本書(shū)是一部深入介紹抽象代數(shù)的入門書(shū)籍，被許多讀者奉為經(jīng)典。本書(shū)假定讀者了解了微積分和線性代數(shù)，旨在讓讀者盡可能多的了解群、環(huán)、以及域理論的有關(guān)知識(shí)。本書(shū)特色之一是基礎(chǔ)部分內(nèi)容詳實(shí)，講解扎實(shí)，可以為讀者打下良好的基礎(chǔ)，對(duì)于讀者更進(jìn)一步的學(xué)習(xí)代數(shù)大有助益。為了滿足更多讀者的要求，本書(shū)還包含了很多有關(guān)拓?fù)渲械耐{(diào)群和同調(diào)群的計(jì)算以加深對(duì)因子群的理解。書(shū)中內(nèi)容淺顯易懂，理論闡述清晰，條理分明，且大都以例子和練習(xí)的形式，便于直觀了解。為了將同調(diào)群講述的更加清楚，本書(shū)在第6版的基礎(chǔ)上減少了自動(dòng)機(jī)、二進(jìn)制線性。
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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