Contemporary abstract algebra - Gallian, Joseph A. (Boston, MA Cengage Learning, 2017)
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 Db5s -- 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 Ab5s -- 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 Rp3s -- 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 Pâolya -- 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.
WCN: 01-100-101--Title page verso.
Includes bibliographical references and indexes.
Keywords: Algebra, Abstract -- Textbooks