View source for Albert–Brauer–Hasse–Noether theorem

{{short description|Theorem in number theory}}
{{no footnotes|date=April 2016}}
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 
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 [[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 (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 | issue=3 | year=1932 | pages=722–726 | zbl=0005.05003 | doi=10.1090/s0002-9947-1932-1501659-x| doi-access=free }}
* {{citation | author1-link=Richard Brauer | first1=R. | last1=Brauer | author2-link=Helmut Hasse | first2=H. | last2=Hasse | author3-link=Emmy Noether | first3=E. | last3=Noether | title=Beweis eines Hauptsatzes in der Theorie der Algebren | journal=J. Reine Angew. Math. | volume=167 | year=1932 | pages=399–404 | doi=10.1515/crll.1932.167.399 }}
*{{Citation
 | doi =  10.1007/s00407-004-0093-6
 | title =  Delicate collaboration: Adrian Albert and Helmut Hasse and the Principal Theorem in Division Algebras 
 | 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.
 }}
* {{citation | first=Richard | last=Pierce | title=Associative algebras | series=[[Graduate Texts in Mathematics]] | volume=88 | publisher=[[Springer-Verlag]] | location=New York-Berlin | year=1982 | isbn=0-387-90693-2 | zbl=0497.16001 | url-access=registration | url=https://archive.org/details/associativealgeb00pier_0 }}
* {{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 | url=http://www.rzuser.uni-heidelberg.de/~ci3/brhano.pdf|title =  The Brauer–Hasse–Noether theorem in historical perspective | citeseerx = 10.1.1.72.4101 | year = 2005 |journal=Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften | last= Roquette | first= Peter | authorlink=Peter Roquette | volume=15 | mr=2222818 | zbl=1060.01009 | accessdate = 2009-07-05 }} Revised version — {{citation | last=Roquette | first=Peter | title=Contributions to the history of number theory in the 20th century | zbl=1276.11001 | series=Heritage of European Mathematics | location=Zürich | publisher=[[European Mathematical Society]] | isbn=978-3-03719-113-2 | year=2013 | pages=1–76 }}
* Albert, Nancy E. (2005), "A<sup>3</sup> & His Algebra, iUniverse, {{ISBN|978-0-595-32817-8}}

== Notes ==
{{reflist}}

{{DEFAULTSORT:Albert-Brauer-Hasse-Noether theorem}}
[[Category:Class field theory]]
[[Category:Theorems in algebraic number theory]]