Number Theory An Introduction to Mathematics Second Edition

Number Theory An Introduction to Mathematics by W. Coppel

Number Theory An Introduction to Mathematics Second Edition by W. A. Coppel

Contents of Number Theory Introduction to Mathematics

  • Part A. The Expanding Universe of Numbers
  • Sets, Relations and Mappings
  • Natural Numbers
  • Integers and Rational Numbers
  • Real Numbers
  • Metric Spaces
  • Complex Numbers
  • Quaternions and Octonions
  • Groups
  • Rings and Fields
  • Vector Spaces and Associative Algebras
  • Inner Product Spaces
  • Further Remarks
  • Selected References
  • Additional References
  • Divisibility
  • Greatest Common Divisors
  • The B´ezout Identity
  • Polynomials
  • Euclidean Domains
  • Congruences
  • Sums of Squares
  • Further Remarks
  • Selected References
  • Additional References
  • More on Divisibility
  • The Law of Quadratic Reciprocity
  • Quadratic Fields
  • Multiplicative Functions
  • Linear Diophantine Equations
  • Further Remarks
  • Selected References
  • Additional References
  • IV Continued Fractions and Their Uses
  • The Continued Fraction Algorithm
  • Diophantine Approximation
  • Periodic Continued Fractions
  • Quadratic Diophantine Equations
  • The Modular Group
  • Non-Euclidean Geometry
  • Complements
  • Further Remarks
  • Selected References
  • Additional References
  • Hadamard’s Determinant Problem
  • What is a Determinant?
  • Hadamard Matrices
  • The Art of Weighing
  • Some Matrix Theory
  • Application to Hadamard’s Determinant Problem
  • Designs
  • Groups and Codes
  • Further Remarks
  • Selected References
  • VI Hensel’s p-adic Numbers
  • Valued Fields
  • Equivalence
  • Completions
  • Non-Archimedean Valued Fields
  • Hensel’s Lemma
  • Locally Compact Valued Fields
  • Further Remarks
  • Selected References
  • Part B The Arithmetic of Quadratic Forms
  • Quadratic Spaces
  • The Hilbert Symbol
  • The Hasse–Minkowski Theorem
  • Supplements
  • Further Remarks
  • Selected References
  • VIII The Geometry of Numbers
  • Minkowski’s Lattice Point Theorem
  • Lattices
  • Proof of the Lattice Point Theorem; Other Results
  • Voronoi Cells
  • Densest Packings
  • Mahler’s Compactness Theorem
  • Further Remarks
  • Selected References
  • Additional References
  • The Number of Prime Numbers
  • Finding the Problem
  • Chebyshev’s Functions
  • Proof of the Prime Number Theorem
  • The Riemann Hypothesis
  • Generalizations and Analogues
  • Alternative Formulations
  • Some Further Problems
  • Further Remarks
  • Selected References
  • Additional References
  • A Character Study
  • Primes in Arithmetic Progressions
  • Characters of Finite Abelian Groups
  • Proof of the Prime Number Theorem for Arithmetic Progressions
  • Representations of Arbitrary Finite Groups
  • Characters of Arbitrary Finite Groups
  • Induced Representations and Examples
  • Applications
  • Generalizations
  • Further Remarks
  • Selected References
  • Uniform Distribution and Ergodic Theory
  • Uniform Distribution
  • Discrepancy
  • Birkhoff’s Ergodic Theorem
  • Applications
  • Recurrence
  • Further Remarks
  • Selected References
  • Additional Reference
  • Elliptic Functions
  • Elliptic Integrals
  • The Arithmetic-Geometric Mean
  • Elliptic Functions
  • Theta Functions
  • Jacobian Elliptic Functions
  • The Modular Function
  • Further Remarks
  • Selected References
  • Connections with Number Theory
  • Sums of Squares
  • Partitions
  • Cubic Curves
  • Mordell’s Theorem
  • Further Results and Conjectures
  • Some Applications
  • Further Remarks
  • Selected References
  • Additional References

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