Description
Book Synopsis Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialised work by emphasising an understanding of the nature of algebraic structures.
Table of Contents
- 0. Sets and Relations.
- I. GROUPS AND SUBGROUPS.
- 1. Introduction and Examples.
- 2. Binary Operations.
- 3. Isomorphic Binary Structures.
- 4. Groups.
- 5. Subgroups.
- 6. Cyclic Groups.
- 7. Generators and Cayley Digraphs.
- I. PERMUTATIONS, COSETS, AND DIRECT PRODUCTS.
- 8. Groups of Permutations.
- 9. Orbits, Cycles, and the Alternating Groups.
- 10. Cosets and the Theorem of Lagrange.
- 11. Direct Products and Finitely Generated Abelian Groups.
- 12. Plane Isometries.
- III. HOMOMORPHISMS AND FACTOR GROUPS.
- 13. Homomorphisms.
- 14. Factor Groups.
- 15. Factor-Group Computations and Simple Groups.
- 16. Group Action on a Set.
- 17. Applications of G-Sets to Counting.
- IV. RINGS AND FIELDS.
- 18. Rings and Fields.
- 19. Integral Domains.
- 20. Fermat's and Euler's Theorems.
- 21. The Field of Quotients of an Integral Domain.
- 22. Rings of Polynomials.
- 23. Factorization of Polynomials over a Field.
- 24. Noncommutative Examples.
- 25. Ordered Rings and Fields.
- V. IDEALS AND FACTOR RINGS.
- 26. Homomorphisms and Factor Rings.
- 27. Prime and Maximal Ideas.
- 28. Gröbner Bases for Ideals.
- VI. EXTENSION FIELDS.
- 29. Introduction to Extension Fields.
- 30. Vector Spaces.
- 31. Algebraic Extensions.
- 32. Geometric Constructions.
- 33. Finite Fields.
- VII. ADVANCED GROUP THEORY.
- 34. Isomorphism Theorems.
- 35. Series of Groups.
- 36. Sylow Theorems.
- 37. Applications of the Sylow Theory.
- 38. Free Abelian Groups.
- 39. Free Groups.
- 40. Group Presentations.
- VIII.. AUTOMORPHISMS AND GALOIS THEORY.
- 41. Automorphisms of Fields.
- 42. The Isomorphism Extension Theorem.
- 43. Splitting Fields.
- 44. Separable Extensions.
- 45. Totally Inseparable Extensions.
- 46. Galois Theory.
- 47. Illustrations of Galois Theory.
- 48. Cyclotomic Extensions.
- 49. Insolvability of the Quintic.
- Appendix: Matrix Algebra.
- Notations.
- Index.
