Binary quadratic forms : classical theory and modern computations

Includes index.

Includes bibliography (p. 213-221)

1 Elementary Concepts
2 Reduction of Positive Definite Forms
3 Indefinite Forms
3.1 Reduction, Cycles
3.2 Automorphs, Pell's Equation
3.3 Continued Fractions and Indefinite Forms
4 The Class Group
4.1 Representation and Genera
4.2 Composition Algorithms
4.3 Generic Characters Revisited
4.4 Representation of Integers
5 Miscellaneous Facts
5.1 Class Number Computations
5.2 Extreme Cases and Asymptotic Results
6 Quadratic Number Fields
6.1 Basic Algebraic Definitions
6.2 Algebraic Numbers and Quadratic Fields
6.3 Ideals in Quadratic Fields
6.4 Binary Quadratic Forms and Classes of Ideals
6.5 History
7 Composition of Forms
7.1 Nonfundamental Discriminants
7.2 The General Problem of Composition
7.3 Composition in Different Orders
8 Miscellaneous Facts II
8.1 The Cohen-Lenstra Heuristics
8.2 Decomposing Class Groups
8.3 Specifying Subgroups of Class Groups
9 The 2-Sylow Subgroup
9.1 Classical Results on the Pell Equation
9.2 Modern Results
9.3 Reciprocity Laws
9.4 Special References for Chapter 9
10 Factoring with Binary Quadratic Forms
10.1 Classical Methods
10.2 SQUFOF
10.3 CLASNO
10.4 SPAR
10.5 CFRAC
10.6 A General Analysis

The 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.