Algebraic Structures

Contents CHAPTER I The Integers 1. Terminology of sets . . . . . . . . . . . . . . . . . . 1 2. Basic properties . . . . . . . . . . . . . . . . . . . . 2 3. Greatest common divisor . . . . . . . . . . . . . . . . 5 4. Unique factorization . . . . . . . . . . . . . . . . . . 6 5. Equivalence relations and congruences. . . . . . . . . . 8 CHAPTER II Groups 1. Groups and examples . . . . . . . . . . . . . . 12 2. Mappings . . . . . . . . . . . . . . . . . . . . . 17 3. Homomorphisms . . . . . . . . . . . . . . . . . . . . . 21 4. Cosets and normal subgroups . . . . . . . . . . . . . . 26 5. Permutation groups. . . . . . . . . . . . . . . . . . . 32 6. Cyclic groups . . . . . . . . . . . . . . . . . . . . . 39 CHAPTER III Rings 1. Rings . . . . . . . . . . . . . . . . . . . 43 2. Ideals . . . . . . . . . . . . . . . . . . . 46 3. Homomorphisms . . . . . . . . . . . . . . . . 48 4. Quotient fields . . . . . . . . . . . . . . . . 54 CHAPTER IV Polynomials 1. Euclidean algorithm . . . . . . . . . . . . . . . 58 2. Greatest common divisor . . . . . . . . . . . . . 63 3. Unique factorization . . . . . . . . . . . . . . 65 4. Partial fractions . . . . . . . . . . . . . . . . 70 5. Polynomials over the integers . . . . . . . . . . 76 6. Transcendental elements . . . . . . . . . . . . . 79 7. Polynomials in several variables . . . . . . . . 84 CHAPTER V Vector Spaces and Modules 1. Vector spaces and bases . . . . . . . . . . . . . . 86 2. Dimension of a vector space . . . . . . . . . . . . 92 3. Modules . . . . . . . . . . . . . . . . . . . . . . 94 CHAPTER VI Field Theory 1. Algebraic extensions . . . . . . . . . . . . . . . 102 2. Embeddings . . . . . . . . . . . . . . . . . . . . 105 3. Splitting fields. . . . . . . . . . . . . . . . . . 110 4. Fundamental theorem. . . . . . . . . . . . . . . . 111 5. Quadratic and cubic extensions. . . . . . . . . . 113 6. Solvability by radicals . . . . . . . . . . . . . 115 7. Infinite extensions . . . . . . . . . . . . . . . . 118 CHAPTER VII The Real and Complex Numbers 1. Ordering of rings. . . . . . . . . . . . . . . . 120 2. Preliminaries. . . . . . . . . . . . . . . . . . 123 3. Construction of the real numbers . . . . . . . . 126 4. Decimal expansions . . . . . . . . . . . . . . . 133 5. The complex numbers. . . . . . . . . . . . . . . 136 CHAPTER VIII Sets 1. More terminology . . . . . . . . . . . . . . . 141 2. Zorn’s lemma . . . . . . . . . . . . . . . . . 144 3. Cardinal numbers . . . . . . . . . . . . . . . 148 4. Well-ordering . . . . . . . . . . . . . . . . . 158 5. Proof of Zorn’s lemma. . . . . . . . . . . . . 160 Appendix 1. The natural numbers . . . . . . . . . . . . . . 164 2. The integers . . . . . . . . . . . . . . . . . 168 Index . . . . . . . . . . . . . . . . . . . 171