Table Of ContentSchriftenderMathematisch-naturwissenschaftlichenKlasse
derHeidelbergerAkademiederWissenschaften
Nr.15(2004)
Peter Roquette
The
Brauer-Hasse-Noether
Theorem
in Historical Perspective
Prof.Dr.Dr.h.c.PeterRoquette
MathematischesInstitutderUniversitätHeidelberg
ImNeuenheimerFeld288
69120Heidelberg,Germany
[email protected]
LibraryofCongressControlNumber:2004111361
ISBN3-540-23005-X SpringerBerlinHeidelbergNewYork
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Contents
1 Introduction................................................. 1
2 TheMainTheorem: CyclicAlgebras ............................ 5
3 ThePaper: DedicationtoHensel ............................... 9
4 TheLocal-GlobalPrinciple..................................... 15
4.1 TheNormTheorem........................................ 15
4.2 TheReductions ........................................... 16
4.3 FactorSystems ........................................... 19
5 FromtheLocal-GlobalPrincipletotheMainTheorem............. 25
5.1 TheSplittingCriterion ..................................... 25
5.2 AnUnprovenTheorem ..................................... 27
5.3 TheGrunwald-WangStory ................................. 29
5.4 Remarks ................................................. 31
5.4.1 TheWeakExistenceTheorem.......................... 31
5.4.2 GroupRepresentations................................ 31
5.4.3 AlgebraswithPureMaximalSubfields .................. 33
5.4.4 Exponent=Index.................................... 34
5.4.5 Grunwald-WangintheSettingofValuationTheory........ 35
6 TheBrauerGroupandClassFieldTheory........................ 37
6.1 TheLocalHasseInvariant .................................. 37
6.2 StructureoftheGlobalBrauerGroup......................... 41
6.3 Remarks ................................................. 45
6.3.1 ArithmeticofAlgebrasandHensel’sMethods ............ 45
6.3.2 ClassFieldTheoryandCohomology.................... 48
7 TheTeam: Noether,BrauerandHasse........................... 51
7.1 Noether’sError ........................................... 51
7.2 Hasse’sCastlesintheAir................................... 54
7.3 TheMarburgSkewCongress................................ 57
VI Contents
8 TheAmericanConnection: Albert.............................. 61
8.1 TheFootnote ............................................. 61
8.2 TheFirstContacts......................................... 62
8.3 Albert’sContributions...................................... 68
8.4 ThePriorityQuestion ...................................... 71
8.5 Remarks ................................................. 75
9 Epilogue: Ka¨teHey........................................... 79
References .................................................... 83
Index ......................................................... 91
1
Introduction
The legacy of Helmut Hasse, consisting of letters, manuscripts and other pa-
pers,iskeptattheHandschriftenabteilungoftheUniversityLibraryatGo¨ttin-
gen.Hassehadanextensivecorrespondence;helikedtoexchangemathematical
ideas,resultsandmethodsfreelywithhiscolleagues.Therearemorethan8000
documents preserved.Although not all of them are of equal mathematical in-
terest, searching through this treasure can help us to assess the development
of NumberTheory through the 1920’s and 1930’s. Unfortunately, most of the
correspondenceispreservedononesideonly,i.e.,theletterssenttoHasseare
availablewhereasmanyoftheletterswhichhadbeensentfromhim,oftenhand-
written,seemtobelost.Sowehavetointerpolate,asfaraspossible,fromthe
repliestoHasseandfromothercontexts,inordertofindoutwhathehadwritten
inhisoutgoingletters.1
Thepresentarticleislargelybasedonthelettersandotherdocumentswhich
Ihavefoundconcerningthe
Brauer-Hasse-NoetherTheorem
inthetheoryofalgebras;thiscoverstheyearsaround1931.Besidesthedocu-
mentsfromtheHasseandtheBrauerlegacyinGo¨ttingen,Ishallalsousesome
lettersfromEmmyNoethertoRichardBrauerwhicharepreservedattheBryn
MawrCollegeLibrary(Pennsylvania,USA).
WeshouldbeawarethattheBrauer-Hasse-NoetherTheorem,althoughtobe
ratedasahighlight,doesnotconstitutethesummitandendpointofadevelop-
ment.We have to regard it as a step, important but not final, in a development
whichleadstotheviewofclassfieldtheoryasweseeittoday.Byconcentrat-
ing on the Brauer-Hasse-Noether Theorem we get only what may be called a
snapshotwithinthegreatedificeofclassfieldtheory.
