An Introduction to Essential Algebraic Structures PDF
Preview An Introduction to Essential Algebraic Structures
(cid:2) (cid:2) “Dixon-Driver” — 2014/9/18 — 19:30 — page ii — #2 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/18 — 19:30 — page i — #1 (cid:2) (cid:2) AN INTRODUCTION TO ESSENTIAL ALGEBRAIC STRUCTURES (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/18 — 19:30 — page ii — #2 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/18 — 19:30 — page iii — #3 (cid:2) (cid:2) AN INTRODUCTION TO ESSENTIAL ALGEBRAIC STRUCTURES MARTYNR.DIXON DepartmentofMathematics TheUniversityofAlabama Tuscaloosa,AL,USA LEONIDA.KURDACHENKO DepartmentofAlgebra NationalUniversityofDnepropetrovsk Dnepropetrovsk,Ukraine IGORYA.SUBBOTIN DepartmentofMathematicsandNaturalSciences NationalUniversity LosAngeles,CA,USA (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/18 — 19:30 — page iv — #4 (cid:2) (cid:2) Copyright©2015byJohnWiley&Sons,Inc.Allrightsreserved PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey PublishedsimultaneouslyinCanada Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400,fax (978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshould beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ 07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permissions. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbesuitable foryoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthepublishernor authorshallbeliableforanylossofprofitoranyothercommercialdamages,includingbutnotlimitedto special,incidental,consequential,orotherdamages. Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactour CustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat (317)572-3993orfax(317)572-4002. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteat www.wiley.com. LibraryofCongressCataloging-in-PublicationData: Dixon,MartynR.(MartynRussell),1955-author. Anintroductiontoessentialalgebraicstructures/MartynR.Dixon,DepartmentofMathematics,The UniversityofAlabama,Tuscaloosa,AL,LeonidA.Kurdachenko,DepartmentofAlgebra,National UniversityofDnepropetrovsk,Dnepropetrovsk,Ukraine,IgorYa.Subbotin,DepartmentofMathematics andNaturalSciences,NationalUniversity,LosAngeles,CA. pages cm Includesbibliographicalreferencesandindex. ISBN978-1-118-45982-9(cloth) 1. Orderedalgebraicstructures.I.Kurdachenko,LeonidA.,1949-author. II. Subbotin,IgorYa., 1950-author. III. Title. QA172.D592015 (cid:2) 511.33–dc23 2014022297 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/18 — 19:30 — page v — #5 (cid:2) (cid:2) CONTENTS Preface vii 1 Sets 1 1.1 OperationsonSets, 1 ExerciseSet1.1, 7 1.2 SetMappings, 9 ExerciseSet1.2, 15 1.3 ProductsofMappingsandPermutations, 16 ExerciseSet1.3, 26 1.4 OperationsonMatrices, 28 ExerciseSet1.4, 35 1.5 BinaryAlgebraicOperationsandEquivalenceRelations, 37 ExerciseSet1.5, 47 2 Numbers 51 2.1 SomePropertiesofIntegers:MathematicalInduction, 51 ExerciseSet2.1, 55 2.2 Divisibility, 56 ExerciseSet2.2, 63 2.3 PrimeFactorization:TheFundamentalTheorem ofArithmetic, 64 ExerciseSet2.3, 67 v (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/18 — 19:30 — page vi — #6 (cid:2) (cid:2) vi CONTENTS 2.4 RationalNumbers,IrrationalNumbers, andRealNumbers, 68 ExerciseSet2.4, 76 3 Groups 79 3.1 GroupsandSubgroups, 79 ExerciseSet3.1, 93 3.2 CosetsandNormalSubgroups, 94 ExerciseSet3.2, 106 3.3 FactorGroupsandHomomorphisms, 108 ExerciseSet3.3, 116 4 Rings 119 4.1 Rings,Subrings,AssociativeRings, 119 ExerciseSet4.1, 131 4.2 RingsofPolynomials, 133 ExerciseSet4.2, 142 4.3 IdealsandQuotientRings, 143 ExerciseSet4.3, 153 4.4 HomomorphismsofRings, 155 ExerciseSet4.4, 165 5 Fields 169 5.1 Fields:BasicPropertiesandExamples, 169 ExerciseSet5.1, 180 5.2 SomeFieldExtensions, 182 ExerciseSet5.2, 187 5.3 FieldsofAlgebraicNumbers, 187 ExerciseSet5.3, 196 HintsandAnswerstoSelectedExercises 199 Chapter1, 199 Chapter2, 205 Chapter3, 210 Chapter4, 214 Chapter5, 222 Index 225 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/25 — 13:14 — page vii — #7 (cid:2) (cid:2) PREFACE Abstract algebra is an essential part of a mathematics program at any uni- versity. It