古画里的中国生活 (中国故事)

A Friendly Introduction to Abstract Algebra offers a new approach to laying a foundation for abstract mathematics. Prior experience with proofs is not assumed, and the book takes time to build proof-writing skills in ways that will serve students through a lifetime of learning and creating mathematics. The author's pedagogical philosophy is that when students abstract from a wide range of examples, they are better equipped to conjecture, formalize, and prove new ideas in abstract algebra. Thus, students thoroughly explore all concepts through illuminating examples before formal definitions are introduced. The instruction in proof writing is similarly grounded in student exploration and experience. Throughout the book, the author carefully explains where the ideas in a given proof come from, along with hints and tips on how students can derive those proofs on their own. Readers of this text are not just consumers of mathematical knowledge. Rather, they are learning mathematics by creating mathematics. The author's gentle, helpful writing voice makes this text a particularly appealing choice for instructors and students alike. The book's website has companion materials that support the active-learning approaches in the book, including in-class modules designed to facilitate student exploration. Cover 1 Title page 5 Copyright 6 Contents 7 Preface 13 For the student 13 For the instructor 13 Note about rings 15 Road map 15 Acknowledgments 16 Unit I: Preliminaries 17 Chapter 1. Introduction to Proofs 19 1.1. Proving an implication 19 1.2. Proof by cases 20 1.3. Contrapositive 22 1.4. Proof by contradiction 23 1.5. If and only if 24 1.6. Counterexample 25 Exercises 25 Chapter 2. Sets and Subsets 29 2.1. What is a set? 29 2.2. Set of integers and its subsets 30 2.3. Closure 31 2.4. Showing set equality 33 Exercises 34 Chapter 3. Divisors 37 3.1. Divisor 37 3.2. GCD theorem 38 3.3. Proofs involving the GCD theorem 39 Exercises 41 Unit II: Examples of Groups 45 Chapter 4. Modular Arithmetic 47 4.1. Number system Z 7 47 4.2. Equality in Z 7 48 4.3. Multiplicative inverses 50 Exercises 53 Chapter 5. Symmetries 57 5.1. Symmetries of a square 57 5.2. Group properties of D4 60 5.3. Centralizer 61 Exercises 63 Chapter 6. Permutations 67 6.1. Permutations of the set {1,2,3} 67 6.2. Group properties of S_{n} 69 6.3. Computations in S_{n} 70 6.4. Associative law in S_{n} (and in D_{n}) 72 Exercises 72 Chapter 7. Matrices 77 7.1. Matrix arithmetic 77 7.2. Matrix group M(Z 10) 78 7.3. Multiplicative inverses 80 7.4. Determinant 81 Exercises 84 Unit III: Introduction to Groups 87 Chapter 8. Introduction to Groups 89 8.1. Definition of a “group” 89 8.2. Essential properties of a group 92 8.3. Proving that a group is commutative 96 8.4. Non-associative operations 97 8.5. Direct product 97 Exercises 99 Chapter 9. Groups of Small Size 103 9.1. Smallest group 103 9.2. Groups with two elements 104 9.3. Groups with three elements 106 9.4. Sudoku property 107 9.5. Groups with four elements 108 Exercises 109 Chapter 10. Matrix Groups 113 10.1. Groups Z 10 and U10 113 10.2. Groups M(Z 10) and G(Z 10) 114 10.3. Group S(Z 10) 116 Exercises 117 Chapter 11. Subgroups 121 11.1. Examples of subgroups 121 11.2. Subgroup proofs 123 11.3. Center and centralizer revisited 125 Exercises 126 Chapter 12. Order of an Element 131 12.1. Motivating example 131 12.2. When does g^{k}=ε? 132 12.3. Conjugates 134 12.4. Order in an additive group 136 12.5. Elements with infinite order 137 Exercises 138 Chapter 13. Cyclic Groups, Part I 141 13.1. Generators of the additive group Z 12 141 13.2. Generators of the multiplicative group U13 143 13.3. Matching Z 12 and U13 144 13.4. Taking positive and negative powers of g 145 13.5. When the group operation is addition 147 Exercises 148 Chapter 14. Cyclic Groups, Part II 151 14.1. Why negative powers are needed 151 14.2. Additive groups revisited 152 14.3. ⟨3⟩ behaves “just like” Z 153 14.4. Subgroups of cyclic groups 154 Exercises 157 Unit IV: Group Homomorphisms 161 Chapter 15. Functions 163 15.1. Domain and codomain 163 15.2. One-to-one function 164 15.3. Onto function 165 15.4. When domain and codomain have the same size 168 Exercises 169 Chapter 16. Isomorphisms 173 16.1. Groups Z 12 and ⟨g⟩: Elements match