A snapshot is not a panoramic view. Accordingly, the reader might miss
severalaspectswhichalsocouldthrowsomelightonthepositionoftheBrauer-
Hasse-Noethertheorem,itssourcesanditsconsequences,notonlywithinalge-
braic number theory but also in other mathematical disciplines. It would have
1AnexceptionisthecorrespondencebetweenHasseandRichardBrauer.Thanksto
Prof. Fred Brauer, the letters from Hasse to Richard Brauer are now available in
Go¨ttingentoo.
2 1.Introduction
been impossible to include all these into this paper. Thus I have decided to
presentitasitisnow,beingawareofitsshortcomingswithrespecttotherange
oftopicstreated,aswellasthetimespantakenintoconsideration.
Apreliminaryversionofthisarticlehadbeenwritteninconnectionwithmy
lectureattheconferenceMarch22–24,2001inStuttgartwhichwasdedicatedto
thememoryofRichardBrauerontheoccasionofhiscentenary.ForBrauer,the
cooperationwithNoetherandHasseinthisprojectconstitutedanunforgettable,
exciting experience. Let us cite from a letter he wrote many years later, on
March3,1961,toHelmutHasse:2
...Itisnow35yearssinceyouintroducedmetoclassfieldtheory.Itbelongs
tomymostdelightfulmemoriesthatItoowasable,incooperationwithyou
andEmmyNoether,togivesomelittlecontribution,andIshallneverforget
theexcitementofthosedayswhenthepapertookshape.
Theavailabledocumentsindicatethatasimilarfeelingofexcitementwaspresent
also in the minds of the other actors in this play. Besides Hasse and Noether
wehavetomentionArtinandalsoAlbertinthisconnection.Othernameswill
appearinduecourse.
AstoA.AdrianAlbert,healsohadanindependentshareintheproofofthe
maintheorem,somuchthatperhapsitwouldbejustifiedtonameitthe
Albert-Brauer-Hasse-NoetherTheorem.
Butsincetheoldname,i.e.,withoutAlbert,hasbecomestandardintheliterature
Ihaveabstainedfromintroducinganewname.Inmathematicsweareusedto
the fact that the names of results do not always reflect the full story of the
historical development. In Section 8, I will describe the role ofAlbert in the
proof of the Brauer-Hasse-Noether theorem, based on the relevant part of the
correspondenceofAlbertwithHasse.
Acknowledgement:Preliminaryversionshadbeeninmyhomepageforsome
time.Iwouldliketoexpressmythankstoallwhosentmetheircommentseach
of which I have carefully examined and taken into consideration. Moreover, I
wishtothankFalkoLorenzandKeithConradfortheirthoroughreadingofthe
lastversion,theircorrectionsandvaluablecomments.LastbutnotleastIwould
liketoexpressmygratitudetoMrs.NancyAlbert,daughterofA.A.Albert,for
lettingmeshareherrecollectionsofherfather.Thiswasparticularlyhelpfulto
mewhilepreparingSection8.
2TheletteriswritteninGerman.Hereandinthefollowing,wheneverweciteaGerman
textfromaletterorfromapaperthenweuseourownfreetranslation.
2
The MainTheorem: Cyclic Algebras
On December 29, 1931 Kurt Hensel, the mathematician who had discovered
p-adicnumbers,celebratedhis70thbirthday.Onthisoccasionaspecialvolume
ofCrelle’sJournalwasdedicatedtohimsincehewasthechiefeditorofCrelle’s
Journalatthattime,andhadbeenforalmost30years.Thededicationvolume
containsthepaper[BrHaNo:1932],authoredjointlybyRichardBrauer,Helmut
HasseandEmmyNoether,withthetitle:
ProofofaMainTheoreminthetheoryofalgebras.3
Thepaperstartswiththefollowingsentence:
At last our joint endeavours have finally been successful, to prove the fol-
lowingtheoremwhichisoffundamentalimportanceforthestructuretheory
ofalgebrasovernumberfields,andalsobeyond...
Thetheoreminquestion,whichhasbecomeknownastheBrauer-Hasse-Noether
Theorem,readsasfollows:
MainTheorem4 Everycentraldivisionalgebraoveranumberfieldiscyclic
(or,asitisalsosaid,ofDicksontype).
