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Introduction to Number Theory
 
Trygve Nagell
Introduction to Number Theory
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6324-3
Product Code:  CHEL/163.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-5459-3
Product Code:  CHEL/163.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-6324-3
eBook: ISBN:  978-1-4704-5459-3
Product Code:  CHEL/163.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
Introduction to Number Theory
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Introduction to Number Theory
Trygve Nagell
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6324-3
Product Code:  CHEL/163.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-5459-3
Product Code:  CHEL/163.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-6324-3
eBook ISBN:  978-1-4704-5459-3
Product Code:  CHEL/163.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 163; 1964; 309 pp
    MSC: Primary 11

    A special feature of Nagell's well-known text is the rather extensive treatment of Diophantine equations of second and higher degree. A large number of non-routine problems are given.
  • Table of Contents
     
     
    • Front Cover
    • PREFACE
    • CONTENTS
    • CHAPTER I: DIVISIBILITY
    • 1. Divisors.
    • 2. Remainders.
    • 3. Primes.
    • 4. The fundamental theorem.
    • 5. Least common multiple and greatest common divisor.
    • 6. Moduls, rings and fields.
    • 7. Euclid's algorithm.
    • 8. Relatively prime numbers. Euler's φ-function.
    • 9. Arithmetical functions.
    • 10. Diophantine equations of the first degree.
    • 11. Lattice points and point lattices.
    • 12. Irrational numbers.
    • 13. Irrationality of the numbers e and :rr.
    • Exercises
    • CHAPTER II: ON THE DISTRIBUTION OF PRIMES
    • 14. Some lemmata.
    • 15. General remarks. The sieve of Eratosthenes.
    • 16. The function π(x).
    • 17. Some elementary results on the distribution of primes.
    • 18. Other problems and results concerning primes.
    • CHAPTER III: THEORY OF CONGRUENCES
    • 19. Definitions and fundamental properties.
    • 20. Residue classes and residue systems.
    • 21. Fermat's theorem and its generalization by Euler.
    • 22. Algebraic congruences and functional congruences.
    • 23. Linear congruences.
    • 24. Algebraic congruences to a prime modulus.
    • 25. Prime divisors of integral polynomials.
    • 26. Algebraic congruences to a composite modulus.
    • 27. Algebraic congruences to a prime-power modulus.
    • 28. Numerical examples of solution of algebraic congruences.
    • 29. Divisibility of integral polynomials with regard to a primemodulus.
    • 30. Wilson's theorem and its generalization.
    • 31. Exponent of an integer modulo n.
    • 32. Moduli having primitive roots.
    • 33. The index calculus
    • 34. Power residues. Binomial congruences.
    • 35. Polynomials representing integers.
    • 36. Thue's remainder theorem and its generalization by Scholz.
    • Exercises
    • CHAPTER IV: THEORY OF QUADRATIC RESIDUES
    • 37. The general quadratic congruence.
    • 38. Euler's criterion and Legendre's symbol.
    • 39. On the solvability of the congruences x2 ≡ ± 2 (mod p ).
    • 40. Gauss's lemma.
    • 41. The quadratic reciprocity law.
    • 42. Jacobi's symbol and the generalization of the reciprocity law.
    • 43. The prime divisors of quadratic polynomials.
    • 44. Primes in special arithmetical progressions.
    • CHAPTER V: ARITHMETICAL PROPERTIES OF THE ROOTS OF UNITY
    • 45. The roots of unity.
    • 46. The cyclotomic polynomial.
    • 47. Irreducibility of the cyclotomic polynomial.
    • 48. The prime divisors of the cyclotomic polynomial.
    • 49. A theorem of Bauer on the prime divisors of certain polynomials.
    • 50. On the primes of the form ny - I.
    • 51. Some trigonometrical products.
    • 52. A polynomial identity of Gauss.
    • 53. The Gaussian sums.
    • Exercises
    • CHAPTER VI: DIOPHANTINE EQUATIONS OF THE SECOND DEGREE
    • 54. The representation of integers as sums of integral squares.
    • 55. Bachet's theorem.
    • 56. The Diophantine equation x2- Dy2= I.
    • 57. The Diophantine equation x2- Dy2 = -1.
    • 58. The Diophantine equation u2- D v2 = C
    • 59. Lattice points on conics.
    • 60. Rational points in the plane and on conics.
    • 61. The Diophantine equation ax2 + by2 + cz2 = 0.
    • CHAPTER VII: DIOPHANTINE EQUATIONS OF HIGHER DEGREE
    • 62. Some Diophantine equations of the fourth degree with three unknowns.
    • 63. The Diophantine equation 2x4- y4= z2.
    • 64. The quadratic fields K (√-1), K (√- 2) and K (√-3).
    • 65. The Diophantine equation ξ3 + n3 + ζ3 = 0 and analogous equations.
    • 66. Diophantine equations of the third degree with an infinityof solutions.
    • 67. The Diophantine equation x7 + y7 + z7 = 0.
    • 68, Fermat's last theorem,
    • 69. Rational points on plane algebraic curves. Mordell's theorem.
    • 70. Lattice points on plane algebraic curves. Theorems of Thue and Siegel.
    • Exercises
    • CHAPTER VIII: THE PRIME NUMBER THEOREM
    • 71. Lemmata on the order of magnitude of some finite sums.
    • 72. Lemmata on the Mohius function and some related functions.
    • 73. Further lemmata. Proof of Selherg's formula.
    • 74. An elementary proof of the prime number theorem.
    • Exercises
    • Table
    • The fundamental solutions of the equations x2- D y2 = ± 1 for D ⫹ 103.
    • NAME INDEX
    • SUBJECT INDEX
    • Back Cover
  • Reviews
     
