The Class Number of Binary Quadratic Forms ...
Author : George Hoffman Cresse
Publisher :
Page : 117 pages
File Size : 37,72 MB
Release : 1923
Category : Forms, Binary
ISBN :
Author : George Hoffman Cresse
Publisher :
Page : 117 pages
File Size : 37,72 MB
Release : 1923
Category : Forms, Binary
ISBN :
Author : George Hoffman Cresse
Publisher :
Page : 105 pages
File Size : 37,27 MB
Release : 1923
Category :
ISBN :
Author : Duncan A. Buell
Publisher : Springer Science & Business Media
Page : 249 pages
File Size : 14,16 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1461245427
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.
Author : George Hoffman Cresse
Publisher :
Page : pages
File Size : 20,63 MB
Release : 2004-01-01
Category :
ISBN : 9781418163419
Author : Johannes Buchmann
Publisher : Springer Science & Business Media
Page : 328 pages
File Size : 39,88 MB
Release : 2007-06-22
Category : Mathematics
ISBN : 3540463682
The book deals with algorithmic problems related to binary quadratic forms. It uniquely focuses on the algorithmic aspects of the theory. The book introduces the reader to important areas of number theory such as diophantine equations, reduction theory of quadratic forms, geometry of numbers and algebraic number theory. The book explains applications to cryptography and requires only basic mathematical knowledge. The author is a world leader in number theory.
Author : George Hoffmann Cresse
Publisher :
Page : pages
File Size : 43,95 MB
Release : 1923
Category :
ISBN :
Author : J. W. S. Cassels
Publisher : Courier Dover Publications
Page : 429 pages
File Size : 23,11 MB
Release : 2008-08-08
Category : Mathematics
ISBN : 0486466701
Exploration of quadratic forms over rational numbers and rational integers offers elementary introduction. Covers quadratic forms over local fields, forms with integral coefficients, reduction theory for definite forms, more. 1968 edition.
Author : Franz Halter-Koch
Publisher : CRC Press
Page : 431 pages
File Size : 18,8 MB
Release : 2013-06-17
Category : Mathematics
ISBN : 1466591846
Quadratic Irrationals: An Introduction to Classical Number Theory gives a unified treatment of the classical theory of quadratic irrationals. Presenting the material in a modern and elementary algebraic setting, the author focuses on equivalence, continued fractions, quadratic characters, quadratic orders, binary quadratic forms, and class groups.T
Author : David A. Cox
Publisher : John Wiley & Sons
Page : 372 pages
File Size : 19,16 MB
Release : 2011-10-24
Category : Mathematics
ISBN : 1118031008
Modern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of field theory and its intimate connection with complex multiplication. While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included. The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively.
Author : Eva Bayer-Fluckiger
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 38,86 MB
Release : 2000
Category : Mathematics
ISBN : 0821827790
This volume outlines the proceedings of the conference on "Quadratic Forms and Their Applications" held at University College Dublin. It includes survey articles and research papers ranging from applications in topology and geometry to the algebraic theory of quadratic forms and its history. Various aspects of the use of quadratic forms in algebra, analysis, topology, geometry, and number theory are addressed. Special features include the first published proof of the Conway-Schneeberger Fifteen Theorem on integer-valued quadratic forms and the first English-language biography of Ernst Witt, founder of the theory of quadratic forms.