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Abstract algebra. An Interactive Approach (2nd Edition) by William Paulsen



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Abstract algebra. An Interactive Approach (2nd Edition) written by William Paulsen , Arkansas State University, Jonesboro, Arkansas, USA

Abstract algebra. An Interactive Approach (2nd Edition) written by William Paulsen cover the following topics.

  • List of Figures
    List of Tables
    Preface
    Acknowledgments
    About the Author
    Symbol Description
    Introduction

  • 0. Preliminaries
    0.1 Integer Factorization
    0.2 Functions
    0.3 Modular Arithmetic
    0.4 Rational and Real Numbers
    1. Understanding the Group Concept
    1.1 Introduction to Groups
    1.2 Modular Congruence
    1.3 The Definition of a Group

  • 2. The Structure within a Group
    2.1 Generators of Groups
    2.2 Defining Finite Groups in Sage
    2.3 Subgroups

  • 3. Patterns within the Cosets of Groups
    3.1 Left and Right Cosets
    3.2 Writing Secret Messages
    3.3 Normal Subgroups
    3.4 Quotient Groups

  • 4. Mappings between Groups
    4.1 Isomorphisms
    4.2 Homomorphisms
    4.3 The Three Isomorphism Theorems

  • 5. Permutation Groups
    5.1 Symmetric Groups
    5.2 Cycles
    5.3 Cayley’s Theorem
    5.4 Numbering the Permutations

  • 6. Building Larger Groups from Smaller Groups
    6.1 The Direct Product
    6.2 The Fundamental Theorem of Finite Abelian Groups
    6.3 Automorphisms
    6.4 Semi-Direct Products

  • 7. The Search for Normal Subgroups
    7.1 The Center of a Group
    7.2 The Normalizer and Normal Closure Subgroups
    7.3 Conjugacy Classes and Simple Groups
    7.4 The Class Equation and Sylow’s Theorems

  • 8. Solvable and Insoluble Groups
    8.1 Subnormal Series and the Jordan-H¨older Theorem
    8.2 Derived Group Series
    8.3 Polycyclic Groups
    8.4 Solving the PyraminxTM

  • 9. Introduction to Rings
    9.1 The Definition of a Ring
    9.2 Entering Finite Rings into Sage
    9.3 Some Properties of Rings

  • 10. The Structure within Rings
    10.1 Subrings
    10.2 Quotient Rings and Ideals
    10.3 Ring Isomorphisms
    10.4 Homomorphisms and Kernels

  • 11. Integral Domains and Fields
    11.1 Polynomial Rings
    11.2 The Field of Quotients
    11.3 Complex Numbers
    11.4 Ordered Commutative Rings

  • 12. Unique Factorization
    12.1 Factorization of Polynomials
    12.2 Unique Factorization Domains
    12.3 Principal Ideal Domains
    12.4 Euclidean Domains

  • 13. Finite Division Rings
    13.1 Entering Finite Fields in Sage
    13.2 Properties of Finite Fields
    13.3 Cyclotomic Polynomials
    13.4 Finite Skew Fields

  • 14. The Theory of Fields
    14.1 Vector Spaces
    14.2 Extension Fields
    14.3 Splitting Fields

  • 15. Galois Theory
    15.1 The Galois Group of an Extension Field
    15.2 The Galois Group of a Polynomial in Q
    15.3 The Fundamental Theorem of Galois Theory
    15.4 Applications of Galois Theory

  • Appendix:
    Sage vs. Mathematica R
    Answers to Odd-Numbered Problems
    Bibliography
    Index

  • Open
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