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M00 FRAL0363 08 LLV C00 pagei A First Course in Abstract Algebra Eighth Edition John B. Fraleigh University of Rhode Island Neal Brand University of North Texas Historical Notes by Victor Katz University of District of Columbia M00 FRAL0363 08 LLV C00 pageii Copyright(cid:2)c 2021,2003,1994byPearsonEducation,Inc.oritsaffiliates,221RiverStreet,Hoboken, NJ07030.AllRightsReserved.ManufacturedintheUnitedStatesofAmerica.Thispublicationis protectedbycopyright,andpermissionshouldbeobtainedfromthepublisherpriortoanyprohibited reproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic, mechanical,photocopying,recording,orotherwise.Forinformationregardingpermissions,request forms,andtheappropriatecontactswithinthePearsonEducationGlobalRightsandPermissions department,pleasevisitwww.pearsoned.com/permissions/. Acknowledgmentsofthird-partycontentappearontheappropriatepagewithinthetext. Coverimagecredit:EdmundSumner/AGEFotostock PEARSON,ALWAYSLEARNING,andMYLABareexclusivetrademarksownedbyPearson Education,Inc.oritsaffiliatesintheU.S.and/orothercountries. Unlessotherwiseindicatedherein,anythird-partytrademarks,logos,oriconsthatmayappearinthis workarethepropertyoftheirrespectiveowners,andanyreferencestothird-partytrademarks,logos, icons,orothertradedressarefordemonstrativeordescriptivepurposesonly.Suchreferencesarenot intendedtoimplyanysponsorship,endorsement,authorization,orpromotionofPearson’sproductsby theownersofsuchmarks,oranyrelationshipbetweentheownerandPearsonEducation,Inc.,orits affiliates,authors,licensees,ordistributors. LibraryofCongressCataloging-in-PublicationData Names:Fraleigh,JohnB.,author.|Katz,VictorJ.,writerofadded commentary. Title:Afirstcourseinabstractalgebra/JohnB.Fraleigh;historical notesbyVictorKatz. Description:Eighthedition.|[Hoboken,NewJersey]:Pearson,[2021]| Series:Worldstudentseries|Includesbibliographicalreferencesand index. Identifiers:LCCN2019038536|ISBN9780135758168(paperback)|ISBN 9780321390363(ebook) Subjects:LCSH:Algebra,Abstract. Classification:LCCQA162.F72020|DDC512/.02–dc23 LCrecordavailableathttps://lccn.loc.gov/2019038536 ScoutAutomatedPrintCode Rental ISBN-10: 0-13-673162-7 ISBN-13:978-0-13-673162-7 LooseLeafVersion ISBN-10: 0-13-575816-5 ISBN-13:978-0-13-575816-8 M00 FRAL0363 08 LLV C00 pageiii Contents Instructor’sPreface vii DependenceChart xii Student’sPreface xv 0 SetsandRelations 1 I GROUPS AND SUBGROUPS 11 1 BinaryOperations 11 2 Groups 19 3 AbelianExamples 32 4 NonabelianExamples 39 5 Subgroups 52 6 CyclicGroups 61 7 GeneratingSetsandCayleyDigraphs 70 II STRUCTURE OF GROUPS 77 8 GroupsofPermutations 77 9 FinitelyGeneratedAbelianGroups 88 10 CosetsandtheTheoremofLagrange 97 †11 PlaneIsometries 105 III HOMOMORPHISMS AND FACTOR GROUPS 113 12 FactorGroups 113 13 Factor-GroupComputationsandSimpleGroups 121 iii M00 FRAL0363 08 LLV C00 pageiv iv Contents ‡14 GroupActiononaSet 132 †15 ApplicationsofG-SetstoCounting 140 IV ADVANCED GROUP THEORY 145 16 IsomorphismTheorems 145 17 SylowTheorems 149 18 SeriesofGroups 157 19 FreeAbelianGroups 166 20 FreeGroups 172 21 GroupPresentations 177 V RINGS AND FIELDS 185 22 RingsandFields 185 23 IntegralDomains 194 24 Fermat’sandEuler’sTheorems 200 25 Encryption 205 VI CONSTRUCTING RINGS AND FIELDS 211 26 TheFieldofQuotientsofanIntegralDomain 211 27 RingsofPolynomials 218 28 FactorizationofPolynomialsoveraField 228 †29 AlgebraicCodingTheory 237 30 HomomorphismsandFactorRings 243 31 PrimeandMaximalIdeals 250 †32 NoncommutativeExamples 258 VII COMMUTATIVE ALGEBRA 267 33 VectorSpaces 267 34 UniqueFactorizationDomains 275 35 EuclideanDomains 286 36 NumberTheory 292 †37 AlgebraicGeometry 297 †38 Gro¨bnerBasesforIdeals 303 VIII EXTENSION FIELDS 311 39 IntroductiontoExtensionFields 311 40 AlgebraicExtensions 319 †41 GeometricConstructions 328 42 FiniteFields 335 M00 FRAL0363 08 LLV C00 pagev Contents v IX GALOIS THEORY 341 43 IntroductiontoGaloisTheory 341 44 SplittingFields 349 45 SeparableExtensions 