Abstract Algebra [Paperback]

The book preserved the emphasis on providing a large number of examples and on helping students learn how to write proofs. The presentation of the sections given at a higher level. Unusual features, for a book is still relatively short, are the inclusion of full proofs of both directions of Gauss theorem on constructible regular polygons and galois theorem on solvability by radicals, a Galoistheoretic proof of the Fundamental Theorem of Algebra, and a proof the Primitive  Element Theorem.

First semester course should probably include the material of Sections 0-13, and some of the material on rings in Section 16 and the following sections, Sections 14 and 15 allow the inclusion of some deeper result on groups. In second semester it should be possible to cover the whole book, possibly omitting Section 21.

The changes include the simplification of some points in the edition of some new exercise, and the updating of some historical material. All the topics are given in step by step method with simple language to understand the concept easily. This book is intended for use in a junior-senior level course in abstract algebra. The students who used the book for the five sections as text and pointed out to me parts of the presentation that needed clarification.

1. Sets and Induction
2. Binary Operations
3. Groups
4. Fundamental Theorems about Groups
5. Powers of a Elements; Cosets
6. Counting the elements of a Finite Group
7. Normal Subgroups
8. Homomorphisms
9. Homomorphisms and Normal Subgroups
10. Direct Products and Finite Abelian Groups
11. Sylow Theorems
12. Rings
13. Subrings, Ideals, and Quotient Rings
14. Ring Homomorphisms
15. Polynomials
16. From Polynomials to Fields
17. Unique Factorization Domains
18. Extensions of Fields
19. Constructions with Straightedge and Compass
20. Normal and Separable Extensions
21. Galois Theory
22. Solvbability
23. Suggestions for Further Reading
24. Answers to Selected Exercises
25. Index