An Open Door to Number Theory (MAA Textbooks) (AMS / MAA Textbooks)

A well-written, inviting textbook designed for a one-semester, junior-level course in elementary number theory. The intended audience will have had exposure to proof writing, but not necessarily to abstract algebra. That audience will be well prepared by this text for a second-semester course focusing on algebraic number theory. The approach throughout is geometric and intuitive; there are over 400 carefully designed exercises, which include a balance of calculations, conjectures, and proofs. There are also nine substantial student projects on topics not usually covered in a first-semester course, including Bernoulli numbers and polynomials, geometric approaches to number theory, the p-adic numbers, quadratic extensions of the integers, and arithmetic generating functions.An instructor's manual for this title is available electronically to those instructors who have already adopted the textbook for classroom use. Please send email to textbooks@ams.org for more information. Cover......Page 1 Title page......Page 4 Contents......Page 8 Preface......Page 12 1. Number systems......Page 14 2. Rings and fields......Page 16 3. Some fundamental facts about \Z and \N......Page 20 4. Proofs by induction......Page 26 5. The binomial theorem......Page 31 6. The fundamental theorem of arithmetic (foreshadowing)......Page 39 7. Divisibility......Page 42 8. Greatest common divisors......Page 44 9. The Euclidean algorithm......Page 46 10. The amazing array......Page 52 11. Convergents......Page 55 12. The amazing super-array......Page 62 13. The modified division algorithm......Page 69 14. Why does the amazing array work?......Page 71 15. Primes......Page 74 16. The proof of the fundamental theorem of arithmetic......Page 77 17. Unique factorization in other rings......Page 81 18. The integers mod , \Z/\Z......Page 84 19. Congruences......Page 89 20. Units and zero-divisors in \Z/\Z......Page 94 21. Cancellation law in \Z/\Z......Page 98 22. Solving linear equations in \Z/\Z......Page 100 23. Solving polynomial equations in \Z/\Z......Page 101 24. Solving systems of linear equations in \Z/\Z......Page 108 25. Lifting roots in \Z/n\Z......Page 116 26. Wilson’s theorem and its converse......Page 121 27. Calculating ()......Page 123 28. Euler’s and Fermat’s theorems......Page 128 29. The order of an integer modulo ......Page 131 30. Divisibility tests......Page 135 31. Divisibility in \Z[]......Page 140 32. The Euclidean algorithm in \Z[]......Page 143 33. Unique factorization in \Z[]......Page 148 34. The structure of \Z[√2]......Page 151 35. The Euclidean algorithm in \Z[√]......Page 153 36. Factoring in \Z[]......Page 157 37. The primes in \Z[]......Page 162 38. The distribution of primes in \Z......Page 166 39. Perfect squares......Page 170 40. Quadratic residues......Page 173 41. Calculating the Legendre symbol (hard way)......Page 180 42. The arithmetic of \Z[√-2] and the Legendre symbol \Leg{-2}......Page 182 43. Gauss’s lemma......Page 184 44. Calculating the Legendre symbol (easier way)......Page 187 45. The arithmetic of \Z[√-3]......Page 193 46. The arithmetic of \Z[]......Page 195 47. Calculating the Legendre symbol (easiest way)......Page 206 48. The Jacobi symbol......Page 210 49. When \Z/\Z has a primitive root......Page 216 50. Minkowski’s theorem (geometry in the aid of algebra)......Page 221 Appendix A. Tables......Page 236 Appendix B. Projects......Page 246 Bibliography......Page 292 Index......Page 294 Back Cover......Page 297

A well-written, inviting textbook designed for a one-semester, junior-level course in elementary number theory. The intended audience will have had exposure to proof writing, but not necessarily to abstract algebra. That audience will be well prepared by this text for a second-semester course focusing on algebraic number theory. The approach throughout is geometric and intuitive; there are over 400 carefully designed exercises, which include a balance of calculations, conjectures, and proofs. There are also nine substantial student projects on topics not usually covered in a first-semester course, including Bernoulli numbers and polynomials, geometric approaches to number theory, the $p$-adic numbers, quadratic extensions of the integers, and arithmetic generating functions.

This textbook is for elementary number theory. The approach throughout is geometric and intuitive; there are over 400 exercises, which include a balance of calculations, conjectures, and proofs. There are also nine substantial student projects on topics including Bernoulli numbers and polynomials, geometric approaches to number theory, the p-adic numbers, quadratic extensions of the integers, and arithmetic generating functions