357 46 GaloisTheory 364 47 IllustrationsofGaloisTheory 372 48 CyclotomicExtensions 378 49 InsolvabilityoftheQuintic 384 Appendix:MatrixAlgebra 391 Bibliography 395 Notations 397 AnswerstoOdd-NumberedExercisesNotAskingforDefinitionsorProofs 401 Index 419 †Notrequiredfortheremainderofthetext. ‡ThissectionisaprerequisiteforSections17and36only. M00 FRAL0363 08 LLV C00 pagevi This page is intentionally left blank M00 FRAL0363 08 LLV C00 pagevii Instructor’s Preface Thisisanintroductiontoabstractalgebra.Itisanticipatedthatthestudentshavestudied calculus and probablylinear algebra. However,these are primarily mathematicalma- turity prerequisites;subjectmatter fromcalculusandlinear algebraappearsmostly in illustrativeexamplesandexercises. Asinpreviouseditionsofthetext,ouraimremainstoteachstudentsasmuchabout groups,rings,andfieldsaswecaninafirstcourse.Formanystudents,abstractalgebra istheirfirstextendedexposuretoanaxiomatictreatmentofmathematics.Recognizing this,wehaveincludedextensiveexplanationsconcerningwhatwearetryingtoaccom- plish, how we are trying to do it, and why we choose these methods. Mastery of this textconstitutesafirmfoundationformorespecializedworkinalgebraandalsoprovides valuableexperienceforanyfurtheraxiomaticstudyofmathematics. New to This Edition [Editor’sNote:Youmayhavenoticedsomethingnewonthecoverofthebook.Another author!IamthrilledthatNealBrandagreedtoupdatethisclassictext.Hehasdoneso carefullyandthoughtfully,stayingtruetothespiritinwhichitwaswritten.Neal’syears of experienceteachingthe coursewith this textatthe Universityof NorthTexashave helpedhimproduceameaningfulandworthwhileupdatetoJohnFraleigh’swork.] Updatesfor the eText Afocusofthisrevisionwastransformingitfromaprimarilyprint-basedlearningtoolto adigitallearningtool.TheeTextisthereforefilledwithcontentandtoolsthatwillhelp bringthecontentofthe courseto life forstudentsin new waysandhelpyouimprove instruction.Specifically, (cid:2) Minilectures.Thesebriefauthor-createdvideosforeachsectionofthetextgive anoverviewtothesectionbutnoteveryexampleorproof.Somesectionswillhave twovideos.Ihaveusedthesevideoseffectivelywithmystudents,whowere assignedtowatchthemaheadofthelectureonthattopic.Studentscametoclass withabasicoverviewofthetopicoftheday,whichhadtheeffectofreducing lecturetimeandincreasingtheclasstimeusedfordiscussionandstudent vii M00 FRAL0363 08 LLV C00 pageviii viii Instructor’sPreface presentations.Studentsreportedthatthevideoswerehelpfulingivinganoverview ofthetopicsandabetterunderstandingoftheconceptsandproofs.Studentswere alsoencouragedtoviewthevideosafterthetopicwascoveredinclasstoreinforce whattheylearned.Manystudentsalsousedthevideostoreviewtopicswhile preparingforexams.AlthoughIhavenotattemptedtofliptheclassroom,my intentionwastoprovidesufficientresourcesintheeTexttomakeitfeasible withoutrequiringotherresources. (cid:2) Keyideaquizzes.Adatabaseofdefinitionsandnamedtheoremswillallow studentstoquizthemselvesonthesekeyideas.Thedatabasecanbeusedinthe waythatflashcardsweretraditionallyused. (cid:2) Self-assessments.Occasionalquestionsinterspersedinthenarrativeallow studentstochecktheirunderstandingofnewideas. (cid:2) Interactivefiguresandutilities.Ihaveaddedanumberofopportunitiesfor studentstointeractwithcontentinadynamicmannerinordertobuildorenhance understanding.Interactivefiguresallowstudentstoexploreconceptsgeometrically orcomputationallyinwaysthatarenotpossiblewithouttechnology. (cid:2) Notes,Labels,andHighlights.Notesallowinstructorstoaddtheirpersonal teachingstyletoimportanttopics,calloutneed-to-knowinformation,orclarify difficultconcepts.StudentscanmaketheireTexttheirownbycreatinghighlights withmeaningfullabelsandnotes,helpingthemfocusonwhattheyneedtostudy. ThecustomizableNotebookallowsstudentstofilter,arrange,andgrouptheirnotes inawaythatmakessensetothem. (cid:2) Dashboard.Instructorscancreatereadingassignmentsandseethetimespentin theeTextsothattheycanplanmoreeffectiveinstruction. (cid:2) Portability.PortableaccessletsstudentsreadtheireTextwhenevertheyhavea