Cornerstones Series Editors CharlesL.Epstein,UniversityofPennsylvania,Philadelphia StevenG.Krantz,WashingtonUniversity,St.Louis Advisory Board AnthonyW.Knapp,StateUniversityofNewYorkatStonyBrook,Emeritus Anthony W. Knapp Advanced Algebra Along with a companion volume Basic 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/a-alg.html CoverdesignbyMaryBurgess. MathematicsSubjectClassicification(2000):11-01,13-01,14-01,16-01,18G99,55U99,11R04,11S15, 12F99,14A05,14H05,12Y05,14A10,14Q99 LibraryofCongressControlNumber:2007936880 ISBN-13:978-0-8176-4522-9 eISBN-13:978-0-8176-4613-4 BasicAlgebra ISBN-13:978-0-8176-3248-9 BasicAlgebraandAdvancedAlgebra(Set) ISBN-13:978-0-8176-4533-5 Printedonacid-freepaper. (cid:1)c2007AnthonyW.Knapp All rights reserved. This work may not be translated or copied in whole or in part without the written permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233Spring Street,NewYork,NY10013,USA)andtheauthor,exceptforbriefexcerptsinconnectionwithreviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped isforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjectto proprietaryrights. 9 8 7 6 5 4 3 2 1 www.birkhauser.com (MP) ToSusan and ToMyAlgebraTeachers: RalphFox,JohnFraleigh,RobertGunning, JohnKemeny,BertramKostant,RobertLanglands, GoroShimura,HaleTrotter,RichardWilliamson CONTENTS ContentsofBasicAlgebra x Preface xi ListofFigures xv DependenceamongChapters xvi GuidefortheReader xvii NotationandTerminology xxi 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 Contents ix VII. INFINITEFIELDEXTENSIONS 403 1. Nullstellensatz 404 2. TranscendenceDegree 408 3. SeparableandPurelyInseparableExtensions 414 4. KrullDimension 423 5. NonsingularandSingularPoints 428 6. InfiniteGaloisGroups 434 7. Problems 445 VIII.BACKGROUNDFORALGEBRAICGEOMETRY 447 1. HistoricalOriginsandOverview 448 2. ResultantandBezout’sTheorem 451 3. ProjectivePlaneCurves 456 4. IntersectionMultiplicityforaLinewithaCurve 466 5. IntersectionMultiplicityforTwoCurves 473 6. GeneralFormofBezout’sTheoremforPlaneCurves 488 7. Gro¨bnerBases 491 8. ConstructiveExistence 499 9. UniquenessofReducedGro¨bnerBases 508 10. SimultaneousSystemsofPolynomialEquations 510 11. Problems 516 IX. THENUMBERTHEORYOFALGEBRAICCURVES 520 1. HistoricalOriginsandOverview 520 2. Divisors 531 3. Genus 534 4. Riemann–RochTheorem 540 5. ApplicationsoftheRiemann–RochTheorem 552 6. Problems 554 X. METHODSOFALGEBRAICGEOMETRY 558 1. AffineAlgebraicSetsandAffineVarieties 559 2. GeometricDimension 563 3. ProjectiveAlgebraicSetsandProjectiveVarieties 570 4. RationalFunctionsandRegularFunctions 579 5. Morphisms 590 6. RationalMaps 595 7. Zariski’sTheoremaboutNonsingularPoints 600 8. ClassificationQuestionsaboutIrreducibleCurves 604 9. AffineAlgebraicSetsforMonomialIdeals 618 10. HilbertPolynomialintheAffineCase 626 x Contents X. METHODSOFALGEBRAICGEOMETRY (Continued) 11. HilbertPolynomialintheProjectiveCase 633 12. IntersectionsinProjectiveSpace 635 13. Schemes 638 14. Problems 644 HintsforSolutionsofProblems 649 SelectedReferences 713 IndexofNotation 717 Index 721 CONTENTS OF BASIC ALGEBRA I. PreliminariesabouttheIntegers,Polynomials,andMatrices II. VectorSpacesoverQ,R,andC III. Inner-ProductSpaces IV. GroupsandGroupActions V. TheoryofaSingleLinearTransformation VI. MultilinearAlgebra VII. AdvancedGroupTheory VIII. CommutativeRingsandTheirModules IX. FieldsandGaloisTheory X. ModulesoverNoncommutativeRings PREFACE Advanced Algebra and its companion volumeBasic Algebra systematically de- velopconceptsandtoolsinalgebrathatarevitaltoeverymathematician,whether pureorapplied,aspiringorestablished. Thetwobookstogetheraimtogivethe reader a global view of algebra, its use, and its role in mathematics as a whole. Theideaistoexplainwhattheyoungmathematicianneedstoknowaboutalgebra inordertocommunicatewellwithcolleaguesinallbranchesofmathematics. The books are written as textbooks, and their primary audience is students whoarelearningthematerialforthefirsttimeandwhoareplanningacareerin whichtheywilluseadvancedmathematicsprofessionally. Muchofthematerial in the two books, including nearly all of Basic Algebra and some of Advanced Algebra,correspondstonormalcoursework,withtheproportionsdependingon theuniversity. Thebooksincludefurthertopicsthatmaybeskippedinrequired coursesbutthattheprofessionalmathematicianwillultimatelywanttolearnby self-study. The test of each topic for inclusion iswhether it issomething that a plenary lecturer at a broad international or national meeting is likely to take as knownbytheaudience. KeytopicsandfeaturesofAdvancedAlgebraareasfollows: • Topicsbuildonthelinearalgebra,grouptheory,factorizationofideals,struc- tureoffields,Galoistheory,andelementarytheoryofmodulesdevelopedin BasicAlgebra. • Individualchapterstreatvarioustopicsincommutativeandnoncommutative algebra,togetherprovidingintroductionstothetheoryofassociativealgebras, homologicalalgebra,algebraicnumbertheory,andalgebraicgeometry. • Thetextemphasizesconnectionsbetweenalgebraandotherbranchesofmath- ematics,particularlytopologyandcomplexanalysis. Allthewhile,itcarries along two themes from Basic Algebra: the analogy between integers and polynomialsinonevariableoverafield,andtherelationshipbetweennumber theoryandgeometry. • SeveralsectionsintwochaptersintroducethesubjectofGro¨bnerbases,which isthemoderngatewaytowardhandlingsimultaneouspolynomialequationsin applications. • Thedevelopmentproceedsfromtheparticulartothegeneral,oftenintroducing exampleswellbeforeatheorythatincorporatesthem. xi