Abstract Algebra: An Introductory Course

Description

This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields. The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions. Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided. Preface Contents Part I Preliminaries 1 Relations and Functions 1.1 Sets and Set Operations 1.2 Relations 1.3 Equivalence Relations 1.4 Functions 2 The Integers and Modular Arithmetic 2.1 Induction and Well Ordering 2.2 Divisibility 2.3 Prime Factorization 2.4 Properties of the Integers 2.5 Modular Arithmetic Part II Groups 3 Introduction to Groups 3.1 An Important Example 3.2 Groups 3.3 A Few Basic Properties 3.4 Powers and Orders 3.5 Subgroups 3.6 Cyclic Groups 3.7 Cosets and Lagrange's Theorem 4 Factor Groups and Homomorphisms 4.1 Normal Subgroups 4.2 Factor Groups 4.3 Homomorphisms 4.4 Isomorphisms 4.5 The Isomorphism Theorems for Groups 4.6 Automorphisms 5 Direct Products and the Classification of Finite Abelian Groups 5.1 Direct Products 5.2 The Fundamental Theorem of Finite Abelian Groups 5.3 Elementary Divisors and Invariant Factors 5.4 A Word About Infinite Abelian Groups 6 Symmetric and Alternating Groups 6.1 The Symmetric Group and Cycle Notation 6.2 Transpositions and the Alternating Group 6.3 The Simplicity of the Alternating Group 7 The Sylow Theorems 7.1 Normalizers and Centralizers 7.2 Conjugacy and the Class Equation 7.3 The Three Sylow Theorems 7.4 Applying the Sylow Theorems 7.5 Classification of the Groups of Small Order Part III Rings 8 Introduction to Rings 8.1 Rings 8.2 Basic Properties of Rings 8.3 Subrings 8.4 Integral Domains and Fields 8.5 The Characteristic of a Ring 9 Ideals, Factor Rings and Homomorphisms 9.1 Ideals 9.2 Factor Rings 9.3 Ring Homomorphisms 9.4 Isomorphisms and Automorphisms 9.5 Isomorphism Theorems for Rings 9.6 Prime and Maximal Ideals 10 Special Types of Domains 10.1 Polynomial Rings 10.2 Euclidean Domains 10.3 Principal Ideal Domains 10.4 Unique Factorization Domains Reference Part IV Fields and Polynomials 11 Irreducible Polynomials 11.1 Irreducibility and Roots 11.2 Irreducibility over the Rationals 11.3 Irreducibility over the Real and Complex Numbers 11.4 Irreducibility over Finite Fields Reference 12 Vector Spaces and Field Extensions 12.1 Vector Spaces 12.2 Basis and Dimension 12.3 Field Extensions 12.4 Splitting Fields 12.5 Applications to Finite Fields Reference Part V Applications 13 Public Key Cryptography 13.1 Private Key Cryptography 13.2 The RSA Scheme 14 Straightedge and Compass Constructions 14.1 Three Ancient Problems 14.2 The Connection to Field Extensions 14.3 Proof of the Impossibility of the Problems A The Complex Numbers Appendix B Matrix Algebra Appendix Solutions Index