Inthisconnection,allalgebrasareassumedtobefinitedimensionaloverafield.
An algebra A over a field K is called “central” if K equals the center of A.
Actually,intheoriginalBrauer-Hasse-Noetherpaper[BrHaNo:1932]theword
“normal” was used instead of “central”; this had gradually come into use at
thattime,followingtheterminologyofAmericanauthors,seee.g.,[Alb:1930].5
Todaythemoreintuitive“central”isstandard.
3“BeweiseinesHauptsatzesinderTheoriederAlgebren.”
4Falko Lorenz [Lor:2004] has criticized the terminology “MainTheorem”. Indeed,
what today is seen as a “Main Theorem” may in the future be looked at just as a
useful lemma. So we should try to invent another name for this theorem, perhaps
“CyclicityTheorem”.Butforthepurposeofthepresentarticle,letuskeeptheauthors’
terminologyandrefertoitasthe“MainTheorem”(incapitals).
5It seems that in 1931 the terminology “normal” was not yet generally accepted.
For, when Hasse had sent Noether the manuscript of their joint paper asking for
her comments, she suggested that for “German readers” Hasse should explain the
notionof“normal”.(LetterofNovember12,1931.)Hassefollowedhersuggestion
andinsertedanexplanation.
6 2.TheMainTheorem:CyclicAlgebras
Cyclicalgebrasaredefinedasfollows.LetL|K beacyclicfieldextension,
ofdegreen,andletσ denoteageneratorofitsGaloisgroupG.Givenanya in
themultiplicativegroupK×,considertheK-algebrageneratedbyLandsome
elementuwiththedefiningrelations:
un =a, xu=uxσ (forx ∈L).
This is a central simple algebra of dimension n2 over K and is denoted by
(L|K,σ,a).ThefieldLisamaximalcommutativesubalgebraof(L|K,σ,a).
ThisconstructionofcyclicalgebrashadbeengivenbyDickson;thereforethey
werealsocalled“ofDicksontype”.6
Thus the Main Theorem asserts that every central division algebra A over
a number field K is isomorphic to (L|K,σ,a) for a suitable cyclic extension
L|K with generating automorphism σ, and suitable a ∈ K×; equivalently, A
contains a maximal commutative subfield L which is a cyclic field extension
ofK.
WhenArtinheardoftheproofoftheMainTheoremhewrotetoHasse:7
...You cannot imagine how ever so pleased I was about the proof, finally
successful, for the cyclic systems. This is the greatest advance in number
theoryofthelastyears.Myheartfeltcongratulationsforyourproof....
Now,giventhebarestatementoftheMainTheorem,Artin’senthusiasticexcla-
mationsoundssomewhatexaggerated.Atfirstglancethetheoremappearsasa
ratherspecialresult.Thedescriptionofcentralsimplealgebrasmayhavebeen
ofimportance,butwoulditqualifyforthe“greatestadvanceinnumbertheory
inthelastyears”?ItseemsthatArtinhadinmindnotonlytheMainTheorem
itself,butalsoitsproof,involvingtheso-calledLocal-GlobalPrincipleandits
manyconsequences,inparticularinclassfieldtheory.
Theauthorsthemselves,inthefirstsentenceoftheirjointpaper,tellusthat
theyseetheimportanceoftheMainTheoreminthefollowingtwodirections:
1. Structureofdivisionalgebras. TheMainTheoremallowsacompleteclassi-
ficationofdivisionalgebrasoveranumberfieldbymeansofwhattodayare
calledHasseinvariants;therebythestructureoftheBrauergroupofanalge-
braicnumberfieldisdetermined.(ThiswaselaboratedinHasse’ssubsequent
paper [Has:1933] which was dedicated to Emmy Noether on the occasion
of her 50th birthday on March 23, 1932.) The splitting fields of a division
algebracanbeexplicitlydescribedbytheirlocalbehavior;thisisimportant
6Dickson himself [Dic:1927] called these “algebras of type D”.Albert [Alb:1930]
gives 1905 as the year when Dickson had discovered this construction. – Dickson
did not yet use the notation (L|K,σ,a) which seems to have been introduced by
Hasse.
7ThisletterfromArtintoHasseisnotdatedbutwehavereasontobelievethatitwas
writtenaroundNovember11,1931.
