A first course in abstract algebra

(*) Not required for the remainder of the text. (**) This section is required only for Chapters 17 and 36.). 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. Notations. Index.