Mark Sepanski''s Algebra is a readable introduction to the delightful world of modern algebra. Beginning with concrete examples from the study of integers and modular arithmetic, the text steadily familiarizes the reader with greater levels of abstraction as it moves through the study of groups, rings, and fields. The book is equipped with over 750 exercises suitable for many levels of student ability. There are standard problems, as well as challenging exercises, that introduce students to topics not normally covered in a first course. Difficult problems are broken into manageable subproblems and come equipped with hints when needed. Appropriate for both self-study and the classroom, the material is efficiently arranged so that milestones such as the Sylow theorems and Galois theory can be reached in one semester
Mark R. Sepanski, Baylor University, Waco, TX
Preface Chapter 1. Arithmetic 1. Integers 1.1. Basic Properties 1.2. Induction 1.3. Division Algorithm 1.4. Divisors 1.5. Fundamental Theorem of Arithmetic 1.6. Exercises 1.1–1.28 2. Modular Arithmetic 2.1. Congruence 2.2. Congruence Classes 2.3. Arithmetic 2.4. Structure 2.5. Applications 2.6. Equivalence Relations 2.7. Exercises 1.29–1.72 Chapter 2. Groups 1. Definitions and Examples 1.1. Binary Operations 1.2. Definition of a Group 1.3. Examples 1.4. Exercises 2.1–2.29 2. Basic Properties and Order 2.1. Exercises 2.30–2.52 3. Subgroups and Direct Products 3.1. Subgroups 3.2. Direct Products 3.3. Exercises 2.53–2.91 4. Morphisms 4.1. Introduction 4.2. Definitions and Examples 4.3. Basic Properties 4.4. Exercises 2.92–2.118 5. Quotients 5.1. Definitions 5.2. Lagrange's Theorem 5.3. Normality 5.4. Direct Products and the Correspondence Theorem 5.5. First Isomorphism Theorem 5.6. Exercises 2.119–2.169 6. Fundamental Theorem of Finite Abelian Groups 6.1. Exercises 2.170–2.188 7. The Symmetric Group 7.1. Cayley's Theorem 7.2. Cyclic Decomposition 7.3. Conjugacy Classes 7.4. Parity and the Alternating Subgroup 7.5. Exercises 2.189–2.229 8. Group Actions 8.1. Exercises 2.230–2.248 9. Sylow Theorems 9.1. Exercises 2.249–2.281 10. Simple Groups and Composition Series 10.1. Simple Groups 10.2. Composition Series 10.3. Exercises 2.282–2.301 Chapter 3. Rings 1. Examples and Basic Properties 1.1. Definition 1.2. Examples 1.3. Basic Properties 1.4. Subrings 1.5. Direct Products 1.6. Exercises 3.1–3.39 2. Morphisms and Quotients 2.1. Morphisms 2.2. Ideals 2.3. Quotients 2.4. Isomorphism Theorem 2.5. Exercises 3.40–3.86 3. Polynomials and Roots 3.1. Division Algorithm 3.2. Roots 3.3. Rational Root Test 3.4. Fundamental Theorem of Algebra 3.5. Exercises 3.87–3.129 4. Polynomials and Irreducibility 4.1. Irreducibility 4.2. Polynomials over a Field 4.3. Exercises 3.130–3.150
5. Factorization 5.1. Unique Factorization 5.2. Quotient Fields 5.3. Unique Factorization in Polynomial Rings 5.4. Exercises 3.151–3.172 6. Principal Ideal and Euclidean Domains 6.1. Principal Ideal Domains 6.2. Euclidean Domains 6.3. Gaussian Integers 6.4. Exercises 3.173–3.201 Chapter 4. Field Theory 1. Finite and Algebraic Extensions 1.1. Vector Spaces 1.2. Finite and Algebraic Extensions 1.3. Exercises 4.1–4.34 2. Splitting Fields 2.1. Splitting Fields 2.2. Algebraic Closures 2.3. Exercises 4.35–4.62 3. Finite Fields 3.1. Exercises 4.63–4.81 4. Galois Theory 4.1. Galois Groups 4.2. Separability 4.3. Galois Correspondence 4.4. Exercises 4.82—4.135 5. Famous Impossibilities 5.1. Compass and Straightedge Constructions 5.2. Solvability of Polynomials 5.3. Exercises 4.136–4.162 6. Cyclotomic Fields 6.1. Exercises 4.163–4.179 Index