Abstract Algebra: A Gentle Introduction

Description

Abstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the ‘everything for everyone’ approach so common in textbooks. Instead, they provide the reader with coverage of numerous algebraic topics to cover the most important areas of abstract algebra. Through a careful selection of topics, supported by interesting applications, the authors Intend the book to be used for a one-semester course in abstract algebra. It is suitable for an introductory course in for mathematics majors. The text is also very suitable for education majors who need to have an introduction to the topic. As textbooks go through various editions and authors employ the suggestions of numerous well-intentioned reviewers, these book become larger and larger and subsequently more expensive. This book is meant to counter that process. Here students are given a "gentle introduction," meant to provide enough for a course, yet also enough to encourage them toward future study of the topic. Features Groups before rings approachInteresting modern applicationsAppendix includes mathematical induction, the well-ordering principle, sets, functions, permutations, matrices, and complex nubers.Numerous exercises at the end of each sectionChapter "Hint and Partial Solutions" offers built in solutions manual Contents Preface Chapter 1. Elementary Number Theory 1.1 Divisibility 1.1 Exercises Solutions 1.1 1.2 Primes and factorization 1.2 Exercises Solutions 1.2 1.3 Congruences 1.3 Exercises Solutions 1.3 1.4 Solving congruences 1.4 Exercises Solutions 1.4 1.5 Theorems of Fermat and Euler 1.5 Exercises Solutions 1.5 1.6 RSA cryptosystem 1.6 Exercises Solutions 1.6 Chapter 2. Groups 2.1 Definition of a group 2.2 Examples of groups 2.2 Exercises Solutions 2.2 2.3 Subgroups 2.3 Exercises Solutions 2.3 2.4 Cosets and Lagrange’s Theorem 2.4 Exercises Solutions 2.4 Chapter 3. Rings 3.1 Definition of a ring 3.1 Exercises Solutions 3.1 3.2 Subrings and ideals 3.2 Exercises Solutions 3.2 3.3 Ring homomorphisms 3.3 Exercises Solutions 3.3 3.4 Integral domains 3.4 Exercises Solutions 3.4 Chapter 4. Fields 4.1 Definition and basic properties of a field 4.1 Exercises Solutions 4.1 Chapter 5. Finite Fields 5.1 Number of elements in a finite field 5.1 Exercises Solutions 5.1 5.2 How to construct finite fields 5.2 Exercises Solutions 5.2 5.3 Properties of finite fields 5.3 Exercises Solutions 5.3 5.4 Polynomials over finite fields 5.4 Exercises Solutions 5.4 5.5 Permutation polynomials 5.5 Exercises Solutions 5.5 5.6 Applications 5.6.1 Orthogonal Latin squares 5.6.2 Diffie/Hellman key exchange 5.6 Exercises Solutions 5.6 Chapter 6. Vector Spaces 6.1 Definition and examples 6.1 Exercises Solutions 6.1 6.2 Basic properties of vector spaces 6.2 Exercises Solutions 6.2 6.3 Subspaces 6.3 Exercises Solutions 6.3 Chapter 7. Polynomials 7.1 Basics 7.1 Exercises Solutions 7.1 7.2 Unique factorization 7.2 Exercises Solutions 7.2 7.3 Polynomials over the real and complex numbers 7.3 Exercises Solutions 7.3 7.4 Root formulas 7.4 Exercises Solutions 7.4 Chapter 8. Linear Codes 8.1 Basics 8.2 Hamming codes 8.3 Encoding 8.4 Decoding 8.5 Further study 8.6 Exercises Solutions 8.6 Chapter 9. Appendix 9.1 Mathematical induction 9.1 Exercises Solutions 9.1 9.2 Well-ordering Principle 9.2 Exercises Solutions 9.2 9.3 Sets 9.3 Exercises Solutions 9.3 9.4 Functions 9.4 Exercises Solutions 9.4 9.5 Permutations 9.5 Exercises Solutions 9.5 9.6 Matrices 9.6 Exercises Solutions 9.6 9.7 Complex numbers 9.7 Exercises Solutions 9.7 Chapter 10. Hints and Partial Solutions to Selected Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Bibliography [14] [30] Index abcd efghi klmnop qrstuv wz