## Binary Quadratic Forms: Classical Theory and Modern ComputationsThe first coherent exposition of the theory of binary quadratic forms was given by Gauss in the Disqnisitiones Arithmeticae. During the nine teenth century, as the theory of ideals and the rudiments of algebraic number theory were developed, it became clear that this theory of bi nary quadratic forms, so elementary and computationally explicit, was indeed just a special case of a much more elega,nt and abstract theory which, unfortunately, is not computationally explicit. In recent years the original theory has been laid aside. Gauss's proofs, which involved brute force computations that can be done in what is essentially a two dimensional vector space, have been dropped in favor of n-dimensional arguments which prove the general theorems of algebraic number the ory. In consequence, this elegant, yet pleasantly simple, theory has been neglected even as some of its results have become extremely useful in certain computations. I find this neglect unfortunate, because binary quadratic forms have two distinct attractions. First, the subject involves explicit computa tion and many of the computer programs can be quite simple. The use of computers in experimenting with examples is both meaningful and enjoyable; one can actually discover interesting results by com puting examples, noticing patterns in the "data," and then proving that the patterns result from the conclusion of some provable theorem. |

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### Contents

Elementary Concepts | 1 |

Reduction of Positive Definite Forms | 13 |

Indefinite Forms | 21 |

32 Automorphs Pells Equation | 31 |

33 Continued Fractions and Indefinite Forms | 35 |

The Class Group | 49 |

42 Composition Algorithms | 55 |

43 Generic Characters Revisited | 66 |

82 Decomposing Class Groups | 141 |

83 Specifying Subgroups of Class Groups | 147 |

832 Exact and Exotic Groups | 153 |

The 2Sylow Subgroup | 159 |

91 Classical Results on the Pell Equation | 162 |

92 Modern Results | 170 |

93 Reciprocity Laws | 183 |

94 Special References for Chapter 9 | 185 |

44 Representation of Integers | 74 |

Miscellaneous Facts | 77 |

52 Extreme Cases and Asymptotic Results | 81 |

Quadratic Number Fields | 87 |

62 Algebraic Numbers and Quadratic Fields | 89 |

63 Ideals in Quadratic Fields | 94 |

64 Binary Quadratic Forms and Classes of Ideals | 103 |

65 History | 107 |

Composition of Forms | 109 |

72 The General Problem of Composition | 120 |

73 Composition in Different Orders | 129 |

Miscellaneous Facts II | 135 |

81 The CohenLenstra Heuristics | 137 |

### Other editions - View all

Binary Quadratic Forms: Classical Theory and Modern Computations Duncan A. Buell Limited preview - 2012 |

Binary Quadratic Forms: Classical Theory and Modern Computations Duncan A. Buell No preview available - 1989 |

### Common terms and phrases

algebraic integers ambiguous class ancipital forms automorph binary quadratic forms biquadratic reciprocity canonical basis Chapter choose class group class number classes of forms congruence continued fraction Crelle cyclic Daniel Shanks define definite forms Dirichlet Disc H S Disc H S Disc Disc H/G G discriminantal divisors elements Emma Lehmer equation x2 exactly exist Ezra Brown form f form of discriminant fundamental discriminants fundamental solution G Disc H/G Gauss genera H S Disc H S Disc H S H/G G Disc identity infinitely integer coefficients Journal of Number matrix modular group modulo multiplication narrow class group negative discriminants negative Pell equation noncyclic Number Theory odd discriminants odd number odd prime p-Sylow subgroup Pell equation positive discriminants prime factors primitive forms principal cycle principal form principal genus Proof Proposition prove quadratic character quadratic fields quadratic number fields rational integers reduced forms relatively prime representation represents solvable