Basic algebra

737 Pages·2006·4.126 MB·English

by  Anthony W. Knapp

Cornerstones
Series Editors
CharlesL.Epstein,UniversityofPennsylvania,Philadelphia
StevenG.Krantz,UniversityofWashington,St.Louis
Advisory Board
AnthonyW.Knapp,StateUniversityofNewYorkatStonyBrook,Emeritus
Anthony W. Knapp
Basic Algebra
Along with a companion volume
Advanced Algebra
Birkha¨user
Boston • Basel • Berlin
AnthonyW.Knapp
81UpperSheepPastureRoad
EastSetauket,NY11733-1729
U.S.A.
e-mail to: [email protected]
http://www.math.sunysb.edu/˜aknapp/books/b-alg.html
CoverdesignbyMaryBurgess.
MathematicsSubjectClassicification(2000):15-01,20-02,13-01,12-01,16-01,08-01,18A05,68P30
LibraryofCongressControlNumber:2006932456
ISBN-100-8176-3248-4 eISBN-100-8176-4529-2
ISBN-13978-0-8176-3248-9 eISBN-13978-0-8176-4529-8
AdvancedAlgebra ISBN0-8176-4522-5
BasicAlgebraandAdvancedAlgebra(Set) ISBN0-8176-4533-0
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(cid:1)c2006AnthonyW.Knapp
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CONTENTS
ContentsofAdvancedAlgebra x
ListofFigures xi
Preface xiii
DependenceAmongChapters xvii
StandardNotation xviii
GuidefortheReader xix
I. PRELIMINARIESABOUTTHEINTEGERS,
POLYNOMIALS,ANDMATRICES 1
1. DivisionandEuclideanAlgorithms 1
2. UniqueFactorizationofIntegers 4
3. UniqueFactorizationofPolynomials 9
4. PermutationsandTheirSigns 15
5. RowReduction 19
6. MatrixOperations 24
7. Problems 30
II. VECTORSPACESOVERQ,R,ANDC 33
1. Spanning,LinearIndependence,andBases 33
2. VectorSpacesDefinedbyMatrices 38
3. LinearMaps 42
4. DualSpaces 50
5. QuotientsofVectorSpaces 54
6. DirectSumsandDirectProductsofVectorSpaces 58
7. Determinants 65
8. EigenvectorsandCharacteristicPolynomials 73
9. BasesintheInfinite-DimensionalCase 77
10. Problems 82
III. INNER-PRODUCTSPACES 88
1. InnerProductsandOrthonormalSets 88
2. Adjoints 98
3. SpectralTheorem 104
4. Problems 111
vii
viii Contents
IV. GROUPSANDGROUPACTIONS 116
1. GroupsandSubgroups 117
2. QuotientSpacesandHomomorphisms 128
3. DirectProductsandDirectSums 134
4. RingsandFields 140
5. PolynomialsandVectorSpaces 147
6. GroupActionsandExamples 158
7. SemidirectProducts 166
8. SimpleGroupsandCompositionSeries 170
9. StructureofFinitelyGeneratedAbelianGroups 174
10. SylowTheorems 183
11. CategoriesandFunctors 188
12. Problems 198
V. THEORYOFASINGLELINEARTRANSFORMATION 209
1. Introduction 209
2. DeterminantsoverCommutativeRingswithIdentity 212
3. CharacteristicandMinimalPolynomials 216
4. ProjectionOperators 224
5. PrimaryDecomposition 226
6. JordanCanonicalForm 229
7. ComputationswithJordanForm 235
8. Problems 239
VI. MULTILINEARALGEBRA 245
1. BilinearFormsandMatrices 246
2. SymmetricBilinearForms 250
3. AlternatingBilinearForms 253
4. HermitianForms 255
5. GroupsLeavingaBilinearFormInvariant 257
6. TensorProductofTwoVectorSpaces 260
7. TensorAlgebra 274
8. SymmetricAlgebra 280
9. ExteriorAlgebra 288
10. Problems 292
VII. ADVANCEDGROUPTHEORY 303
1. FreeGroups 303
2. SubgroupsofFreeGroups 314
3. FreeProducts 319
4. GroupRepresentations 326
Contents ix
VII. ADVANCEDGROUPTHEORY (Continued)
5. Burnside’sTheorem 342
6. ExtensionsofGroups 344
7. Problems 357
VIII.COMMUTATIVERINGSANDTHEIRMODULES 367
1. ExamplesofRingsandModules 367
2. IntegralDomainsandFieldsofFractions 378
3. PrimeandMaximalIdeals 381
