Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae

596 Pages·2007·5.05 MB·English

by  Catherine Goldstein

Goldstein·Schappacher·Schwermer
TheShapingofArithmeticafterC.F.Gauss’sDisquisitionesArithmeticae
Catherine Goldstein
Norbert Schappacher
Joachim Schwermer Editors
The Shaping
of Arithmetic
C. F. Gauss’s
after
Disquisitiones Arithmeticae
With36Figures
123
CatherineGoldstein JoachimSchwermer
Histoiredessciencesmathématiques FakultätfürMathematik
InstitutdemathématiquesdeJussieu UniversitätWien
175rueduChevaleret Nordbergstraße15
75013Paris,France 1090Wien,Austria
E-mail:[email protected] E-mail:[email protected]
NorbertSchappacher
UFRdemathématique
etd’informatique/IRMA
7rueRenéDescartes
67084StrasbourgCedex,France
E-mail:[email protected]
LibraryofCongressControlNumber:2006932291
ISBN 978-3-540-20441-1 SpringerBerlinHeidelbergNewYork
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Inmemoriam
MARTINKNESER
1928–2004
Foreword
In1998,theeditorsconvincedthemselvesthatitwastherighttimetotakestockof
recentresearchconcerningthemodernhistoryofnumbertheory,andtoevaluatein
itslightourcomprehensionofthedevelopmentofthisdisciplineasawhole. One
issueatstakewastobringtogetherhistoriographicalresultscomingfromdifferent
disciplinesandlinguisticdomainswhich,wefelt,hadremainedtoooftenunaware
ofeachother.
WeorganizedtwomeetingsattheMathematischesForschungsinstitutOberwol-
fach: firstasmallRIP-workshopheldJune14–19,1999,amonghistoriansofnumber
theoryandhistoriansofrelatedtopics,andthenalargerconferencewhichtookplace
June17–23,2001,twohundredyearsafterthepublicationofCarlFriedrichGauss’s
DisquisitionesArithmeticae. Thelatterbroughttogetherhistoriansandphilosophers
ofmathematicswithnumbertheoristsinterestedintherecenthistoryoftheirfield.
Two further meetings, organized by one of us in Vienna and Zürich the following
years,continuedourventure.
Twoconcreteprojectsarosefromtheseactivities. Oneconcernedthecreation
of resources, for scholars and students: we initiated a bibliography of secondary
literatureontheHistoryofNumberTheorysince1800.1
Thepresentvolumeisthesecondresultofourwork. Itaimsatansweringthe
question,alreadyraisedduringthefirstworkshop,ontheroleofGauss’sDisquisi-
tiones Arithmeticae in the definition and evolution of number theory. This role is
hereappraisedinacomparativeperspective,withattentionbothtothemathematical
receptionofthetreatise,andtoitsroleasamodelfordoingmathematics. Thevolume
istheresultofacollectivework. Althoughallauthorshavekepttheirpropervoices,
theyhavealsoacceptedquiteabitofeditorialinterferencewithaviewtomakingthe
volumeascoherentaspossible. Wehavenonethelessleftroomfororiginalanalyses
andresults,includingnewlydiscovereddocuments.
1. ApreliminaryversionofthisbibliographyhasbeenkindlyputonlinebyFranzLem-
mermeyeronawebsitehostedbytheUniversityofHeidelberg(http://www.rzuser.uni-
heidelberg.de/ hb3/HINTbib.html).
