## Overview

Considered a classic by many, **A First Course in Abstract Algebra, Seventh Edition** is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures.

**KEY TOPICS:** Sets and Relations; GROUPS AND SUBGROUPS; Introduction and Examples; Binary Operations; Isomorphic Binary Structures; Groups; Subgroups; Cyclic Groups; Generators and Cayley Digraphs; PERMUTATIONS, COSETS, AND DIRECT PRODUCTS; Groups of Permutations; Orbits, Cycles, and the Alternating Groups; Cosets and the Theorem of Lagrange; Direct Products and Finitely Generated Abelian Groups; Plane Isometries; HOMOMORPHISMS AND FACTOR GROUPS; Homomorphisms; Factor Groups; Factor-Group Computations and Simple Groups; Group Action on a Set; Applications of G-Sets to Counting; RINGS AND FIELDS; Rings and Fields; Integral Domains; Fermat's and Euler's Theorems; The Field of Quotients of an Integral Domain; Rings of Polynomials; Factorization of Polynomials over a Field; Noncommutative Examples; Ordered Rings and Fields; IDEALS AND FACTOR RINGS; Homomorphisms and Factor Rings; Prime and Maximal Ideas; Gröbner Bases for Ideals; EXTENSION FIELDS; Introduction to Extension Fields; Vector Spaces; Algebraic Extensions; Geometric Constructions; Finite Fields; ADVANCED GROUP THEORY; Isomorphism Theorems; Series of Groups; Sylow Theorems; Applications of the Sylow Theory; Free Abelian Groups; Free Groups; Group Presentations; GROUPS IN TOPOLOGY; Simplicial Complexes and Homology Groups; Computations of Homology Groups; More Homology Computations and Applications; Homological Algebra; Factorization; Unique Factorization Domains; Euclidean Domains; Gaussian Integers and Multiplicative Norms; AUTOMORPHISMS AND GALOIS THEORY; Automorphisms of Fields; The Isomorphism Extension Theorem; Splitting Fields; Separable Extensions; Totally Inseparable Extensions; Galois Theory; Illustrations of Galois Theory; Cyclotomic Extensions; Insolvability of the Quintic; Matrix Algebra

**MARKET:** For all readers interested in abstract algebra.

## Table of contents

**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. *Gröbner 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. *GROUPS IN TOPOLOGY.**

**41. Simplicial Complexes and Homology Groups.**

**42. Computations of Homology Groups.**

**43. More Homology Computations and Applications.**

**44. Homological Algebra.**

**IX. Factorization.**

**45. Unique Factorization Domains.**

**46. Euclidean Domains.**

**47. Gaussian Integers and Multiplicative Norms.**

**X. AUTOMORPHISMS AND GALOIS THEORY.**

**48. Automorphisms of Fields.**

**49. The Isomorphism Extension Theorem.**

**50. Splitting Fields.**

**51. Separable Extensions.**

**52. *Totally Inseparable Extensions.**

**53. Galois Theory.**

**54. Illustrations of Galois Theory.**

**55. Cyclotomic Extensions.**

**56. Insolvability of the Quintic.**

**Appendix: Matrix Algebra.**

**Notations.**

**Answers to odd-numbered exercises not asking for definitions or proofs.**

**Index.**

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