The Brauer-Hasse-Noether Theorem in Historical Perspective PDF
Preview The Brauer-Hasse-Noether Theorem in Historical Perspective
SchriftenderMathematisch-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 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationor partsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustusemustalwaysbeobtainedfromSpringer-Verlag. ViolationsareliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com ©Springer-VerlagBerlinHeidelberg2005 PrintedinGermany Coverdesign:ErichKirchner,Heidelberg TypesetinLATEXbytheauthor andeditedbyPublicationServiceGiselaKoch,Wiesenbach,usingamodified SpringerLATEXmacro-package. Printedonacid-freepaper 08/3150hs–543210 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.
The list of books you might like
