Contents Preface to Second Edition ix Special Notation xiii Chapter 1. Groups I 1 1.1. Classical Formulas 1 1.2. Permutations 5 1.3. Groups 16 1.4. Lagrange’s Theorem 28 1.5. Homomorphisms 38 1.6. Quotient Groups 47 1.7. Group Actions 60 1.8. Counting 76 Chapter 2. Commutative Rings I 81 2.1. First Properties 81 2.2. Polynomials 91 2.3. Homomorphisms 96 2.4. From Arithmetic to Polynomials 102 2.5. Irreducibility 115 2.6. Euclidean Rings and Principal Ideal Domains 123 2.7. Vector Spaces 133 2.8. Linear Transformations and Matrices 145 2.9. Quotient Rings and Finite Fields 156 Chapter 3. Galois Theory 173 3.1. Insolvability of the Quintic 173 v
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