Number Theory: An Introduction to Mathematics PDF

Content of Number Theory: An Introduction to Mathematics.

Part A
I The Expanding Universe of Numbers.
0 Sets, Relations and Mappings
1 Natural Numbers
2 Integers and Rational Numbers
3 Real Numbers
4 Metric Spaces
5 Complex Numbers
6 Quaternions and Octonions
7 Groups
8 Rings and Fields
9 Vector Spaces and Associative Algebras
10 Inner Product Spaces
11 Further Remarks
12 Selected References

II Divisibility
1 Greatest Common Divisors
2 The B´ezout Identity
3 Polynomials
4 Euclidean Domains
5 Congruences
6 Sums of Squares
7 Further Remarks
8 Selected References

viii Contents
III More on Divisibility
1 The Law of Quadratic Reciprocity
3 Multiplicative Functions
4 Linear Diophantine Equations
5 Further Remarks
6 Selected References

IV Continued Fractions and Their Uses
1 The Continued Fraction Algorithm
2 Diophantine Approximation
3 Periodic Continued Fractions
5 The Modular Group
6 Non-Euclidean Geometry
7 Complements
8 Further Remarks
9 Selected References

1 What is a Determinant?
3 The Art of Weighing
4 Some Matrix Theory
5 Application to Hadamard’s Determinant Problem
6 Designs
7 Groups and Codes
8 Further Remarks
9 Selected References

1 Valued Fields
2 Equivalence
3 Completions
4 Non-Archimedean Valued Fields
5 Hensel’s Lemma
6 Locally Compact Valued Fields
7 Further Remarks
8 Selected References
Contents ix

Part B
VII The Arithmetic of Quadratic Forms
2 The Hilbert Symbol
3 The Hasse–Minkowski Theorem
4 Supplements
5 Further Remarks
6 Selected References
VIII The Geometry of Numbers
1 Minkowski’s Lattice Point Theorem
2 Lattices
3 Proof of the Lattice Point Theorem; Other Results
4 Voronoi Cells
5 Densest Packings
6 Mahler’s Compactness Theorem
7 Further Remarks
8 Selected References

IX The Number of Prime Numbers
1 Finding the Problem
2 Chebyshev’s Functions
3 Proof of the Prime Number Theorem
4 The Riemann Hypothesis
5 Generalizations and Analogues
6 Alternative Formulations
7 Some Further Problems
8 Further Remarks
9 Selected References

X A Character Study
1 Primes in Arithmetic Progressions
2 Characters of Finite Abelian Groups
3 Proof of the Prime Number Theorem for Arithmetic Progressions
4 Representations of Arbitrary Finite Groups
5 Characters of Arbitrary Finite Groups
6 Induced Representations and Examples
7 Applications
8 Generalizations
9 Further Remarks
10 Selected References

x Contents
XI Uniform Distribution and Ergodic Theory
1 Uniform Distribution
2 Discrepancy
3 Birkhoff’s Ergodic Theorem
4 Applications
5 Recurrence
6 Further Remarks
7 Selected References

XII Elliptic Functions
1 Elliptic Integrals
2 The Arithmetic-Geometric Mean
3 Elliptic Functions
4 Theta Functions
5 Jacobian Elliptic Functions .
6 The Modular Function
7 Further Remarks
8 Selected References

XIII Connections with Number Theory
1 Sums of Squares
2 Partitions
3 Cubic Curves
4 Mordell’s Theorem
5 Further Results and Conjectures
6 Some Applications
7 Further Remarks
8 Selected References