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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 [[local field|completion]] ''K''<sub>''v''</sub> is a [[matrix ring|matrix algebra]] over ''K''. The theorem is an example of a [[local-global principle]] in [[algebraic number theory]] and
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leads to a complete description of finite-dimensional [[division algebras]] over algebraic number fields in terms of their [[Hasse invariant of an algebra|local invariants]]. It was proved independently by [[Helmut Hasse]], [[Richard Brauer]], and [[Emmy Noether]] and by [[Abraham Adrian Albert]] and Hasse.


== Statement of the theorem ==
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Let ''A'' be a [[central simple algebra]] of rank ''d'' over an [[algebraic number field]] ''K''. Suppose that for any [[valuation (algebra)|valuation]] ''v'', ''A'' splits over the corresponding local field ''K''<sub>''v''</sub>:
 
: <math> A\otimes_K K_v \simeq M_d(K_v). </math>
 
Then ''A'' is isomorphic to the matrix algebra ''M''<sub>''d''</sub>(''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''<sub>''v''</sub> and ''B''<sub>''v''</sub> are isomorphic over the completion ''K''<sub>''v''</sub> 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 extension|cyclic field extension]] ''L''/''K'' .
 
== See also ==
 
* [[Class field theory]]
 
* [[Hasse norm theorem]]
 
==References==
 
* {{citation | author1-link=Abraham Adrian Albert | first1=A.A. | last1=Albert | author2-link=Helmut Hasse | first2=H. | last2=Hasse | title=A determination of all normal division algebras over an algebraic number field | journal=Trans. Amer. Math. Soc. | volume=34 | year=1932 | pages=722–726 | zbl=0005.05003 }}
* {{citation | author1-link=Helmut Hasse | first1=H. | last1=Hasse | author2-link=Richard Brauer | first2=R. | last2=Brauer | author3-link=Emmy Noether | first3=E. | last3=Noether | title=Beweis eines Hauptsatzes in der Theorie der Algebren | journal=Journal für Mathematik | volume=167 | year=1931 | pages=399–404 }}
*{{Citation
| doi =  10.1007/s00407-004-0093-6
| title =  Delicate collaboration: Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras
| url = http://www.springerlink.com/index/YLLATB4EMBEG6Q3H.pdf
| year = 2005
| journal = Archive for history of exact sciences
| pages = 349–379
| volume = 59
| issue = 4
| last1 = Fenster | first1 =  D.D.
| last2 =  Schwermer | first2 =  J.
| accessdate = 2009-07-05 }}
* {{citation | first=Richard | last=Pierce | title=Associative algebras | series=Graduate Texts in Mathematics | volume=88 | series=Studies in the History of Modern Science | volume=9 | publisher=[[Springer-Verlag]] | location=New York-Berlin | year=1982 | isbn=0-387-90693-2 | zbl=0497.16001  }}
* {{citation | last=Reiner | first=I. | authorlink=Irving Reiner | title=Maximal Orders | series=London Mathematical Society Monographs.  New Series | volume=28 | publisher=[[Oxford University Press]] | year=2003 | isbn=0-19-852673-3 | zbl=1024.16008 | page=276 }}
*{{Citation
| title =  The Brauer–Hasse–Noether theorem in historical perspective
| id = {{citeseerx|10.1.1.72.4101}}
| year = 2005
|journal=Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften
| last= Roquette
|first= Peter
|volume=15
| mr=2222818 | zbl=1060.01009
| accessdate = 2009-07-05
Albert, Nancy E. (2005), "A Cubed & His Algebra, iUniverse, isbn-13: 978-0-595-32817-8
}}
 
== Notes ==
{{reflist}}
 
{{DEFAULTSORT:Albert-Brauer-Hasse-Noether theorem}}
[[Category:Class field theory]]
[[Category:Theorems in algebraic number theory]]
Please add, in the References, a reference to Chapter 10 in the biography "A Cubed & His Algebra," which is devoted to discussion of the Albert-Brauer-Hasse-Noether theorem.

Revision as of 20:13, 15 February 2014

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