would not be an exaggeration to say that this area is one of the mostchallengingandsophisticatedpartsofsuchaprogram.Itrequiresbegin- ners to establish and develop a totally different way of thinking from their previous mathematical experience. Actually, to students, this is a new lan- guagethathasprovedtobeveryeffectiveintheinvestigationanddescription of the most important natural and mathematical laws. The transition from the well-understood ideas of Calculus, aided by its many visual examples, to the abstraction of algebra, less supported by intuition, is perhaps one of the major obstacles that students of mathematics need to overcome. Under these circumstances, it is imperative for students to have a reader-friendly introductory textbook consisting of clearly and carefully explained theoret- ical topics that are essential for algebra and accompanied with thoughtfully selectedexamplesandexercises. Abstract algebra was, until fairly recently, studied for its own sake and becauseithelpedsolvearangeofmathematicalquestionsofinteresttomath- ematicians. It was the province of pure mathematicians. However, much of theriseofinformationtechnologyandtheaccompanyingneedforcomputer securityhasitsbasisinabstractalgebra,whichinturnhasignitedinterestin thisarea.Abstractalgebraisalsoofinteresttophysicists,chemists,andother scientists. There are even applications of abstract algebra to music theory. Additionally,manyfuturehighschoolteachersnowneedtohavesomefamil- iaritywithhigherlevelmathematics.Thereisthereforeaneedforagrowing vii (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “Dixon-Driver” — 2014/9/25 — 13:14 — page viii — #8 (cid:2) (cid:2) viii PREFACE body of undergraduate students to have some knowledge of this beautiful subject. We have tried to write a book that is appropriate for typical students in computerscience,mathematics,mathematicseducation,andotherdisciplines. Such students should already possess a certain degree of general mathemat- ical knowledge pertaining to typical average students at this stage. Ideally, suchstudentsshouldalreadyhavehadamathematicscoursewheretheyhave themselveswrittensomeproofsandalsoworkedwithmatrices.However,the main idea of our book is that it should be as user-friendly to a beginner as possible,andforthisreason,wehaveincludedmaterialaboutmatrices,math- ematical induction, functions, and other such topics. We expect the book to beofinterestnotonlytomathematicsmajors,butalsotoanyonewhowould liketolearnthebasictopicsofmodernalgebra.Undergraduatestudentswho need to take an introductory abstract algebra course will find this book very handy.Wehavemadeeveryefforttomakethebookassimple,understandable, andconciseaspossible,whileleavingroomforrigorousmathematicalproofs. We illustrate the theory with a variety of examples that appeal to the previ- ousexperienceofreaders,whichisusefulinthedevelopmentofanintuitive algebraic way of thinking. We cover only essential topics from the algebra curriculumtypicalforintroductoryabstractalgebracoursesinAmericanuni- versities. Through some of the numerous exercises, we introduce readers to morecomplextopics. The book consists of five chapters. We start our exposition with the ele- mentsofsettheory,functions,andmatrixtheory.InChapter2,wecoverthe mainpropertiesoftheintegers,viewedfromanalgebraicpointofview.This pavesthewayforthefinalthreechapters,coveringGroupsinChapter3,Rings inChapter4,andFieldsinChapter5.Thesechapterscoverthemainbeginning ideasofabstractalgebraaswellassophisticatedideas.Thebookisaccompa- niedbyanInstructor’ssolutionsmanualcontainingsolutionsforallexercises inthebook. TheauthorswouldliketoextendtheirsincereappreciationtotheUniversity of Alabama (Tuscaloosa, USA), National Dnepropetrovsk University (Dne- propetrovsk,Ukraine),andNationalUniversity(LosAngeles,USA)fortheir great support of the authors’ work. The authors also would like thank their familymembersfortheirpatience,understanding,andmuchneededsupport whilethisworkwasinprogress. MartynR.Dixon LeonidA.Kurdachenko IgorYa.Subbotin (cid:2) (cid:2) (cid:2) (cid:2)
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