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Abstract Algebra: An Introduction (3rd Solution) by Thomas W. Hungerford



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Abstract Algebra: An Introduction (3rd Solution) written by Thomas W. Hungerford, Saint Louis University . The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication flavor. The emphasis of this text is on clarity of exposition. The chapters are organized around three themes: arithmetic, congruence, and abstract structures. The interconnections of the basic areas of algebra are frequently pointed out in the text and in the Thematic Table of Contents.

Abstract Algebra: An Introduction (3rd Solution) written by Thomas W. Hungerford cover the following topics.

  • Preface
    To the Instructor
    To the Student
    Thematic Table of Contents for the Core Course

  • Part 1. The Core Course

  • 1. Arithmetic inZRevisited
    1.1 The Division Algorithm
    1.2 Divisibility
    1.3 Primes and Unique Factorization

  • 2. Conuruence inZandModularArithmetic
    2.1 Congruence and Congruence Classes
    2.2 Modular Arithmetic
    2.3 The Structure of Z,, (p Prime) and Zn

  • 3. Rings
    3.1 Definition and Examples of Rings
    3.2 Basic Properties of Rings
    3.3 Isomorphisms and Homomorphisms

  • 4. Arithmetic in f[x]
    4.1 Polynomial Arithmetic and the Division Algorithm
    4.2 Divisibility in f[x]
    4.3 lrreducibles and Unique Factorization
    4.4 Polynomial Functi ons, Roots, and Reducibility
    4.5* lrreducibillty in O[x]
    4.6* lrreducibillty in IR[x] and C[x]

  • 5. Congruence in f[x] and Congruence-Glass Arithmetic
    5.1 Congruence in F[x] and Congruence Classes
    5.2 Congruence-Class Arithmetic
    5.3 The Structure of F{x]/(p(x)) When p(x) Is Irreducible

  • 6. Ideals and Quotient Rings
    6.1 Ideals and Congruence
    6.2 Quotient Rings and Homomorphisms
    6.3* The Structure of Rf /When /Is Prime or Maximal

  • 7. Groups
    7.1 Definition and Examples of Groups
    7.1.A Definition and Examples of Groups
    7.2 Basic Properties of Groups
    7.3 Subgroups
    7.4 Isomorphisms and Homomorphisms
    7.5* The Symmetric and Alternating Groups

  • 8. Normal Subgroups and Quotient Groups
    8.1 Congruence and Lagrange
    s Theorem
    8.2 Normal Subgroups
    8.3 Quotient Groups
    8.4 Quotient Groups and Homomorphisms
    8.5* The Simplicity of An

  • Part 2 Advanced Topics

  • 9. Topics in Group Theory
    9.1 Direct Products
    9.2 Finite Abelian Groups
    9.3 The Sylow Theorems
    9.4 Conjugacy and the Proof of the Sylow Theorems
    9.5 The Structure of Finite Groups

  • 1O. Arithmetic in Integral Domains
    10.1 Euclidean Domains
    10.2 Principal Ideal Domains and Unique Factorization Domains
    10.3 Factorization of Quadratic Integers
    10.4 The Field of Quotients of an Integral Domain
    10.5 Unique Factorization in Polynomial Domains

  • 11. Field Extensions
    11.1 Vector Spaces
    11.2 Simple Extensions
    11.3 Algebraic Extensions
    11.4 Splitting Fields
    11.5 Separability
    11.6 Finite Fields

  • 12. GaloisTheory
    12.1 The Galois Group
    12.2 The Fundamental Theorem of Galois Theory
    12.3 Solvability by Radicals

  • Part 3 Excursions and Applications

  • 13. Public-Key Cryptography
    Prerequisite: Section 2.3

  • 14. The Chinese RemainderTheorem
    14.1 Proof of the Chinese Remainder Theorem Prerequisites: Section 2.1, Appendix C
    14.2 Applications of the Chinese Remainder Theorem Prerequisite: Section 3.1
    14.3 The Chinese Remainder Theorem for Rings Prerequisite: Section 6.2

  • 15. Geometric Constructions
    Prerequisites: Sections 4.1, 4.4, and 4.5

  • 16. Algebraic CodingTheory
    16.1 Linear Codes, Prerequisites: Section 7.4, Appendix F
    16.2 DecodingTechniques, Prerequisite; Section 8.4
    16.3 BCH Codes, Prerequisite: Section 11.6

  • 4. Appendices
    A. Logic and Proof
    B. Sets and Functions
    C.Well Ordering And Induction
    D.Equivalence Relations
    E. The Binomial Theorem
    F.Matrix Algebra
    Bibliography
    Answers and Suggestions for Selected Odd-Numbered
    Exercises
    Index

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