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