4. UniqueFactorization 384
5. Gauss’sLemma 390
6. FinitelyGeneratedModules 396
7. OrientationforAlgebraicNumberTheoryand
AlgebraicGeometry 408
8. NoetherianRingsandtheHilbertBasisTheorem 414
9. IntegralClosure 417
10. LocalizationandLocalRings 425
11. DedekindDomains 434
12. Problems 439
IX. FIELDSANDGALOISTHEORY 448
1. AlgebraicElements 449
2. ConstructionofFieldExtensions 453
3. FiniteFields 457
4. AlgebraicClosure 460
5. GeometricConstructionsbyStraightedgeandCompass 464
6. SeparableExtensions 469
7. NormalExtensions 476
8. FundamentalTheoremofGaloisTheory 479
9. ApplicationtoConstructibilityofRegularPolygons 483
10. ApplicationtoProvingtheFundamentalTheoremofAlgebra 486
11. ApplicationtoUnsolvabilityofPolynomialEquationswith
NonsolvableGaloisGroup 488
12. ConstructionofRegularPolygons 493
13. SolutionofCertainPolynomialEquationswithSolvable
GaloisGroup 501
14. ProofThatπ IsTranscendental 510
15. NormandTrace 514
16. SplittingofPrimeIdealsinExtensions 521
17. TwoToolsforComputingGaloisGroups 527
18. Problems 534
x Contents
X. MODULESOVERNONCOMMUTATIVERINGS 544
1. SimpleandSemisimpleModules 544
2. CompositionSeries 551
3. ChainConditions 556
4. HomandEndforModules 558
5. TensorProductforModules 565
6. ExactSequences 574
7. Problems 579
APPENDIX 583
A1. SetsandFunctions 583
A2. EquivalenceRelations 589
A3. RealNumbers 591
A4. ComplexNumbers 594
A5. PartialOrderingsandZorn’sLemma 595
A6. Cardinality 599
HintsforSolutionsofProblems 603
SelectedReferences 697
IndexofNotation 699
Index 703
CONTENTS OF ADVANCEDALGEBRA
I. TransitiontoModernNumberTheory
II. Wedderburn–ArtinRingTheory
III. BrauerGroup
IV. HomologicalAlgebra
V. ThreeTheoremsinAlgebraicNumberTheory
VI. ReinterpretationwithAdelesandIdeles
VII. InfiniteFieldExtensions
VIII. BackgroundforAlgebraicGeometry
IX. TheNumberTheoryofAlgebraicCurves
X. MethodsofAlgebraicGeometry
LIST OF FIGURES
2.1. Thevectorspaceoflinesv+U inR2 paralleltoagivenlineU
throughtheorigin 55
2.2. Factorizationoflinearmapsviaaquotientofvectorspaces 56
2.3. Three1-dimensionalvectorsubspacesofR2 suchthateachpair
hasintersection0 62
2.4. Universalmappingpropertyofadirectproductofvectorspaces 64
2.5. Universalmappingpropertyofadirectsumofvectorspaces 65
3.1. Geometricinterpretationoftheparallelogramlaw 91
3.2. Resolutionofavectorintoaparallelcomponentandan
orthogonalcomponent 93
4.1. Factorizationofhomomorphismsofgroupsviathequotientofa
groupbyanormalsubgroup 132
4.2. Universalmappingpropertyofanexternaldirectproductofgroups 136
4.3. Universalmappingpropertyofadirectproductofgroups 136
4.4. Universalmappingpropertyofanexternaldirectsumofabelian
groups 138
4.5. Universalmappingpropertyofadirectsumofabeliangroups 139
4.6. Factorizationofhomomorphismsofringsviathequotientofaring
byanideal 146
4.7. Substitutionhomomorphismforpolynomialsinoneindeterminate 150
4.8. Substitutionhomomorphismforpolynomialsinn indeterminates 156
4.9. Asquarediagram 193
4.10. Diagramsobtainedbyapplyingacovariantfunctoranda
contravariantfunctor 193
4.11. Universalmappingpropertyofaproductinacategory 195
4.12. Universalmappingpropertyofacoproductinacategory 197
5.1. ExampleofanilpotentmatrixinJordanform 231
5.2. PowersofthenilpotentmatrixinFigure5.1 232
6.1. Universalmappingpropertyofatensorproduct 261
6.2. Diagramsforuniquenessofatensorproduct 261
xi