Quadratic irrationals : an introduction to classical number theory / Franz Halter-Koch.

Includes bibliographical references (p. 407-410) and index.

Chapter 1. Quadratic Irrationals
1.1 Quadratic irrationals, quadratic number fields and discriminants
1.2 The modular group
1.3 Reduced quadratic irrationals
1.4 Two short tables of class numbers
Chapter 2. Continued Fractions
2.1 General theory of continued fractions
2.2 Continued fractions of quadratic irrationals I: General theory
2.3 Continued fractions of quadratic irrationals II: Special types
Chapter 3 Quadratic Residues and Gauss Sums
3.1 Elementary theory of power residues
3.2 Gauss and Jacobi sums
3.3 The quadratic reciprocity law
3.4 Sums of two squares
3.5 Kronecker and quadratic symbols
Chapter 4. L-Series and Dirichlet's Prime Number Theorem
4.1 Preliminaries and some elementary cases
4.2 Multiplicative functions
4.3 Dirichlet L-functions and proof of Dirichlet's theorem
4.4 Summation of L-series
Chapter 5. Quadratic Orders
5.1 Lattices and orders in quadratic number fields
5.2 Units in quadratic orders
5.3 Lattices and (invertible) fractional ideals in quadratic orders 5.4 Structure of ideals in quadratic orders
5.5 Class groups and class semigroups
5.6 Ambiguous ideals and ideal classes
5.7 An application: Some binary Diophantine equations
5.8 Prime ideals and multiplicative ideal theory
5.9 Class groups of quadratic orders
Chapter 6. Binary Quadratic Forms
6.1 Elementary definitions and equivalence relations
6.2 Representation of integers
6.3 Reduction
6.4 Composition
6.5 Theory of genera
6.6 Ternary quadratic forms
6.7 Sums of squares
Chapter 7. Cubic and biquadratic residues
7.1 The cubic Jacobi symbol
7.2 The cubic reciprocity law
7.3 The biquadratic Jacobi symbol
7.4 The biquadratic reciprocity law
7.5 Rational biquadratic reciprocity laws
7.6 A biquadratic class group character and applications
Chapter 8. Class Groups
8.1 The analytic class number formula
8.2 L-functions of quadratic orders
8.3 Ambiguous classes and classes of order divisibility by 4
8.4 Discriminants with cyclic 2-class group: Divisibility by 8 and 16
Appendix A: Review of Elementary Algebra and Number Theory Appendix B: Some Results from Analysis
Bibliography
List of Symbols
Subject Index

"This work focuses on the number theory of quadratic irrationalities in various forms, including continued fractions, orders in quadratic number fields, and binary quadratic forms. It presents classical results obtained by the famous number theorists Gauss, Legendre, Lagrange, and Dirichlet. Collecting information previously scattered in the literature, the book covers the classical theory of continued fractions, quadratic orders, binary quadratic forms, and class groups based on the concept of a quadratic irrational"--