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.
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.
Table of Contents
(*) Not required for the remainder of the text. (**) This section is required only for Chapters 17 and 36.).
0. Sets and Relations. I. GROUPS AND SUBGROUPS.
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.
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. *GROUPS IN TOPOLOGY.
41. Simplicial Complexes and Homology Groups.
42. Computations of Homology Groups.
43. More Homology Computations and Applications.
44. Homological Algebra.