Quadratic irrationals : An introduction to classical number theory

Includes index

Includes bibliographical references

Chapter 1 Quadratic irrationals 1 1.1 Quadratic irrational̀s, quadratic number fields and discriminants 1 1.2 The modular group 7 1.3 Reduced quadratic irrationals 16 1.4 Two short tables of class numbers 21 Chapter 2 Continued fractions 25 2.1 General theory of continued fractions 25 2.2 Continued fractions of quadratic irrationals I: General theory 38 2.3 Continued fractions of quadratic irrationals II: Special types 50 Chapter 3 Quadratic residues and Gauss sums 63 3.1 Elementary theory of power residues 63 3.2 Gauss and Jacobi sums 67 3.3 The quadratic reciprocity law 72 3.4 Sums of two squares 79 3.5 Kronecker and quadratic symbols 82 Chapter 4 L-series and Dirichlet's prime number theorem 99 4.1 Preliminaries and some elementary cases 99 4.2 Multiplicative functions 102 4.3 Dirichlet L-functions and proof of Dirichlet's theorem 106 4.4 Summation of L-series 112 Chapter 5 Quadratic orders 115 5.1 Lattices and orders in quadratic number fields 115 5.2 Units in quadratic orders 121 5.3 Lattices and (invertible) fractional ideals in quadratic orders 129 5.4 Structure of ideals in quadratic orders 132 5.5 Class groups and class semigroups 140 5.6 Ambiguous ideals and ideal classes 147 5.7 An application: Some binary Diophantine equations 160 5.8 Prime ideals and multiplicative ideal theory 174 5.9 Class groups of quadratic orders 179 Chapter 6 Binary quadratic forms 191 6.1 Elementary definitions and equivalence relations 191 6.2 Representation of integers 199 6.3 Reduction 210 6.4 Composition 213 6.5 Theory of genera 221 6.6 Ternary quadratic forms 240 6.7 Sums of squares 248 Chapter 7 Cubic and biquadratic residues 257 7.1 The cubic Jacobi symbol 257 7.2 The cubic reciprocity law 263 7.3 The biquadratic Jacobi symbol 271 7.4 The biquadratic reciprocity law 279 7.5 Rational biquadratic reciprocity laws 289 7.6 A biquadratic class group character and applications 306 Chapter 8 Class groups 321 8.1 The analytic class number formula 322 8.2 L-functions of quadratic orders 332 8.3 Ambiguous classes and classes of order divisibility by 4 340 8.4 Discriminants with cyclic 2-class group: Divisibility by 8 and 16 345.

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"-- Provided by publisher.