This Book Is Primarily Intended For An Abstract Algebra Course Whose Main Purpose Is To Enable Students To Do Computations And Write Proofs. The Text Stresses The Importance Of Obtaining A Solid Introduction To The Traditional Topics Of Abstract Algebra, While At The Same Time Presenting It As A Contemporary And Very Much An Active Subject Which Is Currently Being Used By Working Physicists, Chemists, And Computer Scientists. Part 1. Integers And Equivalence Relations. Preliminaries. Properties Of Integers -- Modular Arithmetic -- Complex Numbers -- Mathematical Induction -- Equivalence Relations -- Functions (mappings) -- Exercises -- 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 -- 5. Permutation Groups. Definition And Notation -- Cycle Notation -- Properties Of Permutations -- A Check-digit Scheme Based On D₅ -- Exercises -- Biography Of Augustin Cauchy -- Biography Of Alan Turing -- 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 -- 9. Normal Subgroups And Factor Groups. Normal Subgroups -- Factor Groups -- Applications Of Factor Groups -- Internal Direct Products -- Exercises -- Biography Of Evariste 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 -- 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 -- 15. Ring Homomorphisms. Definition And Examples -- Properties Of Ring Homomorphisms -- The Field Of Quotients -- Exercises -- Biography Of Irving Kaplansky -- 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 -- Biography Of Pierre De Fermat -- 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 Ernst Steinitz -- 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 -- Part 5. Special Topics. 24. Sylow Theorems. Conjugacy Classes -- The Class Equation -- The Sylow Theorems -- Applications Of Sylow Theorems -- Exercises -- Biography Of Oslo 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 Erdos -- 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 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. Joseph A. Gallian. Wcn:01-100-101--title Page Verso. Includes Bibliographical References And Indexes.

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