est ce que je:10.1007/s00407-004-0093-6 Pierce, Richard (1982), Associative algebras, Textes d'études supérieures en mathématiques, volume. 88, New York-Berlin: Springer Verlag, ISBN 0-387-90693-2, Zbl 0497.16001 Reiner, je. (2003), Maximal Orders, Monographies de la London Mathematical Society. Nouvelle série, volume. 28, Presse universitaire d'Oxford, p. 276, ISBN 0-19-852673-3, Zbl 1024.16008 Roquette, Pierre (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, M 2222818, Zbl 1060.01009, récupéré 2009-07-05 Revised version — Roquette, Pierre (2013), Contributions to the history of number theory in the 20th century, Heritage of European Mathematics, Zurich: Société mathématique européenne, pp. 1–76, ISBN 978-3-03719-113-2, Zbl 1276.11001 Albert, Nancy E. (2005), "A Cubed & His Algebra, iUniverse, ISBN 978-0-595-32817-8 Notes Categories: Théorie des corps de classes Théorèmes en théorie algébrique des nombres