Basic Algebra

762 Pages·2015·8.97 MB·English

by  Anthony Knapp

BasicAlgebra
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
BasicAlgebra
Along with a Companion VolumeAdvancedAlgebra
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: BasicAlgebra
Cover: Constructionofaregularheptadecagon,thestepsshownincolorsequence;seepage505.
MathematicsSubjectClassification(2010): 15–01,20–01,13–01,12–01,16–01,08–01,18A05,
68P30.
FirstEdition,ISBN-13978-0-8176-3248-9
c2006AnthonyW.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 AdvancedAlgebra x
PrefacetotheSecondEdition xi
PrefacetotheFirstEdition xiii
ListofFigures xvii
DependenceAmongChapters xix
StandardNotation xx
GuidefortheReader xxi
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 78
10. Problems 82
III. INNER-PRODUCTSPACES 89
1. InnerProductsandOrthonormalSets 89
2. Adjoints 99
3. SpectralTheorem 105
4. Problems 112
vii
viii Contents
IV. GROUPSANDGROUPACTIONS 117
1. GroupsandSubgroups 118
2. QuotientSpacesandHomomorphisms 129
3. DirectProductsandDirectSums 135
4. RingsandFields 141
5. PolynomialsandVectorSpaces 148
6. GroupActionsandExamples 159
7. SemidirectProducts 167
8. SimpleGroupsandCompositionSeries 171
9. StructureofFinitelyGeneratedAbelianGroups 176
10. SylowTheorems 185
11. CategoriesandFunctors 189
12. Problems 200
V. THEORYOFASINGLELINEARTRANSFORMATION 211
1. Introduction 211
2. DeterminantsoverCommutativeRingswithIdentity 215
3. CharacteristicandMinimalPolynomials 218
4. ProjectionOperators 226
5. PrimaryDecomposition 228
6. JordanCanonicalForm 231
7. ComputationswithJordanForm 238
8. Problems 241
VI. MULTILINEARALGEBRA 248
1. BilinearFormsandMatrices 249
2. SymmetricBilinearForms 253
3. AlternatingBilinearForms 256
4. HermitianForms 258
5. GroupsLeavingaBilinearFormInvariant 260
6. TensorProductofTwoVectorSpaces 263
7. TensorAlgebra 277
8. SymmetricAlgebra 283
9. ExteriorAlgebra 291
10. Problems 295
VII. ADVANCEDGROUPTHEORY 306
1. FreeGroups 306
2. SubgroupsofFreeGroups 317
3. FreeProducts 322
4. GroupRepresentations 329
Contents ix
VII. ADVANCEDGROUPTHEORY (Continued)
5. Burnside’sTheorem 345
6. ExtensionsofGroups 347
7. Problems 360
VIII.COMMUTATIVERINGSANDTHEIRMODULES 370
1. ExamplesofRingsandModules 370
2. IntegralDomainsandFieldsofFractions 381
3. PrimeandMaximalIdeals 384
4. UniqueFactorization 387
5. Gauss’sLemma 393
6. FinitelyGeneratedModules 399
7. OrientationforAlgebraicNumberTheoryand
AlgebraicGeometry 411
8. NoetherianRingsandtheHilbertBasisTheorem 417
9. IntegralClosure 420
10. LocalizationandLocalRings 428
11. DedekindDomains 437
12. Problems 443
IX. FIELDSANDGALOISTHEORY 452
1. AlgebraicElements 453
2. ConstructionofFieldExtensions 457
3. FiniteFields 461
4. AlgebraicClosure 464
5. GeometricConstructionsbyStraightedgeandCompass 468
6. SeparableExtensions 474
7. NormalExtensions 481
8. FundamentalTheoremofGaloisTheory 484
9. ApplicationtoConstructibilityofRegularPolygons 489
10. ApplicationtoProvingtheFundamentalTheoremofAlgebra 492
11. ApplicationtoUnsolvabilityofPolynomialEquationswith
NonsolvableGaloisGroup 493
12. ConstructionofRegularPolygons 499
13. SolutionofCertainPolynomialEquationswithSolvable
GaloisGroup 506
14. ProofThatπ IsTranscendental 515
15. NormandTrace 519
16. SplittingofPrimeIdealsinExtensions 526
17. TwoToolsforComputingGaloisGroups 532
18. Problems 539
x Contents
X. MODULESOVERNONCOMMUTATIVERINGS 553
1. SimpleandSemisimpleModules 553
2. CompositionSeries 560
3. ChainConditions 565
4. HomandEndforModules 567
5. TensorProductforModules 574
6. ExactSequences 583
7. Problems 587
APPENDIX 593
A1. SetsandFunctions 593
A2. EquivalenceRelations 599
A3. RealNumbers 601
A4. ComplexNumbers 604
A5. PartialOrderingsandZorn’sLemma 605
A6. Cardinality 610
HintsforSolutionsofProblems 615
SelectedReferences 715
IndexofNotation 717
Index 721
CONTENTS OF ADVANCED ALGEBRA
I. TransitiontoModernNumberTheory
II. Wedderburn–ArtinRingTheory
III. BrauerGroup
IV. HomologicalAlgebra
V. ThreeTheoremsinAlgebraicNumberTheory
VI. ReinterpretationwithAdelesandIdeles
VII. InfiniteFieldExtensions
VIII. BackgroundforAlgebraicGeometry
IX. TheNumberTheoryofAlgebraicCurves
X. MethodsofAlgebraicGeometry