Contemporary abstract algebra
著者
書誌事項
Contemporary abstract algebra
Cengage Learning, c2017
9th ed
大学図書館所蔵 全3件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
Includes bibliographical references and index
内容説明・目次
内容説明
CONTEMPORARY ABSTRACT ALGEBRA, NINTH EDITION is primarily intended for an abstract algebra course whose main purpose is to enable students to do computations and write proofs. Gallian s 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.
目次
- PART I: INTEGERS AND EQUIVALENCE RELATIONS. Preliminaries. Properties of Integers. Complex Numbers. Modular Arithmetic. Mathematical Induction. Equivalence Relations. Functions (Mappings). Exercises. PART I!: GROUPS. 1. Introduction to Groups. Symmetries of a Square. The Dihedral Groups. Exercises. Biography of Neils 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 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. 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. 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 variste 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 III: 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 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 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 V: 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 Ludvig 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 Group of Rotations in R . 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 P lya. 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 Erd 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 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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