Table Of Content

AMS / MAA TEXTBOOKS VOL 65 Thinking Algebraically VOL An Introduction to Abstract Algebra AMS / MAA TEXTBOOKS 65 Thomas Q. Sibley Thinking Algebraically presents the insights of abstract algebra in a welcoming and accessible way. It succeeds in combining the advantages of rings-fi rst and groups-fi rst approaches while avoiding the disadvantages. After an historical overview, the fi rst chapter studies familiar examples and elementary properties of groups and rings simultaneously to motivate the modern understanding of algebra. The text builds intuition for abstract algebra starting from high school algebra. In addition to the standard number systems, polynomials, vectors, and matrices, the fi rst chapter introduces modular arithmetic and dihedral groups. The second chapter builds on these basic examples and properties, enabling students to T h learn structural ideas common to rings and groups: isomorphism, homomorphism, and direct i product. The third chapter investigates introductory group theory. Later chapters delve more n k deeply into groups, rings, and fi elds, including Galois theory, and they also introduce other T i h n topics, such as lattices. The exposition is clear and conversational throughout. o m g The book has numerous exercises in each section as well as supplemental exercises and a A s projects for each chapter. Many examples and well over 100 fi gures provide support for Q lg learning. Short biographies introduce the mathematicians who proved many of the results. . S e The book presents a pathway to algebraic thinking in a semester- or year-long algebra i b b l r course. e y a i c a l l y For additional information and updates on this book, visit www.ams.org/bookpages/text-65 A M S / M A A P TEXT/65 R E S S 4-Color Process 496 pages on 50lb stock • Softcover • Spine 1" Thinking Algebraically An Introduction to Abstract Algebra AMS/MAA TEXTBOOKS VOL 65 Thinking Algebraically An Introduction to Abstract Algebra Thomas Q. Sibley MAATextbooksEditorialBoard StanleyE.Seltzer,Editor MatthiasBeck SuzanneLynneLarson JeffreyL.Stuart DebraSusanCarney MichaelJ.McAsey RonD.Taylor,Jr. HeatherAnnDye VirginiaA.Noonburg ElizabethThoren WilliamRobertGreen ThomasC.Ratliff RuthVanderpool 2020MathematicsSubjectClassification.Primary20-XX,16-XX,12-XX,06-XX. CoverphotographusedwithpermissionbyToddRosso©2020. Figure6.25iscourtesyofDouglasDunham. Ken-Kenpuzzles,©MathematicalAssociationofAmerica,2015.Allrightsreserved. Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/text-65 LibraryofCongressCataloging-in-PublicationData Names:Sibley,ThomasQ.,author. Title:Thinkingalgebraically:anintroductiontoabstractalgebra/ThomasQ.Sibley. Description:Providence:AmericanMathematicalSociety,2020.|Series:AMS/MAAtextbooks,2577-1205; volume65.|Includesbibliographicalreferencesandindex. Identifiers:LCCN2020031328|ISBN9781470460303(paperback)|(ebook) Subjects:LCSH:Algebra,Abstract–Textbooks.|AMS:Grouptheoryandgeneralizations.|Associativerings andalgebras.|Fieldtheoryandpolynomials.|Order,lattices,orderedalgebraicstructures. Classification:LCCQA162|DDC512/.02–dc23 LCrecordavailableathttps://lccn.loc.gov/2020031328 Copyingandreprinting. Individualreadersofthispublication,andnonprofitlibrariesactingforthem, arepermittedtomakefairuseofthematerial,suchastocopyselectpagesforuseinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationinreviews,providedthecustomaryac- knowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublicationispermit- tedonlyunderlicensefromtheAmericanMathematicalSociety.Requestsforpermissiontoreuseportions ofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. Formoreinformation,please visitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. ©2021bytheAmericanMathematicalSociety.Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. ⃝1Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 262524232221 Contents Preface ix Topics x Features xii Prologue 1 Exercises 1 1 ATransitiontoAbstractAlgebra 3 1.1 AnHistoricalViewofAlgebra 3 Exercises 8 1.2 BasicAlgebraicSystemsandProperties 14 Exercises 23 1.3 Functions,Symmetries,andModularArithmetic 28 Exercises 37 SupplementalExercises 43 Projects 46 2 RelationshipsbetweenSystems 51 2.1 Isomorphisms 51 Exercises 56 2.2 ElementsandSubsets 60 Exercises 66 2.3 DirectProducts 71 Exercises 76 2.4 Homomorphisms 81 Exercises 88 SupplementalExercises 94 Projects 95 3 Groups 99 3.1 CyclicGroups 99 Exercises 103 3.2 AbelianGroups 108 Exercises 113 3.3 CayleyDigraphs 120 Exercises 124 3.4 GroupActionsandFiniteSymmetryGroups 127 Exercises 134 v vi Contents 3.5 PermutationGroups,PartI 140 Exercises 145 3.6 NormalSubgroupsandFactorGroups 150 Exercises 158 3.7 PermutationGroups,PartII 162 Exercises 165 SupplementalExercises 169 Projects 172 Appendix: TheFundamentalTheoremofFiniteAbelianGroups 177 4 Rings,IntegralDomains,andFields 181 4.1 RingsandIntegralDomains 181 Exercises 187 4.2 IdealsandFactorRings 190 Exercises 194 4.3 PrimeandMaximalIdeals 197 Exercises 203 4.4 PropertiesofIntegralDomains 207 Exercises 215 4.5 GröbnerBasesinAlgebraicGeometry 219 Exercises 225 4.6 PolynomialDynamicalSystems 228 Exercises 232 SupplementalExercises 234 Projects 236 5 VectorSpacesandFieldExtensions 243 5.1 VectorSpaces 244 Exercises 250 5.2 LinearCodesandCryptography 255 Exercises 261 5.3 AlgebraicExtensions 266 Exercises 273 5.4 GeometricConstructions 277 Exercises 286 5.5 SplittingFields 290 Exercises 297 5.6 AutomorphismsofFields 302 Exercises 308 5.7 GaloisTheoryandtheInsolvabilityoftheQuintic 312 Exercises 319 SupplementalExercises 322 Projects 325 6 TopicsinGroupTheory 327 6.1 FiniteSymmetryGroups 327 Exercises 335 6.2 Frieze,Wallpaper,andCrystalPatterns 341 Contents vii Exercises 351 6.3 MatrixGroups 356 Exercises 364 6.4 SemidirectProductsofGroups 370 Exercises 376 6.5 TheSylowTheorems 381 Exercises 388 SupplementalExercises 391 Projects 395 7 TopicsinAlgebra 399 7.1 LatticesandPartialOrders 399 Exercises 405 7.2 BooleanAlgebras 408 Exercises 414 7.3 Semigroups 417 Exercises 422 7.4 UniversalAlgebraandPreservationTheorems 426 Exercises 431 SupplementalExercises 433 Projects 435 Epilogue 439 SelectedAnswers 443 Terms 469 Symbols 475 Names 477