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Modern Algebra
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Table of Contents

Introduction 1 I. Mappings and Operations 9 1 Mappings 9 2 Composition. Invertible Mappings 15 3 Operations 19 4 Composition as an Operation 25 II. Introduction to Groups 30 5 Definition and Examples 30 6 Permutations 34 7 Subgroups 41 8 Groups and Symmetry 47 III. Equivalence. Congruence. Divisibility 52 9 Equivalence Relations 52 10 Congruence. The Division Algorithm 57 11 Integers Modulo n 61 12 Greatest Common Divisors. The Euclidean Algorithm 65 13 Factorization. Euler's Phi-Function 70 IV. Groups 75 14 Elementary Properties 75 15 Generators. Direct Products 81 16 Cosets 85 17 Lagrange's Theorem. Cyclic Groups 88 18 Isomorphism 93 19 More on Isomorphism 98 20 Cayley's Theorem 102 Appendix: RSA Algorithm 105 V. Group Homomorphisms 106 21 Homomorphisms of Groups. Kernels 106 22 Quotient Groups 110 23 The Fundamental Homomorphism Theorem 114 VI. Introduction to Rings 120 24 Definition and Examples 120 25 Integral Domains. Subrings 125 26 Fields 128 27 Isomorphism. Characteristic 131 VII. The Familiar Number Systems 137 28 Ordered Integral Domains 137 29 The Integers 140 30 Field of Quotients. The Field of Rational Numbers 142 31 Ordered Fields. The Field of Real Numbers 146 32 The Field of Complex Numbers 149 33 Complex Roots of Unity 154 VIII. Polynomials 160 34 Definition and Elementary Properties 160 Appendix to Section 34 162 35 The Division Algorithm 165 36 Factorization of Polynomials 169 37 Unique Factorization Domains 173 IX. Quotient Rings 178 38 Homomorphisms of Rings. Ideals 178 39 Quotient Rings 182 40 Quotient Rings of F[X] 184 41 Factorization and Ideals 187 X. Galois Theory: Overview 193 42 Simple Extensions. Degree 194 43 Roots of Polynomials 198 44 Fundamental Theorem: Introduction 203 XI. Galois Theory 207 45 Algebraic Extensions 207 46 Splitting Fields. Galois Groups 210 47 Separability and Normality 214 48 Fundamental Theorem of Galois Theory 218 49 Solvability by Radicals 219 50 Finite Fields 223 XII. Geometric Constructions 229 51 Three Famous Problems 229 52 Constructible Numbers 233 53 Impossible Constructions 234 XIII. Solvable and Alternating Groups 237 54 Isomorphism Theorems and Solvable Groups 237 55 Alternating Groups 240 XIV. Applications of Permutation Groups 243 56 Groups Acting on Sets 243 57 Burnside's Counting Theorem 247 58 Sylow's Theorem 252 XV. Symmetry 256 59 Finite Symmetry Groups 256 60 Infinite Two-Dimensional Symmetry Groups 263 61 On Crystallographic Groups 267 62 The Euclidean Group 274 XVI. Lattices and Boolean Algebras 279 63 Partially Ordered Sets 279 64 Lattices 283 65 Boolean Algebras 287 66 Finite Boolean Algebras 291 A. Sets 296 B. Proofs 299 C. Mathematical Induction 304 D. Linear Algebra 307 E. Solutions to Selected Problems 312 Photo Credit List 326 Index of Notation 327 Index 330

About the Author

Dr. John R. Durbin is a professor of Mathematics at The University of Texas Austin. A native Kansan, he received B.A. and M.A. degrees from the University of Wichita (now Wichita State University), and a Ph.D. from the University of Kansas. He came to UT immediately thereafter. Professor Durbin has been active in faculty governance at the University for many years. He served as chair of the Faculty Senate, 1982-84 and 1991-92, and as Secretary of the General Faculty, 1975-76 and 1998-2003. In September of 2003 he received the University & Civitatis Award,in recognition of dedicated and meritorious service to the University above and beyond the regular expectations of teaching, research, and writing. He has received a Teaching Excellence Award from the College of Natural Sciences and an Outstanding Teaching Award from the Department of Mathematics.

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