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Binary Quadratic Forms by Johannes Buchmann,Ulrich Vollmer Pdf
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.
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.
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.
The Algebraic and Geometric Theory of Quadratic Forms by Richard S. Elman,Nikita Karpenko,Alexander Merkurjev Pdf
This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given. Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.
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.
The Sensual (quadratic) Form by John Horton Conway Pdf
John Horton Conway's unique approach to quadratic forms was the subject of the Hedrick Lectures that he gave in August of 1991 at the Joint Meetings of the Mathematical Association of America and the American Mathematical Society in Orono, Maine. This book presents the substance of those lectures. The book should not be thought of as a serious textbook on the theory of quadratic forms. It consists rather of a number of essays on particular aspects of quadratic forms that have interested the author. The lectures are self-contained and will be accessible to the generally informed reader who has no particular background in quadratic form theory. The minor exceptions should not interrupt the flow of ideas. The afterthoughts to the lectures contain discussion of related matters that occasionally presuppose greater knowledge.
An Illustrated Theory of Numbers by Martin H. Weissman Pdf
News about this title: — Author Marty Weissman has been awarded a Guggenheim Fellowship for 2020. (Learn more here.) — Selected as a 2018 CHOICE Outstanding Academic Title — 2018 PROSE Awards Honorable Mention An Illustrated Theory of Numbers gives a comprehensive introduction to number theory, with complete proofs, worked examples, and exercises. Its exposition reflects the most recent scholarship in mathematics and its history. Almost 500 sharp illustrations accompany elegant proofs, from prime decomposition through quadratic reciprocity. Geometric and dynamical arguments provide new insights, and allow for a rigorous approach with less algebraic manipulation. The final chapters contain an extended treatment of binary quadratic forms, using Conway's topograph to solve quadratic Diophantine equations (e.g., Pell's equation) and to study reduction and the finiteness of class numbers. Data visualizations introduce the reader to open questions and cutting-edge results in analytic number theory such as the Riemann hypothesis, boundedness of prime gaps, and the class number 1 problem. Accompanying each chapter, historical notes curate primary sources and secondary scholarship to trace the development of number theory within and outside the Western tradition. Requiring only high school algebra and geometry, this text is recommended for a first course in elementary number theory. It is also suitable for mathematicians seeking a fresh perspective on an ancient subject.
The Arithmetic Theory of Quadratic Forms by Burton W Jones Pdf
This monograph presents the central ideas of the arithmetic theory of quadratic forms in self-contained form, assuming only knowledge of the fundamentals of matric theory and the theory of numbers. Pertinent concepts of p -adic numbers and quadratic ideals are introduced. It would have been possible to avoid these concepts, but the theory gains elegance as well as breadth by the introduction of such relationships. Some results, and many of the methods, are here presented for the first time. The development begins with the classical theory in the field of reals from the point of view of representation theory; for in these terms, many of the later objectives and methods may be revealed. The successive chapters gradually narrow the fields and rings until one has the tools at hand to deal with the classical problems in the ring of rational integers. The analytic theory of quadratic forms is not dealt with because of the delicate analysis involved. However, some of the more important results are stated and references are given.
Quaternion Orders, Quadratic Forms, and Shimura Curves by Montserrat Alsina,Pilar Bayer i Isant Pdf
Shimura curves are a far-reaching generalization of the classical modular curves. They lie at the crossroads of many areas, including complex analysis, hyperbolic geometry, algebraic geometry, algebra, and arithmetic. This monograph presents Shimura curves from a theoretical and algorithmic perspective. The main topics are Shimura curves defined over the rational number field, the construction of their fundamental domains, and the determination of their complex multiplicationpoints. The study of complex multiplication points in Shimura curves leads to the study of families of binary quadratic forms with algebraic coefficients and to their classification by arithmetic Fuchsian groups. In this regard, the authors develop a theory full of new possibilities that parallels Gauss'theory on the classification of binary quadratic forms with integral coefficients by the action of the modular group. This is one of the few available books explaining the theory of Shimura curves at the graduate student level. Each topic covered in the book begins with a theoretical discussion followed by carefully worked-out examples, preparing the way for further research.
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