     

    • This is a very readable introduction to number theory, with particular emphasis on diophantine equations, and requires only a school knowledge of mathematics. The exposition is admirably clear. More advanced or recent work is cited as background, where relevant ... [T]here are welcome novelties: Gauss's own evaluation of Gauss's sums, which is still perhaps the most elegant, is reproduced apparently for the first time. There are 180 examples, many of considerable interest, some of these being little known.
      Mathematical Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
AMS Chelsea Publishing
Volume: 163; 1964; 309 pp
MSC: Primary 11

A special feature of Nagell's well-known text is the rather extensive treatment of Diophantine equations of second and higher degree. A large number of non-routine problems are given.
  • Front Cover
  • PREFACE
  • CONTENTS
  • CHAPTER I: DIVISIBILITY
  • 1. Divisors.
  • 2. Remainders.
  • 3. Primes.
  • 4. The fundamental theorem.
  • 5. Least common multiple and greatest common divisor.
  • 6. Moduls, rings and fields.
  • 7. Euclid's algorithm.
  • 8. Relatively prime numbers. Euler's φ-function.
  • 9. Arithmetical functions.
  • 10. Diophantine equations of the first degree.
  • 11. Lattice points and point lattices.
  • 12. Irrational numbers.
  • 13. Irrationality of the numbers e and :rr.
  • Exercises
  • CHAPTER II: ON THE DISTRIBUTION OF PRIMES
  • 14. Some lemmata.
  • 15. General remarks. The sieve of Eratosthenes.
  • 16. The function π(x).
  • 17. Some elementary results on the distribution of primes.
  • 18. Other problems and results concerning primes.
  • CHAPTER III: THEORY OF CONGRUENCES
  • 19. Definitions and fundamental properties.
  • 20. Residue classes and residue systems.
  • 21. Fermat's theorem and its generalization by Euler.
  • 22. Algebraic congruences and functional congruences.
  • 23. Linear congruences.
  • 24. Algebraic congruences to a prime modulus.
  • 25. Prime divisors of integral polynomials.
  • 26. Algebraic congruences to a composite modulus.
  • 27. Algebraic congruences to a prime-power modulus.
  • 28. Numerical examples of solution of algebraic congruences.
  • 29. Divisibility of integral polynomials with regard to a primemodulus.
  • 30. Wilson's theorem and its generalization.
  • 31. Exponent of an integer modulo n.
  • 32. Moduli having primitive roots.
  • 33. The index calculus
  • 34. Power residues. Binomial congruences.
  • 35. Polynomials representing integers.
  • 36. Thue's remainder theorem and its generalization by Scholz.
  • Exercises
  • CHAPTER IV: THEORY OF QUADRATIC RESIDUES
  • 37. The general quadratic congruence.
  • 38. Euler's criterion and Legendre's symbol.
  • 39. On the solvability of the congruences x2 ≡ ± 2 (mod p ).
  • 40. Gauss's lemma.
