First Course in Abstract Algebra

Title Preface Contents I. Basic Concepts 1. Sets and elements 2. Mappings and operations 3. Relations II. The Real Number System 4. The natural numbers 5. The integers 6. The rational numbers 7. The real numbers III. Integral Domains and Fields 8. Integral domains 9. The integers modulo k 10. The characteristic 11. Fields 12. The quotient field of an integral domain IV. Polynomial Domains 13. The polynomial domain of an integral domain 14. The polynomial domain of a field V. The Complex Number Field 15. The field of polynomials modulo a given prime 16. The complex number field 17. The polynomial domain C[x] VI. Groups 18. Permutation groups 19. Abstract groups 20. The symmetric group Pn 21. Cosets of a subgroup in a group 22. Invariant subgroups and quotient groups VII. Vector Spaces 23. Definition and elementary properties of a vector space 24. Linear independence in a vector space 25. Bases of vector spaces 26. Algebra of subspaces of a vector space VIII. Linear Transformations and Matrices 27. Elementary properties of linear transformations 28. The algebra of linear transformations 29. The algebra of matrices 30. Further properties of matrices IX. Linear Equations and Determinants 31. Systems of linear equations 32. Determinants 33. Further properties of determinants 34. On solving a system of linear equations X. Other Algebraic Systems 35. Linear algebras 36. Rings 37. Boolean algebras Index