Contemporary Abstract Algebra by Joseph GallianContemporary Abstract Algebra by Joseph Gallian

Contemporary Abstract Algebra

byJoseph Gallian

Hardcover | January 8, 2009

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Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.
Title:Contemporary Abstract AlgebraFormat:HardcoverDimensions:656 pages, 9.1 × 6.4 × 1.2 inPublished:January 8, 2009Publisher:Brooks ColeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0547165099

ISBN - 13:9780547165097

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Table of Contents

PART I. Integers and Equivalence Relations.Preliminaries. Properties of Integers. Modular Arithmetic. Mathematical Induction. Equivalence Relations. Functions (Mappings). Exercises. Computer Exercises.PART II. Groups.1. Introduction to Groups. Symmetries of a Square. The Dihedral Groups. Exercises. Biography of Neils Abel2. Groups.Definition and Examples of Groups. Elementary Properties of Groups. Historical Note. Exercises. Computer Exercises.3. Finite Groups; Subgroups.Terminology and Notation. Subgroup Tests. Examples of Subgroups. Exercises. Computer Exercises.4. Cyclic Groups.Properties of Cyclic Groups. Classification of Subgroups of Cyclic Groups. Exercises. Computer Exercises. Biography of J. J. 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. Computer Exercises. Biography of Augustin Cauchy.6. Isomorphisms.Motivation. Definition and Examples. Cayley''s Theorem. Properties of Isomorphisms.Automorphisms. Exercises. Computer 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. Exercises.Computer 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. Computer Exercises.Biography of Leonard Adleman. Supplementary Exercises for Chapters 5-89. Normal Subgroups and Factor Groups.Normal Subgroups. Factor Groups. Applications of Factor Groups. Internal Direct Products. Exercises. Biography of Évariste Galois10. Group Homomorphisms.Definition and Examples. Properties of Homomorphisms. The First Isomorphism Theorem. Exercises. Computer 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. Computer Exercises. Supplementary Exercises for Chapters 9-11.PART III. Rings.12. Introduction to Rings.Motivation and Definition. Examples of Rings. Properties of Rings. Subrings. Exercises. Computer Exercises. Biography of I. N. Herstein.13. Integral Domains.Definition and Examples. Fields. Characteristic of a Ring. Exercises. Computer Exercises. Biography of Nathan Jacobson.14. Ideals and Factor Rings.Ideals. Factor Rings. Prime Ideals and Maximal Ideals. Exercises. Computer 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. Computer 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. Computer Exercises.Biography of Sophie Germain. Biography of Andrew Wiles. Supplementary Exercises for Chapters 15-18.PART IV. 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 ExtensionsExercises. Biography of Irving Kaplansky.22. Fin