Table Of ContentAdvancedAlgebra
Digital Second Editions
By Anthony W. Knapp
Basic Algebra
Advanced Algebra
Basic Real Analysis,
with an appendix “ElementaryComplex Analysis”
Advanced Real Analysis
Anthony W. Knapp
AdvancedAlgebra
Along with a Companion VolumeBasicAlgebra
Digital Second Edition, 2016
Published by the Author
East Setauket, New York
AnthonyW.Knapp
81UpperSheepPastureRoad
EastSetauket,N.Y.11733–1729,U.S.A.
Emailto: [email protected]
Homepage:www.math.stonybrook.edu/ aknapp
∼
Title: AdvancedAlgebra
Cover: ContentoftheSnakeLemma;seepage185.
MathematicsSubjectClassification(2010): 11–01,13–01,14–01,16–01,18G99,55U99,11R04,
11S15,12F99,14A05,14H05,12Y05,14A10,14Q99.
FirstEdition,ISBN-13978-0-8176-4522-9
c2007AnthonyW.Knapp
"
PublishedbyBirkha¨userBoston
DigitalSecondEdition,nottobesold,noISBN
c2016AnthonyW.Knapp
"
PublishedbytheAuthor
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iv
ToSusan
and
ToMyChildren,SarahandWilliam,
and
ToMyAlgebraTeachers:
RalphFox,JohnFraleigh,RobertGunning,
JohnKemeny,BertramKostant,RobertLanglands,
GoroShimura,HaleTrotter,RichardWilliamson
CONTENTS
Contentsof BasicAlgebra x
PrefacetotheSecondEdition xi
PrefacetotheFirstEdition xiii
ListofFigures xvii
DependenceamongChapters xviii
GuidefortheReader xix
NotationandTerminology xxiii
I. TRANSITIONTOMODERNNUMBERTHEORY 1
1. HistoricalBackground 1
2. QuadraticReciprocity 8
3. EquivalenceandReductionofQuadraticForms 12
4. CompositionofForms,ClassGroup 24
5. Genera 31
6. QuadraticNumberFieldsandTheirUnits 35
7. RelationshipofQuadraticFormstoIdeals 38
8. PrimesintheProgressions4n 1and4n 3 50
+ +
9. DirichletSeriesandEulerProducts 56
10. Dirichlet’sTheoremonPrimesinArithmeticProgressions 61
11. Problems 67
II. WEDDERBURN–ARTINRINGTHEORY 76
1. HistoricalMotivation 77
2. SemisimpleRingsandWedderburn’sTheorem 81
3. RingswithChainConditionandArtin’sTheorem 87
4. Wedderburn–ArtinRadical 89
5. Wedderburn’sMainTheorem 94
6. SemisimplicityandTensorProducts 104
7. Skolem–NoetherTheorem 111
8. DoubleCentralizerTheorem 114
9. Wedderburn’sTheoremaboutFiniteDivisionRings 117
10. Frobenius’sTheoremaboutDivisionAlgebrasovertheReals 118
11. Problems 120
vii
viii Contents
III. BRAUERGROUP 123
1. DefinitionandExamples,RelativeBrauerGroup 124
2. FactorSets 132
3. CrossedProducts 135
4. Hilbert’sTheorem90 145
5. DigressiononCohomologyofGroups 147
6. RelativeBrauerGroupwhentheGaloisGroupIsCyclic 158
7. Problems 162
IV. HOMOLOGICALALGEBRA 166
1. Overview 167
2. ComplexesandAdditiveFunctors 171
3. LongExactSequences 184
4. ProjectivesandInjectives 192
5. DerivedFunctors 202
6. LongExactSequencesofDerivedFunctors 210
7. ExtandTor 223
8. AbelianCategories 232
9. Problems 250
V. THREETHEOREMSINALGEBRAICNUMBERTHEORY 262
1. Setting 262
2. Discriminant 266
3. DedekindDiscriminantTheorem 274
4. CubicNumberFieldsasExamples 279
5. DirichletUnitTheorem 288
6. FinitenessoftheClassNumber 298
7. Problems 307
VI. REINTERPRETATIONWITHADELESANDIDELES 313
1. p-adicNumbers 314
2. DiscreteValuations 320
3. AbsoluteValues 331
4. Completions 342
5. Hensel’sLemma 349
6. RamificationIndicesandResidueClassDegrees 353
7. SpecialFeaturesofGaloisExtensions 368
8. DifferentandDiscriminant 371
9. GlobalandLocalFields 382
10. AdelesandIdeles 388
11. Problems 397
