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Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope topics from analysis, modern algebra, and discrete mathematics are all included.
The book is divided into two parts. Part A covers key concepts of number theory and could serve as a first course on the subject. Part B delves into more advanced topics and an exploration of related mathematics. Part B contains, for example, complete proofs of the Hasse-Minkowski theorem and the prime number theorem, as well as self-contained accounts of the character theory of finite groups and the theory of elliptic functions.
The prerequisites for this self-contained text are elements from linear algebra. Valuable references for the reader are collected at the end of each chapter. It is suitable as an introduction to higher level mathematics for undergraduates, or for self-study.
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Preface to the Second Edition
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
Divisibility
Greatest Common Divisors
The Bézout Identity
Polynomials
Euclidean Domains
Congruences
Sums of Squares
More on Divisibility
The Law of Quadratic Reciprocity
Quadratic Fields
Multiplicative Functions
Linear Diophantine Equations
Continued Fractions and Their Uses
The Continued Fraction Algorithm
Diophantine Approximation
Periodic Continued Fractions
Quadratic Diophantine Equations
The Modular Group
Non-Euclidean Geometry
Complements
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
Hensel's p-adic Numbers
Valued Fields
Equivalence
Completions
Non-Archimedean Valued Fields
Hensel's Lemma
Locally Compact Valued Fields
The Arithmetic of Quadratic Forms
Quadratic Spaces
The Hilbert Symbol
The Hasse–Minkowski Theorem
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
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
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
Uniform Distribution and Ergodic Theory
Uniform Distribution
Discrepancy
Birkhoff's Ergodic Theorem
Elliptic Functions
Elliptic Integrals
The Arithmetic-Geometric Mean
Elliptic Functions
Theta Functions
Jacobian Elliptic Functions
The Modular Function
Connections with Number Theory
Sums of Squares
Partitions
Cubic Curves
Mordell's Theorem
Notations
Axioms