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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
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
Bibliography
Index

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