A First Course in Abstract Algebra

Description

Title Contents Instructor's Preface Student’s Preface 0. Sets and Relations I. Groups and Subgroups 1. Binary Operations 2. Groups 3. Abelian Examples 4. Nonabelian Examples 5. Subgroups 6. Cyclic Groups 7. Generating Sets and Cayley Digraphs II. Structure of Groups 8. Groups of Permutations 9. Finitely Generated Abelian Groups 10. Cosets and the Theorem of Lagrange 11. Plane Isometries III. Homomorphisms and Factor Groups 12. Factor Groups 13. Factor-Group Computations and Simple Groups 14. Group Action on a Set 15. Applications of G-Sets to Counting IV. Advanced Group Theory 16. Isomorphism Theorems 17. Sylow Theorems 18. Series of Groups 19. Free Abelian Groups 20. Free Groups 21. Group Presentations V. Rings and Fields 22. Rings and Fields 23. Integral Domains 24. Fermat’s and Euler’s Theorems 25. Encryption VI. Constructing Rings and Fields 26. The Field of Quotients of an Integral Domain 27. Rings of Polynomials 28. Factorization of Polynomials over a Field 29. Algebraic Coding Theory 30. Homomorphisms and Factor Rings 31. Prime and Maximal Ideals 32. Noncommutative Examples VII. Commutative Algebra 33. Vector Spaces 34. Unique Factorization Domains 35. Euclidean Domains 36. Number Theory 37. Algebraic Geometry 38. Gröbner Bases for Ideals VIII. Extension Fields 39. Introduction to Extension Fields 40. Algebraic Extensions 41. Geometric Constructions 42. Finite Fields IX. Galois Theory 43. Introduction to Galois Theory 44. Splitting Fields 45. Separable Extensions 46. Galois Theory 47. Illustrations of Galois Theory 48. Cyclotomic Extensions 49. Insolvability of the Quintic Appendix: Matrix Algebra Bibliography Notations Answers to Odd-Numbered Exercises Not Asking for Definitions or Proofs Index