∼
vii
viii
During its rather long elaboration, the present book has greatly profited from
thehelpofmanyindividualsandinstitutionswhichweheregratefullyacknowledge:
theMathematischesForschungsinstitutOberwolfachanditsdirectoratthetimeof
the meetings, Matthias Kreck; the Erwin-Schroedinger International Institute for
Mathematical Physics at Vienna; the ETH at Zürich, and special encouragement
providedbyUrsStammbach;theCIRMatLuminy,whichgaveusaweek’srefuge
foroureditorialworkinthesummerof2003;theLaboratoiredemathématiquesde
l’UniversitéParis-Sud,theInstitutdemathématiquesdeJussieu,aswellastheInsti-
tutdelarecherchemathématiqueavancéeatStrasbourgwhichactivelysupportedour
jointworkforthepreparationofthisbook. SpecialthanksgototheAbteilungHand-
schriftenundAlteDruckederNiedersächsischenStaats-undUniversitätsbibliothek
Göttingen, andinparticulartoJürgenRohlfing, fortheexpertcollaborationwhich
madesomanydocumentsavailabletous,someofthemashighqualityscans. We
also express our sincere gratitude to Springer-Verlag and their associated staff at
HeidelbergandLeipzig,especiallytoJoachimHeinze,whobelievedinthisproject
atanearlystage. OurwarmestthanksgotoFrazerJarvisforlinguisticworkonthe
texts,andtoJimRitterforconstanttechnicalandmoralsupport.
Andlastbutnotleast,wethankalltheparticipantsoftheOberwolfachmeetings
whohavesharedtheirinsightsandknowledgewithusforthebenefitoftheproject:
besides the authors of this book, Leo Corry, Hélène Gispert, Jeremy Gray, Ralf
Haubrich, HelmutKoch, MartinaSchneider, TakaseMasahito, ErhardScholz, Urs
Stammbach,andHansWussing.
Oneoftheparticipantsatthe2001conferencewasMartinKneser. Despitehis
serious illness, his intense, visible passion for number theory and its history was
a challenging inspiration to all of us, historians and mathematicians alike. Martin
KneserdiedonFebruary16,2004. Wededicatethisbooktohismemory.
July2006 CatherineGoldstein
NorbertSchappacher
JoachimSchwermer
Table of Contents
Foreword vii
TableofContents ix
EditionsofC.F.Gauss’sDisquisitionesArithmeticae xi
PartI.ABook’sHistory 1
ChapterI.1 ABookinSearchofaDiscipline(1801–1860) 3
CatherineGoldstein&NorbertSchappacher
ChapterI.2 SeveralDisciplinesandaBook(1860–1901) 67
CatherineGoldstein&NorbertSchappacher
PartII.AlgebraicEquations,QuadraticForms,HigherCongruences:
KeyMathematicalTechniquesoftheDisquisitiones 105
ChapterII.1 TheDisquisitionesArithmeticaeandtheTheoryofEquations 107
OlafNeumann
ChapterII.2 CompositionofBinaryQuadraticFormsandthe
FoundationsofMathematics 129
HaroldM.Edwards
ChapterII.3 CompositionofQuadraticForms: AnAlgebraicPerspective 145
DellaFenster&JoachimSchwermer
ChapterII.4 TheUnpublishedSectionEight: OntheWaytoFunction
FieldsoveraFiniteField 159
GüntherFrei
PartIII.TheGermanReceptionoftheDisquisitionesArithmeticae:
InstitutionsandIdeas 199
ChapterIII.1 ANetworkofScientificPhilanthropy: Humboldt’sRelations
withNumberTheorists 201
HerbertPieper
ChapterIII.2 ^O Je‰c >Arijmht–zei: TheRiseofPureMathematics
asArithmeticwithGauss 235
Jose´ Ferreiro´s
PartIV.ComplexNumbersandComplexFunctionsinArithmetic 269
ChapterIV.1 FromReciprocityLawstoIdealNumbers: An(Un)Known
ManuscriptbyE.E.Kummer 271
ReinhardBölling
ix
x
