Introduction to Abstract Algebra

This Third Edition of the acclaimed self-study text lets you learn abstract algebra at your own pace The Third Edition of Introduction to Abstract Algebra continues to provide an accessible introduction to the basic structures of abstract algebra: groups, rings, and fields.The text's unique approach helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text. As in the previous editions, the Third Edition offers many special features to help readers learn and apply their developing knowledge of abstract algebra: * Historical notes and biographies of mathematicians provide context and perspective * Some 500 worked examples help readers understand key concepts and their applications * Almost 1,500 computational and theoretical exercises ranging from basic to complex challenge readers to apply their knowledge to solve problems. Many answers are provided in the text * Applications to real-world problems in such areas as coding theory and combinatorics help readers grasp the topic's relevancy * Special topics such as symmetric polynomials, nilpotent groups, semidirect products of groups, and the Wedderburn-Artin theorem for rings are included for more advanced study This Third Edition includes thoroughly revised chapters and much new material, including new sections covering free, semisimple, and projective modules; modules over principal ideal domains; as well as semidirect products and the Wedderburn-Artin theorem. Two new appendices have been added on Zorn's lemma and the proof of the recursion theorem. Numerous worked examples, exercises, and real-world applications make this text perfect for upper-level undergraduate and graduate courses. Moreover, with this book's self-contained chapters, students can learn at their own pace, and instructors can adapt the text to meet a variety of course objectives.

Table of Contents:
Preface. Acknowledgment. Notations Used in the Text. A Sketch of the History of Algebra to 1929. 0. Preliminaries. 0.1 Proofs. 0.2 Sets. 0.3 Mappings. 0.4 Equivalences. 1. Integers and Permutations. 1.1 Induction. 1.2 Divisors and Prime Factorization. 1.3 Integers Modulo n. 1.4 Permutations. 1.5 An Application to Cryptography. 2. Groups. 2.1 Binary Operations. 2.2 Groups. 2.3 Subgroups. 2.4 Cyclic Groups and the Order of an Element. 2.5 Homomorphisms and Isomorphisms. 2.6 Cosets and Lagrange's Theorem. 2.7 Groups of Motions and Symmetries. 2.8 Normal Subgroups. 2.9 Factor Groups. 2.10 The Isomorphism Theorem. 2.11 An Application to Binary Linear Codes. 3. Rings. 3.1 Examples and Basic Properties. 3.2 Integral Domains and Fields. 3.3 Ideals and Factor Rings. 3.4 Homomorphisms. 3.5 Ordered Integral Domains. 4. Polynomials. 4.1 Polynomials. 4.2 Factorization of Polynomials over a Field. 4.3 Factor Rings of Polynomials over a Field. 4.4 Partial Fractions. 4.5 Symmetric Polynomials. 4.6 Formal Construction of Polynomials. 5. Factorization in Integral Domains. 5.1 Irrducibles and Unique Factorization. 5.2 Principal Ideal Domains. 6. Fields. 6.1 Vector Spaces. 6.2 Algebraic Extensions. 6.3 Splitting Fields. 6.4 Finite Fields. 6.5 Geometric Constructions. 6.6 The Fundamental Theorem of Algebra. 6.7 An Application to Cyclic and BCH Codes. 7. Modules over Principal Ideal Domains. 7.1 Modules. 7.2 Modules over a PID. 8. p-Groups and the Sylow Theorems. 8.1 Factors and Products. 8.2 Cauchy 's Theorem. 8.3 Group Actions. 8.4 The Sylow Theorems. 8.5 Semidirect Products. 8.6 An Application to Combinatorics. 9. Series of Subgroups. 9.1 The Jordan-Holder Theorem. 9.2 Solvable Groups. 9.3 Nilpotent Groups. 10. Galois Theory. 10.1 Galois Groups and Separability. 10.2 The Main Theorem of Galois Theory. 10.3 Insolvability of Polynomials. 10.4 Cyclotomic Polynomials and Wedderburn's Theorem. 11. Finiteness Conditions for Rings and Modules. 11.1 Wedderburn's Theorem. 11.2 The Wedderburn-Artin Theorem. Appendices. App.A Complex Numbers. App.B Matrix Arithmetic. App.C Zorn's Lemma. App.D Proof of the Recursion Theorem. Bibliography. Selected Answers. Index.