Thinking Algebraically: An Introduction to Abstract Algebra

496 Pages·2021·18.75 MB·English

by  Thomas Q. Sibley

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
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deeply into groups, rings, and fi elds, including Galois theory, and they also introduce other T
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topics, such as lattices. The exposition is clear and conversational throughout. o
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The book has numerous exercises in each section as well as supplemental exercises and a A
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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
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and updates on this book, visit
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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
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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