Description

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.

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.

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.