momentintheirday,onAndroidandiOSmobilephonesandtablets.Evenwithout anInternetconnection,offlinereadingensuresstudentsnevermissachanceto learn. (cid:2) Ease-of-Use.Straightforwardsetupmakesiteasyforinstructorstogettheirclass upandreadingquicklyonthefirstdayofclass.Inaddition,LearningManagement System(LMS)integrationprovidesinstitutions,instructors,andstudentswith singlesign-onaccesstotheeTextviamanypopularLMSs. Exercises Many exercises in the text have been updated, and many are new. In order to prevent studentsfromusingsolutionsfromthepreviousedition,Ipurposefullyreplacedorre- wordedsomeexercises. IcreatedanInstructorSolutionsManual,whichisavailableonlineatwww.pearson.com to instructorsonly.Solutionsto exercisesinvolvingproofsareoftensketchesorhints, whichwouldnotbeintheproperformtoturnin. TextOrganizationModifications Foreachpartofthetext,Iprovideanoverviewofthechangesfollowedbysignificant changestosections.Incaseswherechangestopartsorsectionswereminor,Ihavenot includedalistofchanges. PartI:GroupsandSubgroups (cid:2) Overviewofchanges:Mymaingoalsweretodefinegroupsandtointroducethe symmetricanddihedralgroupsasearlyaspossible.Theearlyintroductionofthese M00 FRAL0363 08 LLV C00 pageix Instructor’sPreface ix groupsprovidesstudentswithexamplesoffinitegroupsthatareconsistentlyused throughoutthebook. (cid:2) Section1(BinaryOperations).FormerSection2.Addeddefinitionofanidentity forabinaryoperation. (cid:2) Section2(Groups).FormerSection4.Includedtheformaldefinitionofagroup isomorphism. (cid:2) Section3(AbelianExamples).FormerSection1.Includeddefinitionofcircle group,R ,andZ .UsedcirclegrouptoshowassociativityofZ andR . a n n a (cid:2) Section4(NonabelianExamples).BasedonpartsofformerSections5,8,and9. Defineddihedralgroupandsymmetricgroup.Gaveastandardizednotationforthe dihedralgroupthatisusedconsistentlythroughoutthebook.Introducedboth two-rowandcyclenotationforthesymmetricgroup (cid:2) Section5(Subgroups).FormerSection5.Includedstatementoftwoother conditionsthatimplyasubsetisasubgroupandkepttheproofsintheexercise section.MademinormodificationsusingexamplesfromnewSection4. (cid:2) Section6(CyclicGroups).FormerSection6.Addedexamplesusingdihedral groupandsymmetricgroup. (cid:2) Section7(GeneratingSetsandCayleyDigraphs).Minormodificationofformer Section7. PartII:StructureofGroups (cid:2) Overviewofchanges:Themaingoalwastogivetheformaldefinitionof homomorphismearlierinordertosimplifytheproofsofCayley’sandLagrange’s theorems. (cid:2) Section8(GroupsofPermutations).Includedformaldefinitionofhomomorphism. BasedonpartsofformerSections8,9,and13.Usedtwo-rowpermutation notationtomotivateCayley’stheorembeforeproof.Deletedfirstpartofsection13 (coveredinSection4).Omitteddeterminantproofofeven/oddpermutationssince definitionofdeterminantusuallyusessignofapermutation.Keptorbitcounting proof.Putdeterminantproofandinversioncountingproofinexercises. (cid:2) Section9(FinitelyGeneratedAbelianGroups).FormerSection11.Addedthe invariantfactorversionofthetheorem.Showedhowtogobackandforthbetween thetwoversionsofthefundamentaltheorem. (cid:2) Section10(CosetsandtheTheoremofLagrange).FormerSection10.Changed theorderbyputtingLagrange’sTheoremfirst,motivatingG/Hlaterinthesection. (cid:2) Section11(PlaneIsometries).MinormodificationofformerSection12. PartIII:HomomorphismsandFactorGroups (cid:2) Overviewofchanges:Mymaingoalwastoincludeafewmoreexamplesto motivatethetheoryandgiveanintroductiontousinggroupactionstoprove propertiesofgroups. (cid:2) Sections12-15arebasedonformerSections14-17,respectively. (cid:2) Section12(FactorGroups).StartedsectionwithZ/nZexampletomotivate generalconstruction.Definedfactorgroupsfromnormalsubgroupsfirstinsteadof fromhomomorphisms.Afterdevelopingfactorgroups,showedhowtheyare formedfromhomomorphisms. (cid:2) Section13(Factor-GroupComputationsandSimpleGroups).Addedafewmore examplesofcomputingfactorgroups.Explicitlyusedthefundamental homomorphismtheoremincomputationexamples.