◎图书目录
Preface
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
II. 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. Groebner 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
Bibliography
Notations
Index
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