Foreword Introduction and Preface to the Reader Notations 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 sumsThe 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 Composition 6.5 Theory of genera 6.6 Ternary quadrat?c 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 Amblguous classes and classes of order divisibility by
8.4 Discriminants with cyclic 2-class group: Divisibility by 8 and 16 Appendix A Review of elementary algebra and number theory A.1 Fundamentals of group theory A.2 Fundamentals of ring theory A.3 Elementary arithmetic in Z A.4 Lattices A.5 Finite abelian groups A.6 Prime residue class groups A.7 Roots of unity and characters of finite abelian groups A.8 Factorization in integral domains A.9 Algebraic integers Appendix B Some results from analysis B.1 Notational conventions and results from complex analysis B.2 Further analytic tools Bibliography List of Symbols Subject Index 編輯手記
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