  • 41. The quadratic reciprocity law.
  • 42. Jacobi's symbol and the generalization of the reciprocity law.
  • 43. The prime divisors of quadratic polynomials.
  • 44. Primes in special arithmetical progressions.
  • CHAPTER V: ARITHMETICAL PROPERTIES OF THE ROOTS OF UNITY
  • 45. The roots of unity.
  • 46. The cyclotomic polynomial.
  • 47. Irreducibility of the cyclotomic polynomial.
  • 48. The prime divisors of the cyclotomic polynomial.
  • 49. A theorem of Bauer on the prime divisors of certain polynomials.
  • 50. On the primes of the form ny - I.
  • 51. Some trigonometrical products.
  • 52. A polynomial identity of Gauss.
  • 53. The Gaussian sums.
  • Exercises
  • CHAPTER VI: DIOPHANTINE EQUATIONS OF THE SECOND DEGREE
  • 54. The representation of integers as sums of integral squares.
  • 55. Bachet's theorem.
  • 56. The Diophantine equation x2- Dy2= I.
  • 57. The Diophantine equation x2- Dy2 = -1.
  • 58. The Diophantine equation u2- D v2 = C
  • 59. Lattice points on conics.
  • 60. Rational points in the plane and on conics.
  • 61. The Diophantine equation ax2 + by2 + cz2 = 0.
  • CHAPTER VII: DIOPHANTINE EQUATIONS OF HIGHER DEGREE
  • 62. Some Diophantine equations of the fourth degree with three unknowns.
  • 63. The Diophantine equation 2x4- y4= z2.
  • 64. The quadratic fields K (√-1), K (√- 2) and K (√-3).
  • 65. The Diophantine equation ξ3 + n3 + ζ3 = 0 and analogous equations.
  • 66. Diophantine equations of the third degree with an infinityof solutions.
  • 67. The Diophantine equation x7 + y7 + z7 = 0.
  • 68, Fermat's last theorem,
  • 69. Rational points on plane algebraic curves. Mordell's theorem.
  • 70. Lattice points on plane algebraic curves. Theorems of Thue and Siegel.
  • Exercises
  • CHAPTER VIII: THE PRIME NUMBER THEOREM
  • 71. Lemmata on the order of magnitude of some finite sums.
  • 72. Lemmata on the Mohius function and some related functions.
  • 73. Further lemmata. Proof of Selherg's formula.
  • 74. An elementary proof of the prime number theorem.
  • Exercises
  • Table
  • The fundamental solutions of the equations x2- D y2 = ± 1 for D ⫹ 103.
  • NAME INDEX
  • SUBJECT INDEX
  • Back Cover

  • This is a very readable introduction to number theory, with particular emphasis on diophantine equations, and requires only a school knowledge of mathematics. The exposition is admirably clear. More advanced or recent work is cited as background, where relevant ... [T]here are welcome novelties: Gauss's own evaluation of Gauss's sums, which is still perhaps the most elegant, is reproduced apparently for the first time. There are 180 examples, many of considerable interest, some of these being little known.
    Mathematical Reviews
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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