ChapterIV.2 EllipticFunctionsandArithmetic 291
ChristianHouzel
PartV.NumbersasModelObjectsofMathematics 313
ChapterV.1 TheConceptofNumberfromGausstoKronecker 315
JacquelineBoniface
ChapterV.2 OnArithmetization 343
BirgitPetri&NorbertSchappacher
PartVI.NumberTheoryandtheDisquisitionesinFranceafter1850 375
ChapterVI.1 TheHermitianFormofReadingtheDisquisitiones 377
CatherineGoldstein
ChapterVI.2 NumberTheoryattheAssociationfrançaisepour
l’avancementdessciences 411
Anne-MarieDécaillot
PartVII.SpotlightingSomeLaterReactions 429
ChapterVII.1 AnOverviewonItalianArithmeticaftertheDisquisitiones
Arithmeticae 431
AldoBrigaglia
ChapterVII.2 Zolotarev’sTheoryofAlgebraicNumbers 453
PaolaPiazza
ChapterVII.3 GaussGoesWest: TheReceptionoftheDisquisitiones
ArithmeticaeintheUSA 463
DellaFenster
PartVIII.Gauss’sTheoremsintheLongRun: ThreeCaseStudies 481
ChapterVIII.1 ReductionTheoryofQuadraticForms: TowardRäumliche
AnschauunginMinkowski’sEarlyWork 483
JoachimSchwermer
ChapterVIII.2 GaussSums 505
SamuelJ.Patterson
ChapterVIII.3 TheDevelopmentofthePrincipalGenusTheorem 529
FranzLemmermeyer
ListofIllustrations 563
Index 565
Authors’Addresses 577
Editions of Carl Friedrich Gauss’s
Disquisitiones Arithmeticae
TheDisquisitionesArithmeticaehasbeenomittedfromthelistofreferencesoftheindividual
chapters:welistunderneathitsvariouseditions.Throughoutthisbook,passagesfromGauss’s
DisquisitionesArithmeticaearereferredtoonlybythearticlenumber. ThetitleofGauss’s
workisroutinelyabbreviatedas“D.A.”Forallworks,amentionof[Author1801a]refersto
theitem“AUTHOR.1801a”inthebibliography,amentionof[Author1801/1863]referstothe
1863editioninthisitem.
1801. Disquisitiones Arithmeticae. Leipzig: Fleischer. Repr. Bruxelles: Culture
et civilisation, 1968. Repr. Hildesheim: Olms, 2006. Rev. ed. in Werke, vol. 1,
ed. Königliche Gesellschaft zu Göttingen [E. Schering]. Göttingen: Universitäts-
Druckerei,1863;2ndrev.ed.,1870;repr.Hildesheim: Olms,1973.
http://gallica.bnf.fr
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D137206.html
1807. Recherches arithmétiques. French transl. A.-C.-M. Poullet-Delisle. Paris:
Courcier. Repr.Sceaux: Gabay,1989.
http://gallica.bnf.fr
1889. ArithmetischeUntersuchungen. Germantransl.H.Maser. InUntersuchungen
über höhere Arithmetik, pp. 1–453. Berlin: Springer. Repr. New York: Chelsea,
1965;2nded.,1981.
http://dz-srv1.sub.uni-goettingen.de/cache/toc/D232699.html
1959. Arifmeticˇeskie issledovaniya. Russian transl. V. B. Dem’yanov. In Trudi
po teorii cˇisel [Works on number theory], ed. I. M. Vinogradov, B. N. Delone,
pp.7–583. Moscow: AcademyofSciencesUSSR.
1966. DisquisitionesArithmeticae. Englishtransl.A.A.Clarke. NewHaven: Yale
UniversityPress. Rev. ed. W.C.Waterhouse. NewYork: Springer,1986.
1995. DisquisitionesArithmeticae. Spanishtransl.H.BarrantesCampos,M.Josephy,
A. Ruiz Zu`ñiga. Coleccio´n Enrique Pérez Arbelaez 10. Santa Fe de Bogota´:
AcademiaColombianadeCienciasExactas,FisicasyNaturales.
1995. Seisuuron. Japanesetransl.TakaseMasahito. Tokyo: Asakura-Shoten.
1996. Disquisicions aritmètiques. Catalan transl. G. Pascual Xufré. Barcelona:
Institutd’EstudisCatalans,SocietatCatalanadeMatemàtiques.
xi
Fig.I. TitlepageofDisquisitionesArithmeticae,1801edition
(Privatecopy)