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BL
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| Record Number
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850309
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| Main Entry
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Campbell, Duff,1959-
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| Title & Author
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An open door to number theory /\ Duff Campbell.
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| Publication Statement
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Providence, Rhode Island :: MAA Press, an imprint of the American Mathematical Society,, [2018]
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| Series Statement
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AMS/MAA textbooks ;; vol. 39
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| Page. NO
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1 online resource
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| ISBN
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1470446847
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: 9781470446840
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1470443481
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9781470443481
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Cover; Title page; 1. The Integers, \Z; 1. Number systems; 2. Rings and fields; 3. Some fundamental facts about \Z and \N; 4. Proofs by induction; 5. The binomial theorem; 6. The fundamental theorem of arithmetic (foreshadowing); 7. Divisibility; 8. Greatest common divisors; 9. The Euclidean algorithm; 10. The amazing array; 11. Convergents; 12. The amazing super-array; 13. The modified division algorithm; 14. Why does the amazing array work?; 15. Primes; 16. The proof of the fundamental theorem of arithmetic; 17. Unique factorization in other rings; 2. Modular Arithmetic in \Z/ \Z.
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18. The integers mod , \Z/ \Z19. Congruences; 20. Units and zero-divisors in \Z/ \Z; 21. Cancellation law in \Z/ \Z; 22. Solving linear equations in \Z/ \Z; 23. Solving polynomial equations in \Z/ \Z; 24. Solving systems of linear equations in \Z/ \Z; 25. Lifting roots in \Z/ ⁿ\Z; 26. Wilson's theorem and its converse; 27. Calculating ( ); 28. Euler's and Fermat's theorems; 29. The order of an integer modulo; 30. Divisibility tests; 3. Quadratic Extensions of the Integers, \Z[√ ]; 31. Divisibility in \Z[ ]; 32. The Euclidean algorithm in \Z[ ]
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33. Unique factorization in \Z[ ]34. The structure of \Z[√2]; 35. The Euclidean algorithm in \Z[√ ]; 36. Factoring in \Z[ ]; 37. The primes in \Z[ ]; 4. An Interlude of Analytic Number Theory; 38. The distribution of primes in \Z; 5. Quadratic Residues; 39. Perfect squares; 40. Quadratic residues; 41. Calculating the Legendre symbol (hard way); 42. The arithmetic of \Z[√-2] and the Legendre symbol \Leg{-2}; 43. Gauss's lemma; 44. Calculating the Legendre symbol (easier way); 45. The arithmetic of \Z[√-3]; 46. The arithmetic of \Z[ ]; 47. Calculating the Legendre symbol (easiest way)
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48. The Jacobi symbol6. Further Topics; 49. When \Z/ \Z has a primitive root; 50. Minkowski's theorem (geometry in the aid of algebra); Appendix A. Tables; Appendix B. Projects; Bibliography; Index; Back Cover.
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| Abstract
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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 cou.
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| Subject
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Number theory, Textbooks.
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| Subject
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MATHEMATICS-- Algebra-- Intermediate.
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| Subject
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Number theory-- Elementary number theory-- Congruences; primitive roots; residue systems.
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Number theory-- Elementary number theory-- Continued fractions.
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Number theory-- Elementary number theory-- Factorization; primality.
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Number theory-- Elementary number theory-- Multiplicative structure; Euclidean algorithm; greatest common divisors.
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Number theory-- Elementary number theory-- Power residues, reciprocity.
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Number theory-- Elementary number theory-- Primes.
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Number theory-- Instructional exposition (textbooks, tutorial papers, etc.)
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| Subject
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Number theory.
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| Dewey Classification
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512.72
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| LC Classification
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QA241
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| NLM classification
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11-0111A0511A0711A1511A4111A5111A55msc
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