Binary Quadratic Forms: Classical Theory and Modern Computations

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Springer Science & Business Media, Aug 25, 1989 - Mathematics - 248 pages
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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. 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
Factoring with Binary Quadratic Forms
191
102 SQUFOF
197
103 CLASNO
202
104 SPAR
206
1042 SPAR
208
105 CFRAC
209
106 A General Analysis
212
Bibliography
213
Tables Negative Discriminants
223
Tables Positive Discriminants
235
Index
245
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