A history of abstract algebra

175 Pages·2007·3.637 MB·English

by  Israel Kleiner

Israel Kleiner
A History of
Abstract Algebra
Birkha¨user
Boston • Basel • Berlin
IsraelKleiner
DepartmentofMathematicsandStatistics
YorkUniversity
Toronto,ONM3J1P3
Canada
[email protected]
CoverdesignbyAlexGerasev,Revere,MA.
MathematicsSubjectClassification(2000):00-01,00-02,01-01,01-02,01A55,01A60,01A70,
12-03,13-03,15-03,16-03,20-03,97-03
LibraryofCongressControlNumber:2007932362
ISBN-13:978-0-8176-4684-4 e-ISBN-13:978-0-8176-4685-1
Printedonacid-freepaper.
(cid:1)c2007Birkha¨userBoston
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit-
tenpermissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233
SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor
scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic
adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde-
velopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
subjecttoproprietaryrights.
9 8 7 6 5 4 3 2 1
www.birkhauser.com (LAP/EB)
With much love to my family
Nava
Ronen, Melissa, Leeor, Tania, Ayelet, Tamir
Tia, Jordana, Jake
Contents
Preface ......................................................... xi
Permissions ..................................................... xv
1 HistoryofClassicalAlgebra ................................... 1
1.1 Earlyroots ................................................ 1
1.2 TheGreeks................................................ 2
1.3 Al-Khwarizmi ............................................. 3
1.4 Cubicandquarticequations.................................. 5
1.5 Thecubicandcomplexnumbers.............................. 7
1.6 Algebraicnotation:VièteandDescartes ........................ 8
1.7 ThetheoryofequationsandtheFundamentalTheoremofAlgebra . 10
1.8 Symbolicalalgebra ......................................... 13
References..................................................... 14
2 HistoryofGroupTheory ...................................... 17
2.1 Sourcesofgrouptheory ..................................... 17
2.1.1 ClassicalAlgebra .................................... 18
2.1.2 NumberTheory ..................................... 19
2.1.3 Geometry........................................... 20
2.1.4 Analysis............................................ 21
2.2 Developmentof“specialized”theoriesofgroups ................ 22
2.2.1 PermutationGroups .................................. 22
2.2.2 AbelianGroups...................................... 26
2.2.3 TransformationGroups ............................... 28
2.3 Emergenceofabstractioningrouptheory ...................... 30
2.4 Consolidationoftheabstractgroupconcept;dawnofabstract
grouptheory............................................... 33
2.5 Divergenceofdevelopmentsingrouptheory.................... 35
References..................................................... 38
viii Contents
3 HistoryofRingTheory ....................................... 41
3.1 Noncommutativeringtheory ................................. 42
3.1.1 ExamplesofHypercomplexNumberSystems ............ 42
3.1.2 Classification........................................ 43
3.1.3 Structure ........................................... 45
3.2 Commutativeringtheory .................................... 47
3.2.1 AlgebraicNumberTheory............................. 48
3.2.2 AlgebraicGeometry.................................. 54
3.2.3 InvariantTheory..................................... 57
3.3 Theabstractdefinitionofaring............................... 58
3.4 EmmyNoetherandEmilArtin ............................... 59
3.5 Epilogue.................................................. 60
References..................................................... 60
4 HistoryofFieldTheory ....................................... 63
4.1 Galoistheory .............................................. 63
4.2 Algebraicnumbertheory .................................... 64
4.2.1 Dedekind’sideas..................................... 65
4.2.2 Kronecker’sideas.................................... 67
4.2.3 DedekindvsKronecker ............................... 68
4.3 Algebraicgeometry......................................... 68
4.3.1 FieldsofAlgebraicFunctions .......................... 68
4.3.2 FieldsofRationalFunctions ........................... 70
4.4 Congruences .............................................. 70
4.5 Symbolicalalgebra ......................................... 71
4.6 Theabstractdefinitionofafield .............................. 71
4.7 Hensel’sp-adicnumbers..................................... 73
4.8 Steinitz ................................................... 74
4.9 Aglanceahead............................................. 76
References..................................................... 77
5 HistoryofLinearAlgebra ..................................... 79
5.1 Linearequations ........................................... 79
5.2 Determinants .............................................. 81
5.3 Matricesandlineartransformations ........................... 82
5.4 Linearindependence,basis,anddimension ..................... 84
5.5 Vectorspaces .............................................. 86
References..................................................... 89
6 EmmyNoetherandtheAdventofAbstractAlgebra ............... 91
6.1 Invarianttheory ............................................ 92
6.2 Commutativealgebra ....................................... 94
6.3 Noncommutativealgebraandrepresentationtheory .............. 97
6.4 Applicationsofnoncommutativetocommutativealgebra ......... 98
6.5 Noether’slegacy ........................................... 99
References..................................................... 101
