Ebook: Binary Quadratic Forms: Classical Theory and Modern Computations
Author: Duncan A. Buell (auth.)
- Tags: Number Theory, Combinatorics
- Year: 1989
- Publisher: Springer-Verlag New York
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-x
Elementary Concepts....Pages 1-12
Reduction of Positive Definite Forms....Pages 13-20
Indefinite Forms....Pages 21-48
The Class Group....Pages 49-75
Miscellaneous Facts....Pages 77-85
Quadratic Number Fields....Pages 87-107
Composition of Forms....Pages 109-133
Miscellaneous Facts II....Pages 135-157
The 2-Sylow Subgroup....Pages 159-189
Factoring with Binary Quadratic Forms....Pages 191-212
Back Matter....Pages 213-247
Content:
Front Matter....Pages i-x
Elementary Concepts....Pages 1-12
Reduction of Positive Definite Forms....Pages 13-20
Indefinite Forms....Pages 21-48
The Class Group....Pages 49-75
Miscellaneous Facts....Pages 77-85
Quadratic Number Fields....Pages 87-107
Composition of Forms....Pages 109-133
Miscellaneous Facts II....Pages 135-157
The 2-Sylow Subgroup....Pages 159-189
Factoring with Binary Quadratic Forms....Pages 191-212
Back Matter....Pages 213-247
....