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
• Divisibility
• Greatest Common Divisors
• The B´ezout Identity
• Polynomials
• Euclidean Domains
• Congruences
• Sums of Squares
• Further Remarks
• Selected References
• More on Divisibility
• The Law of Quadratic Reciprocity
• Multiplicative Functions
• Linear Diophantine Equations
• Further Remarks
• Selected References
• IV Continued Fractions and Their Uses
• The Continued Fraction Algorithm
• Diophantine Approximation
• Periodic Continued Fractions
• The Modular Group
• Non-Euclidean Geometry
• Complements
• Further Remarks
• Selected References
• What is a Determinant?
• The Art of Weighing
• Some Matrix Theory
• Application to Hadamard’s Determinant Problem
• Designs
• Groups and Codes
• Further Remarks
• Selected References
• 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
• 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
• 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
• 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
• 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