Abstract algebra /
I. N. Herstein.
- 3rd.
- Hoboken, NJ: John Wiley and Sons, 1999
- 249 p.
CONTENIDO Preface ix 1. Things Familiar and Less Familiar 1 A Few Preliminary Remarks 1 Set Theory 3 Mappings 8 A(S) (The Set of 1-1 Mappings of S onto Itself) 16 The Integers 21 Mathematical Induction 29 Complex Numbers 32 2. Groups 40 Definitions and Examples of Groups 40 Some Simple Remarks 48 Subgroups 51 Lagrange's Theorem 56 Homomorphisms and Normal Subgroups 66 Factor Groups 77 The Homomorphism Theorems 84 Cauchy's Theorem 88 Direct Products 92 Finite Abelian Groups (Optional) 96 Conjugacy and Sylow's Theorem (Optional) 101 3. The Symmetric Group 108 Preliminaries 108 Cycle Decomposition 111 Odd and Even Permutations 119 4. Ring Theory 125 Definitions and Examples 125 Some Simple Results 137 Ideals, Homomorphisms, and Quotient Rings 139 Maximal Ideals 148 Polynomial Rings 151 Polynomials over the Rationals 166 Field of Quotients of an Integral Domain 172 5. Fields 176 Examples of Fields 176 A Brief Excursion into Vector Spaces 180 Field Extensions 191 Finite Extensions 198 Constructibility 201 Roots of Polynomials 207 6. Special Topics (Optional) 215 The Simplicity of An 215 Finite Fields I 221 Finite Fields II: Existence 224 Finite Fields III: Uniqueness 227 Cyclotomic Polynomials 229 Liouville's Criterion 236 The Irrationality of Pi 239 Index 243
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ALGEBRA ABSTRAC GROUPS SYMMETRIC GROUP RING THEORY FIELDS FINITE FIELDS LIOUVILLES CRITERION