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A Comprehensive Course in Number Theory

Book Description

Developed from the author's popular text, A Concise Introduction to the Theory of Numbers, this book provides a comprehensive initiation to all the major branches of number theory. Beginning with the rudiments of the subject, the author proceeds to more advanced topics, including elements of cryptography and primality testing, an account of number fields in the classical vein including properties of their units, ideals and ideal classes, aspects of analytic number theory including studies of the Riemann zeta-function, the prime-number theorem and primes in arithmetical progressions, a description of the Hardy-Littlewood and sieve methods from respectively additive and multiplicative number theory and an exposition of the arithmetic of elliptic curves. The book includes many worked examples, exercises and further reading. Its wider coverage and versatility make this book suitable for courses extending from the elementary to beginning graduate studies.

Table of Contents

  1. Cover
  2. About The Book
  3. Title
  4. Copyright
  5. Contents
  6. Preface
  7. Introduction
  8. 1 Divisibility
    1. 1.1 Foundations
    2. 1.2 Division algorithm
    3. 1.3 Greatest common divisor
    4. 1.4 Euclid’s algorithm
    5. 1.5 Fundamental theorem
    6. 1.6 Properties of the primes
    7. 1.7 Further reading
    8. 1.8 Exercises
  9. 2 Arithmetical functions
    1. 2.1 The function [x]
    2. 2.2 Multiplicative functions
    3. 2.3 Euler’s (totient) function ϕ(n)
    4. 2.4 The Möbius function μ(n)
    5. 2.5 The functions τ(n) and σ(n)
    6. 2.6 Average orders
    7. 2.7 Perfect numbers
    8. 2.8 The Riemann zeta-function
    9. 2.9 Further reading
    10. 2.10 Exercises
  10. 3 Congruences
    1. 3.1 Definitions
    2. 3.2 Chinese remainder theorem
    3. 3.3 The theorems of Fermat and Euler
    4. 3.4 Wilson’s theorem
    5. 3.5 Lagrange’s theorem
    6. 3.6 Primitive roots
    7. 3.7 Indices
    8. 3.8 Further reading
    9. 3.9 Exercises
  11. 4 Quadratic residues
    1. 4.1 Legendre’s symbol
    2. 4.2 Euler’s criterion
    3. 4.3 Gauss’ lemma
    4. 4.4 Law of quadratic reciprocity
    5. 4.5 Jacobi’s symbol
    6. 4.6 Further reading
    7. 4.7 Exercises
  12. 5 Quadratic forms
    1. 5.1 Equivalence
    2. 5.2 Reduction
    3. 5.3 Representations by binary forms
    4. 5.4 Sums of two squares
    5. 5.5 Sums of four squares
    6. 5.6 Further reading
    7. 5.7 Exercises
  13. 6 Diophantine approximation
    1. 6.1 Dirichlet’s theorem
    2. 6.2 Continued fractions
    3. 6.3 Rational approximations
    4. 6.4 Quadratic irrationals
    5. 6.5 Liouville’s theorem
    6. 6.6 Transcendental numbers
    7. 6.7 Minkowski’s theorem
    8. 6.8 Further reading
    9. 6.9 Exercises
  14. 7 Quadratic fields
    1. 7.1 Algebraic number fields
    2. 7.2 The quadratic field
    3. 7.3 Units
    4. 7.4 Primes and factorization
    5. 7.5 Euclidean fields
    6. 7.6 The Gaussian field
    7. 7.7 Further reading
    8. 7.8 Exercises
  15. 8 Diophantine equations
    1. 8.1 The Pell equation
    2. 8.2 The Thue equation
    3. 8.3 The Mordell equation
    4. 8.4 The Fermat equation
    5. 8.5 The Catalan equation
    6. 8.6 The abc-conjecture
    7. 8.7 Further reading
    8. 8.8 Exercises
  16. 9 Factorization and primality testing
    1. 9.1 Fermat pseudoprimes
    2. 9.2 Euler pseudoprimes
    3. 9.3 Fermat factorization
    4. 9.4 Fermat bases
    5. 9.5 The continued-fraction method
    6. 9.6 Pollard’s method
    7. 9.7 Cryptography
    8. 9.8 Further reading
    9. 9.9 Exercises
  17. 10 Number fields
    1. 10.1 Introduction
    2. 10.2 Algebraic numbers
    3. 10.3 Algebraic number fields
    4. 10.4 Dimension theorem
    5. 10.5 Norm and trace
    6. 10.6 Algebraic integers
    7. 10.7 Basis and discriminant
    8. 10.8 Calculation of bases
    9. 10.9 Further reading
    10. 10.10 Exercises
  18. 11 Ideals
    1. 11.1 Origins
    2. 11.2 Definitions
    3. 11.3 Principal ideals
    4. 11.4 Prime ideals
    5. 11.5 Norm of an ideal
    6. 11.6 Formula for the norm
    7. 11.7 The different
    8. 11.8 Further reading
    9. 11.9 Exercises
  19. 12 Units and ideal classes
    1. 12.1 Units
    2. 12.2 Dirichlet’s unit theorem
    3. 12.3 Ideal classes
    4. 12.4 Minkowski’s constant
    5. 12.5 Dedekind’s theorem
    6. 12.6 The cyclotomic field
    7. 12.7 Calculation of class numbers
    8. 12.8 Local fields
    9. 12.9 Further reading
    10. 12.10 Exercises
  20. 13 Analytic number theory
    1. 13.1 Introduction
    2. 13.2 Dirichlet series
    3. 13.3 Tchebychev’s estimates
    4. 13.4 Partial summation formula
    5. 13.5 Mertens’ results
    6. 13.6 The Tchebychev functions
    7. 13.7 The irrationality of ζ(3)
    8. 13.8 Further reading
    9. 13.9 Exercises
  21. 14 On the zeros of the zeta-function
    1. 14.1 Introduction
    2. 14.2 The functional equation
    3. 14.3 The Euler product
    4. 14.4 On the logarithmic derivative of ζ(s)
    5. 14.5 The Riemann hypothesis
    6. 14.6 Explicit formula for ζ′(s)/ζ(s)
    7. 14.7 On certain sums
    8. 14.8 The Riemann–von Mangoldt formula
    9. 14.9 Further reading
    10. 14.10 Exercises
  22. 15 On the distribution of the primes
    1. 15.1 The prime-number theorem
    2. 15.2 Refinements and developments
    3. 15.3 Dirichlet characters
    4. 15.4 Dirichlet L-functions
    5. 15.5 Primes in arithmetical progressions
    6. 15.6 The class number formulae
    7. 15.7 Siegel’s theorem
    8. 15.8 Further reading
    9. 15.9 Exercises
  23. 16 The sieve and circle methods
    1. 16.1 The Eratosthenes sieve
    2. 16.2 The Selberg upper-bound sieve
    3. 16.3 Applications of the Selberg sieve
    4. 16.4 The large sieve
    5. 16.5 The circle method
    6. 16.6 Additive prime number theory
    7. 16.7 Further reading
    8. 16.8 Exercises
  24. 17 Elliptic curves
    1. 17.1 Introduction
    2. 17.2 The Weierstrass ℘-function
    3. 17.3 The Mordell–Weil group
    4. 17.4 Heights on elliptic curves
    5. 17.5 The Mordell–Weil theorem
    6. 17.6 Computing the torsion subgroup
    7. 17.7 Conjectures on the rank
    8. 17.8 Isogenies and endomorphisms
    9. 17.9 Further reading
    10. 17.10 Exercises
  25. Bibliography
  26. Index