Abstract Algebra by Joseph A. Gallian

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Abstract Algebra written by Joseph A. Gallian, University of Minnesota Duluth. This book include 120 new exercises, new theorems and examples, and a freshening of the quotations and biographies. I have also expanded the supplemental material for abstract algebra available at my website. Extensive coverage of groups, rings, and fields, plus a variety of non-traditional special topics.

Abstract Algebra written by Joseph A. Gallian cover the following topics.

PART 1 Integers and Equivalence Relations

0. Preliminaries
Properties of Integers, Modular Arithmetic, Complex Numbers, Mathematical Induction, Equivalence Relations, Functions (Mappings)

PART 2 Groups

1. Introduction to Groups
Symmetries of a Square, The Dihedral Groups, Exercises, Biography of Niels Abel

2. Groups
Definition and Examples of Groups, Elementary, Properties of Groups, Historical Note, Exercises

3. Finite Groups; Subgroups
Terminology and Notation, Subgroup Tests, Examples of Subgroups, Exercises

4. Cyclic Groups
Properties of Cyclic Groups, Classification of Subgroups of Cyclic Groups, Exercises, Biography of James Joseph Sylvester, Supplementary Exercises for Chapters 1–4

5. Permutation Groups
Definition and Notation, Cycle Notation, Properties of Permutations, A Check-Digit Scheme Based on D5, Exercises, Biography of Augustin Cauchy

6. Isomorphisms
Motivation, Definition and Examples, Cayley’s Theorem, Properties of Isomorphisms, Automorphisms, Exercises, Biography of Arthur Cayley

7. Cosets and Lagrange’s Theorem
Properties of Cosets, Lagrange’s Theorem and Consequences, An Application of Cosets to Permutation Groups, The Rotation Group of a Cube and a Soccer Ball, An Application of Cosets to the Rubik’s Cube, Exercises, Biography of Joseph Lagrange

8. External Direct Products
Definition and Examples, Properties of External Direct Products, The Group of Units Modulo n as an External Direct Product, Applications, Exercises, Biography of Leonard Adleman, Supplementary Exercises for Chapters 5–8

9. Normal Subgroups and Factor Groups
Normal Subgroups, Factor Groups, Applications of Factor Groups, Internal Direct Products, Exercises, Biography of Évariste Galois

10. Group Homomorphisms
Definition and Examples, Properties of Homomorphisms, The First Isomorphism Theorem, Exercises, Biography of Camille Jordan

11. Fundamental Theorem of Finite Abelian Groups
The Fundamental Theorem, The Isomorphism Classes of Abelian Groups, Proof of the Fundamental Theorem, Exercises, Supplementary Exercises for Chapters 9–11

PART 3 Rings

12. Introduction to Rings
Motivation and Definition, Examples of Rings, Properties of Rings, Subrings, Exercises, Biography of I. N. Herstein

13. Integral Domains
Definition and Examples, Fields, Characteristic of a Ring, Exercises, Biography of Nathan Jacobson

14. Ideals and Factor Rings
Ideals, Factor Rings, Prime Ideals and Maximal Ideals, Exercises, Biography of Richard Dedekind, Biography of Emmy Noether, Supplementary Exercises for Chapters 12–14

15. Ring Homomorphisms
Definition and Examples, Properties of Ring Homomorphisms, The Field of Quotients, Exercises

16. Polynomial Rings
Notation and Terminology, The Division Algorithm and Consequences, Exercises, Biography of Saunders Mac Lane

17. Factorization of Polynomials
Reducibility Tests Irreducibility Tests Unique Factorization in Z[x], Weird Dice: An Application of Unique Factorization, Exercises, Biography of Serge Lang

18. Divisibility in Integral Domains
Irreducibles, Primes, Historical Discussion of Fermat’s Last Theorem, Unique Factorization Domains, Euclidean Domains, Exercises, Biography of Sophie Germain, Biography of Andrew Wiles, Supplementary Exercises for Chapters 15–18

PART 4 Fields

19. Vector Spaces
Definition and Examples, Subspaces, Linear Independence, Exercises, Biography of Emil Artin, Biography of Olga Taussky-Todd

20. Extension Fields
The Fundamental Theorem of Field Theory, Splitting Fields, Zeros of an Irreducible Polynomial, Exercises, Biography of Leopold Kronecker

21. Algebraic Extensions
Characterization of Extensions, Finite Extensions, Properties of Algebraic Extensions, Exercises, Biography of Irving Kaplansky

22. Finite Fields
Classification of Finite Fields, Structure of Finite Fields, Subfields of a Finite Field, Exercises, Biography of L. E. Dickson

23. Geometric Constructions
Historical Discussion of Geometric Constructions, Constructible Numbers, Angle-Trisectors and Circle-Squarers, Exercises, Supplementary Exercises for Chapters 19–23

PART 5 Special Topics

24. Sylow Theorems
Conjugacy Classes, The Class Equation, The Probability That Two Elements Commute, The Sylow Theorems, Applications of Sylow Theorems, Exercises, Biography of Ludwig Sylow

25. Finite Simple Groups
Historical Background, Nonsimplicity Tests, The Simplicity of A5, The Fields Medal, The Cole Prize, Exercises, Biography of Michael Aschbacher, Biography of Daniel Gorenstein, Biography of John Thompson

26. Generators and Relations
Motivation, Definitions and Notation, Free Group, Generators and Relations, Classification of Groups of Order Up to 15, Characterization of Dihedral Groups, Realizing the Dihedral Groups with Mirrors, Exercises, Biography of Marshall Hall, Jr.

27. Symmetry Groups
Isometries, Classification of Finite Plane Symmetry Groups, Classification of Finite Groups of Rotations in R3, Exercises

28. Frieze Groups and Crystallographic Groups
The Frieze Groups, The Crystallographic Groups, Identification of Plane Periodic Patterns, Exercises, Biography of M. C. Escher, Biography of George Pólya, Biography of John H. Conway

29. Symmetry and Counting
Motivation, Burnside’s Theorem, Applications, Group Action, Exercises, Biography of William Burnside

30. Cayley Digraphs of Groups
Motivation, The Cayley Digraph of a Group, Hamiltonian Circuits and Paths, Some Applications, Exercises, Biography of William Rowan Hamilton, Biography of Paul Erdó´s

31. Introduction to Algebraic Coding Theory
Motivation, Linear Codes, Parity-Check Matrix Decoding, Coset Decoding, Historical Note: The Ubiquitous Reed–Solomon Codes, Exercises, Biography of Jessie MacWilliams, Biography of Richard W. Hamming, Biography of Jessie MacWilliams, Biography of Vera Pless

32. An Introduction to Galois Theory
Fundamental Theorem of Galois Theory, Solvability of Polynomials by Radicals, Insolvability of a Quintic, Exercises, Biography of Philip Hall

33. Cyclotomic Extensions
Motivation, Cyclotomic Polynomials, The Constructible Regular n-gons, Exercises, Biography of Carl Friedrich Gauss, Biography of Manjul Bhargava, Supplementary Exercises for Chapters 24–33

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