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 specialized work by emphasizing an understanding of the nature of algebraic structures.

Features

• This classical approach to abstract algebra focuses on applications.
• The text is geared toward high-level courses at schools with strong mathematics programs.
• Accessible pedagogy includes historical notes written by Victor Katz, an authority on the history of math.
• By opening with a study of group theory, this text provides students with an easy transition to axiomatic mathematics.

Contents

1. Introduction and Examples.
2. Binary Operations.
3. Isomorphic Binary Structures.
4. Groups.
5. Subgroups.
6. Cyclic Groups.
7. Generators and Cayley Digraphs.
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.
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.
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.
26. Homomorphisms and Factor Rings.
27. Prime and Maximal Ideas.
28. Gröbner Bases for Ideals.
29. Introduction to Extension Fields.
30. Vector Spaces.
31. Algebraic Extensions.
32. Geometric Constructions.
33. Finite Fields.
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.
41. Simplicial Complexes and Homology Groups.
42. Computations of Homology Groups.
43. More Homology Computations and Applications.
44. Homological Algebra.
45. Unique Factorization Domains.
46. Euclidean Domains.
47. Gaussian Integers and Multiplicative Norms.
48. Automorphisms of Fields.
49. The Isomorphism Extension Theorem.
50. Splitting Fields.
51. Separable Extensions.
52. Totally Inseparable Extensions.
53. Galois Theory.
54. Illustrations of Galois Theory.
55. Cyclotomic Extensions.
56. Insolvability of the Quintic.