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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