Description
Taylor & Francis Ltd Numbers And Symmetry 1997 Edition by Bernard L. Johnston, Fred Richman
This textbook presents modern algebra from the ground up using numbers and symmetry. The idea of a ring and of a field are introduced in the context of concrete number systems. Groups arise from considering transformations of simple geometric objects. The analysis of symmetry provides the student with a visual introduction to the central algebraic notion of isomorphism.Designed for a typical one-semester undergraduate course in modern algebra, it provides a gentle introduction to the subject by allowing students to see the ideas at work in accessible examples, rather than plunging them immediately into a sea of formalism. The student is involved at once with interesting algebraic structures, such as the Gaussian integers and the various rings of integers modulo n, and is encouraged to take the time to explore and become familiar with those structures.In terms of classical algebraic structures, the text divides roughly into three parts: Chapter 1. New NumbersA Planeful of Integers, Z[i]Circular Numbers, ZnMore Integers on the Number Line, Z [v2]NotesChapter 2. The Division AlgorithmRational IntegersNormsGaussian NumbersQ (v2)PolynomialsNotesChapter 3. The Euclidean AlgorithmBezout's EquationRelatively Prime NumbersGaussian IntegersNotesChapter 4. UnitsElementary PropertiesBezout's EquationsWilson's TheoremOrders of Elements: Fermat and EulerQuadratic ResiduesZ [v2]NotesChapter 5. PrimesPrime Numbers Gaussian PrimesZ [v2]Unique Factorization into PrimesZnNotesChapter 6. SymmetriesSymmetries of Figures in the PlaneGroupsThe Cycle Structure of a PermutationCyclic GroupsThe Alternating GroupsNotesChapter 7. MatricesSymmetries and CoordinatesTwo-by-Two MatricesThe Ring of Matrices M2(R)UnitsComplex Numbers and QuaternionsNotesChapter 8. GroupsAbstract GroupsSubgroups and CosetsIsomorphismThe Group of Units of a Finite FieldProducts of GroupsThe Euclidean Groups E (1), E (2), and E (3)NotesChapter 9. Wallpaper PatternsOne-Dimensional PatternsPlane LatticesFrieze PatternsSpace GroupsThe 17 Plane GroupsNotesChapter 10. FieldsPolynomials Over a FieldKronecker's Construction of Simple Field ExtensionsFinite FieldsNotesChapter 11. Linear AlgebraVector SpacesMatricesRow Space and Echelon FormInverses and Elementary MatricesDeterminantsNotesChapter 12. Error-Correcting CodesCoding for RedundancyLinear CodesParity-Check MatricesCyclic CodesBCH CodesCDsNotesChapter 13. Appendix: InductionFormulating the n-th StatementThe Domino Theory: IterationFormulating the Induction StatementSquaresTemplatesRecursionNotesChapter 14. Appendix: The Usual RulesRingsNotesIndex