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Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary Abstract Algebra, 6/e, includes challenging topics in abstract algebra as well as numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings that give the subject a current feel and makes the content interesting and relevant for students.
Contents
NoteI. Integers and Equivalence RelationsPreliminariesProperties of IntegersModular Arithmetic Mathematical InductionEquivalence RelationsFunctions (Mappings)Computer ExercisesII. Groups1. Introduction to GroupsSymmetries of a SquareThe Dihedral Groups Biography of Neils Abel2. GroupsDefinition and Examples of GroupsElementary Properties of GroupsHistorical NoteComputer Exercises3. Finite Groups; SubgroupsTerminology and NotationSubgroup TestsExamples of SubgroupsComputer Exercises4. Cyclic GroupsProperties of Cyclic GroupsClassification of Subgroups of Cyclic GroupsComputer ExercisesBiography of J. J. SylvesterSupplementary Exercises for Chapters 1-45. Permutation GroupsDefinition and NotationCycle NotationProperties of Permutations A Check-Digit Scheme Based on D5Computer ExercisesBiography of Augustin Cauchy6. IsomorphismsMotivationDefinition and ExamplesCayley's Theorem Properties of IsomorphismsAutomorphisms Biography of Arthur Cayley7. Cosets and Lagrange's TheoremProperties of Cosets Lagrange's Theorem and Consequences An Application of Cosets to Permutation GroupsThe Rotation Group of a Cube and a Soccer Ball Computer ExercisesBiography of Joseph Lagrange8. External Direct ProductsDefinition and ExamplesProperties of External Direct ProductsThe Group of Units Modulo n as an External Direct ProductApplicationsComputer ExercisesBiography of Leonard Adleman Supplementary Exercises for Chapters 5-89. Normal Subgroups and Factor GroupsNormal Subgroups Factor GroupsApplications of Factor GroupsInternal Direct ProductsBiography of Evariste Galois 10. Group HomomorphismsDefinition and ExamplesProperties of HomomorphismsThe First Isomorphism Theorem Computer ExercisesBiography of Camille Jordan11. Fundamental Theorem of Finite Abelian GroupsThe Fundamental TheoremThe Isomorphism Classes of Abelian Groups Proof of the Fundamental TheoremComputer ExercisesSupplementary Exercises for Chapters 9-11III. Rings12. Introduction to RingsMotivation and DefinitionExamples of Rings Properties of RingsSubringsComputer ExercisesBiography of I. N. Herstein13. Integral DomainsDefinition and Examples Fields Characteristic of a RingComputer ExercisesBiography of Nathan Jacobson14. Ideals and Factor RingsIdeals Factor Rings Prime Ideals and Maximal Ideals Biography of Richard DedekindBiography of Emmy Noether Supplementary Exercises for Chapters 12-1415. Ring HomomorphismsDefinition and ExamplesProperties of Ring HomomorphismsThe Field of Quotients 16. Polynomial RingsNotation and Terminology The Division Algorithm and ConsequencesBiography of Saunders Mac Lane17. Factorization of PolynomialsReducibility Tests Irreducibility TestsUnique Factorization in Z [x]Weird Dice: An Application of Unique FactorizationComputer Exercises18. Divisibility in Integral DomainsIrreducibles, PrimesHistorical Discussion of Fermat's Last Theorem Unique Factorization DomainsEuclidean DomainsBiography of Sophie GermainBiography of Andrew WilesSupplementary Exercises for Chapters 15-18IV. Fields19. Vector SpacesDefinition and ExamplesSubspaces Linear IndependenceBiography of Emil ArtinBiography of Olga Taussky-Todd20. Extension FieldsThe Fundamental Theorem of Field TheorySplitting Fields Zeros of an Irreducible Polynomial Biography of Leopold Kronecker21. Algebraic ExtensionsCharacterization of ExtensionsFinite Extensions Properties of Algebraic ExtensionsBiography of Irving Kaplansky22. Finite FieldsClassification of Finite FieldsStructure of Finite Fields Subfields of a Finite FieldComputer ExercisesBiography of L. E. Dickson23. Geometric ConstructionsHistorical Discussion of Geometric ConstructionsConstructible NumbersAngle-Trisectors and Circle-SquarersSupplementary Exercises for Chapters 19-23V. Special Topics24. Sylow TheoremsConjugacy ClassesThe Class EquationThe Probability That Two Elements CommuteThe Sylow Theorems Applications of Sylow TheoremsBiography of Ludvig Sylow25. Finite Simple GroupsHistorical Background Nonsimplicity TestsThe Simplicity of A5The Fields MedalThe Cole PrizeComputer ExercisesBiography of Michael AschbacherBiography of Daniel GorensteinBiography of John Thompson26. Generators and RelationsMotivationDefinitions and NotationFree GroupGenerators and Relations Classification of Groups of Order up to 15Characterization of Dihedral GroupsRealizing the Dihedral Groups with MirrorsBiography of Marshall Hall, Jr. 27. Symmetry GroupsIsometriesClassification of Finite Plane Symmetry GroupsClassification of Finite Group of Rotations in R328. Frieze Groups and Crystallographic GroupsThe Frieze GroupsThe Crystallographic Groups Identification of Plane Periodic Patterns Biography of M. C. EscherBiography of George PolyaBiography of John H. Conway29. Symmetry and CountingMotivationBurnside's TheoremApplications Group ActionBiography of William Burnside 30. Cayley Digraphs of GroupsMotivationThe Cayley Digraph of a GroupHamiltonian Circuits and Paths Some Applications Biography of William Rowan HamiltonBiography of Paul Erdoes 31. Introduction to Algebraic Coding TheoryMotivationLinear CodesParity-Check Matrix DecodingCoset DecodingHistorical Note: Reed-Solomon Codes Biography of Richard W. HammingBiography of Jessie MacWilliams Biography of Vera Pless32. An Introduction to Galois TheoryFundamental Theorem of Galois TheorySolvability of Polynomials by RadicalsInsolvability of a QuinticBiography of Philip Hall33. Cyclotomic Extensions MotivationCyclotomic PolynomialsThe Constructible Regular n-gonsComputer ExerciseBiography of Carl Friedrich GaussBiography of Manjul BhargavaSupplementary Exercises for Chapters 24-33