Contents ix
7 ACourseinAbstractAlgebraInspiredbyHistory ................. 103
ProblemI:Whyis(−1)(−1)=1?................................. 104
ProblemII:Whataretheintegersolutionsofx2+2=y3?............. 105
ProblemIII:Canwetrisecta60◦ angleusingonlystraightedgeand
compass? ................................................. 106
ProblemIV:Canwesolvex5−6x+3=0byradicals? .............. 107
ProblemV:“Papa,canyoumultiplytriples?” ........................ 108
Generalremarksonthecourse .................................... 109
References..................................................... 110
8 BiographiesofSelectedMathematicians ......................... 113
8.1 ArthurCayley(1821–1895) .................................. 113
8.1.1 Invariants........................................... 115
8.1.2 Groups............................................. 116
8.1.3 Matrices............................................ 117
8.1.4 Geometry........................................... 118
8.1.5 Conclusion ......................................... 119
References..................................................... 120
8.2 RichardDedekind(1831–1916)............................... 121
8.2.1 AlgebraicNumbers................................... 124
8.2.2 RealNumbers....................................... 126
8.2.3 NaturalNumbers .................................... 128
8.2.4 OtherWorks ........................................ 129
8.2.5 Conclusion ......................................... 131
References..................................................... 132
8.3 EvaristeGalois(1811–1832) ................................. 133
8.3.1 Mathematics ........................................ 135
8.3.2 Politics............................................. 135
8.3.3 Theduel............................................ 137
8.3.4 Testament .......................................... 137
8.3.5 Conclusion ......................................... 138
References..................................................... 139
8.4 CarlFriedrichGauss(1777–1855) ............................ 139
8.4.1 Numbertheory ...................................... 140
8.4.2 DifferentialGeometry,Probability,andStatistics.......... 142
8.4.3 Thediary ........................................... 142
8.4.4 Conclusion ......................................... 143
References..................................................... 144
8.5 WilliamRowanHamilton(1805–1865) ........................ 144
8.5.1 Optics.............................................. 146
8.5.2 Dynamics .......................................... 147
8.5.3 ComplexNumbers ................................... 149
8.5.4 FoundationsofAlgebra ............................... 150
8.5.5 Quaternions......................................... 152
8.5.6 Conclusion ......................................... 156
References..................................................... 156
x Contents
8.6 EmmyNoether(1882–1935) ................................. 157
8.6.1 EarlyYears ......................................... 157
8.6.2 UniversityStudies ................................... 158
8.6.3 Göttingen........................................... 159
8.6.4 NoetherasaTeacher ................................. 160
8.6.5 BrynMawr ......................................... 161
8.6.6 Conclusion ......................................... 162
References..................................................... 162
Index........................................................... 165
Preface
Mygoalinwritingthisbookwastogiveanaccountofthehistoryofmanyofthebasic
concepts, results, and theories of abstract algebra, an account that would be useful
forteachersofrelevantcourses,fortheirstudents,andforthebroadermathematical
public.
Thecoreofafirstcourseinabstractalgebradealswithgroups,rings,andfields.
ThesearethecontentsofChapters2,3,and4,respectively.Butabstractalgebragrew
outofanearlierclassicaltradition,whichmeritsanintroductorychapterinitsown
right (Chapter 1). In this tradition, which developed before the nineteenth century,
“algebra” meant the study of the solution of polynomial equations. In the twenti-
ethcenturyitmeantthestudyofabstract,axiomaticsystemssuchasgroups,rings,
and fields. The transition from “classical” to “modern” occurred in the nineteenth
century.Abstractalgebracameintoexistencelargelybecausemathematicianswere
unabletosolveclassical(pre-nineteenth-century)problemsbyclassicalmeans.The
classicalproblemscamefromnumbertheory,geometry,analysis,thesolvabilityof
polynomialequations,andtheinvestigationofpropertiesofvariousnumbersystems.
Amajorthemeofthisbookistoshowhow“abstract”algebrahasariseninattemptsto
solvesomeofthese“concrete”problems,thusprovidingconfirmationofWhitehead’s
paradoxicaldictumthat“theutmostabstractionsarethetrueweaponswithwhichto
controlourthoughtofconcretefact.”Putanotherway:thereisnothingsopractical
asagoodtheory.
Although linear algebra is not normally taught in a course in abstract algebra,
itsevolutionhasoftenbeenconnectedwiththatofgroups,rings,andfields.And,of
course, vector spaces are among the fundamental notions of abstract algebra. This
warrantsa(short)chapteronthehistoryoflinearalgebra(Chapter5).
Abstractalgebraisessentiallyacreationofthenineteenthcentury,butitbecamean
independentandflourishingsubjectonlyintheearlydecadesofthetwentieth,largely
throughthepioneeringworkofEmmyNoether,whohasbeencalled“thefather”of
abstractalgebra.ThusthechapteronNoether’salgebraicwork(Chapter6).
Itismyfirmbelief,buttressedbymyownteachingexperience,thatthehistoryof
mathematicscanmakeanimportantcontributiontoour—teachers’andstudents’—
understanding and appreciation of mathematics. It can act as a useful integrating