up 173 16.2. Groups Z 12 and ⟨g⟩: Operations match up 174 16.3. Elements with infinite order revisited 177 16.4. Inverse isomorphisms 178 Exercises 180 Chapter 17. Homomorphisms, Part I 185 17.1. Group homomorphism 185 17.2. Properties of homomorphisms 188 17.3. Order of an element 190 Exercises 191 Chapter 18. Homomorphisms, Part II 195 18.1. Kernel of a homomorphism 195 18.2. Image of a homomorphism 198 18.3. Partitioning the domain 199 18.4. Finding homomorphisms 200 Exercises 201 Unit V: Quotient Groups 205 Chapter 19. Introduction to Cosets 207 19.1. Multiplicative group example 207 19.2. Additive group example 209 19.3. Right cosets 211 19.4. Properties of cosets 212 19.5. When are cosets equal? 214 Exercises 216 Chapter 20. Lagrange’s Theorem 221 20.1. Motivating Lagrange’s theorem 221 20.2. Proving Lagrange’s theorem 223 20.3. Applications of Lagrange’s theorem 225 Exercises 227 Chapter 21. Multiplying/Adding Cosets 229 21.1. Turning a set of cosets into a group 229 21.2. Coset multiplication shortcut 232 21.3. Cosets of H=5Z in Z revisited 233 Exercises 235 Chapter 22. Quotient Group Examples 239 22.1. Quotient group U13/H revisited 239 22.2. Quotient group U37/H 240 22.3. Quotient group G/H (generalization) 241 Exercises 243 Chapter 23. Quotient Group Proofs 247 23.1. Sample quotient group proofs 247 23.2. Collapsing G into G/H 250 Exercises 252 Chapter 24. Normal Subgroups 255 24.1. How does the shortcut fail and work? 255 24.2. Normal subgroups: What and why 257 24.3. Examples of normal subgroups 257 24.4. Normal subgroup test 258 Exercises 261 Chapter 25. First Isomorphism Theorem 265 25.1. Familiar homomorphism 265 25.2. Another homomorphism 267 25.3. First Isomorphism Theorem 269 25.4. Finding and building homomorphisms 269 Exercises 271 Unit VI: Introduction to Rings 275 Chapter 26. Introduction to Rings 277 26.1. Examples and definition 277 26.2. Fundamental properties 280 26.3. Units and zero divisors 282 26.4. Subrings 283 26.5. Group of units 284 Exercises 285 Chapter 27. Integral Domains and Fields 287 27.1. Integral domains 287 27.2. Fields 289 27.3. Idempotent elements 292 Exercises 293 Chapter 28. Polynomial Rings, Part I 297 28.1. Examples and definition 297 28.2. Degree of a polynomial 299 28.3. Units and zero divisors 302 Exercises 303 Chapter 29. Polynomial Rings, Part II 305 29.1. Division algorithm in F[x] 305 29.2. Factor theorem 307 29.3. Nilpotent elements 309 Big picture stuff 311 Exercises 311 Chapter 30. Factoring Polynomials 315 30.1. Examples and definition 315 30.2. Factorable or unfactorable? 317 Big picture stuff 320 Exercises 320 Unit VII: Quotient Rings 325 Chapter 31. Ring Homomorphisms 327 31.1. Evaluation map 327 31.2. Properties of ring homomorphisms 330 31.3. Kernel and image 331 31.4. Examples and definition of an ideal 332 31.5. Ideals in Z and in F[x] 335 Big picture stuff 335 Exercises 336 Chapter 32. Introduction to Quotient Rings 339 32.1. From a quotient group to a quotient ring 339 32.2. Role of an ideal in a quotient ring 340 32.3. Quotient ring Z 3[x]/⟨x2⟩ 343 32.4. First Isomorphism Theorem for rings 344 Big picture stuff 345 Exercises 345 Chapter 33. Quotient Ring Z 7[x]/⟨x2-1⟩ 349 33.1. Division algorithm revisited 349 33.2. Another way to reduce in Z 7[x]/⟨x2-1⟩ 352 33.3. F[x]/⟨g(x)⟩ is not a field 353 33.4. F[x]/⟨g(x)⟩ is a field 354 Big picture stuff 354 Exercises 354 Chapter 34. Quotient Ring R [x]/⟨x2+1⟩ 359 34.1. Reducing elements in R [x]/⟨x2+1⟩ 359 34.2. Field of complex numbers 360 34.3. F[x]/⟨g(x)⟩ is a field revisited 363 Exercises 363 Chapter 35. F[x]/⟨g(x)⟩ Is/Isn’t a Field, Part I 367 35.1. Translate from F[x] to Z 367 35.2. Translate (back) from Z to F[x] 369 35.3. Proof of Theorem 35.1(b) 371 Big picture stuff 371 Exercises 372 Chapter 36. Maximal Ideals 375 36.1. Examples and definition 376 36.2. Maximality of ⟨g(x)⟩ 378 Big picture stuff 380 Exercises 380 Chapter 37. F[x]/⟨g(x)⟩ Is/Isn’t a Field, Part II 383 37.1. Maximal ideals and quotient rings 383 37.2. Putting it all together 385 37.3. Oh wait, but there’s more! 386 37.4. Prime ideals 386 Exercises 387 Appendix A. Proof of the GCD Theorem 389 Appendix B. Composition Table for D4 393 Appendix C. Symbols and Notations 395 Appendix D. Essential Theorems 397 Index of Terms 401 Back Cover 404