The Composition Of Binary Quadratic Forms

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Binary Quadratic Forms

Author : Duncan A. Buell
Publisher : Springer Science & Business Media
ISBN 13 : 1461245427
Total Pages : 249 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Binary Quadratic Forms by : Duncan A. Buell

Download or read book Binary Quadratic Forms written by Duncan A. Buell and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Binary Quadratic Forms

Author : Johannes Buchmann
Publisher : Springer Science & Business Media
ISBN 13 : 3540463682
Total Pages : 328 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Binary Quadratic Forms by : Johannes Buchmann

Download or read book Binary Quadratic Forms written by Johannes Buchmann and published by Springer Science & Business Media. This book was released on 2007-06-22 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Rational Quadratic Forms

Author : J. W. S. Cassels
Publisher : Courier Dover Publications
ISBN 13 : 0486466701
Total Pages : 429 pages
Book Rating : 4.4/5 (864 download)

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Book Synopsis Rational Quadratic Forms by : J. W. S. Cassels

Download or read book Rational Quadratic Forms written by J. W. S. Cassels and published by Courier Dover Publications. This book was released on 2008-08-08 with total page 429 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Author : Richard S. Elman
Publisher : American Mathematical Soc.
ISBN 13 : 9780821873229
Total Pages : 456 pages
Book Rating : 4.8/5 (732 download)

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Book Synopsis The Algebraic and Geometric Theory of Quadratic Forms by : Richard S. Elman

Download or read book The Algebraic and Geometric Theory of Quadratic Forms written by Richard S. Elman and published by American Mathematical Soc.. This book was released on 2008-07-15 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Quaternion Algebras

Author : John Voight
Publisher : Springer Nature
ISBN 13 : 3030566943
Total Pages : 877 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Quaternion Algebras by : John Voight

Download or read book Quaternion Algebras written by John Voight and published by Springer Nature. This book was released on 2021-06-28 with total page 877 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.

Selected Papers and Other Writings

Author : Irving Kaplansky
Publisher : Springer Science & Business Media
ISBN 13 : 9780387944067
Total Pages : 290 pages
Book Rating : 4.9/5 (44 download)

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Book Synopsis Selected Papers and Other Writings by : Irving Kaplansky

Download or read book Selected Papers and Other Writings written by Irving Kaplansky and published by Springer Science & Business Media. This book was released on 1995-04-13 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is not often that one gets to write a preface to a collection of one's own papers. The most urgent task is to thank the people who made this book possible. That means first of all Hy Bass who, on behalf of Springer-Verlag, approached me about the idea. The late Walter Kaufmann-Biihler was very encouraging; Paulo Ribenboim helped in an important way; and Ina Lindemann saw the project through with tact and skill that I deeply appreciate. My wishes have been indulged in two ways. First, I was allowed to follow up each selected paper with an afterthought. Back in my student days I became aware of the Gesammelte Mathematische Werke of Dedekind, edited by Fricke, Noether, and Ore. I was impressed by the editors' notes that followed most of the papers and found them very usefuL A more direct model was furnished by the collected papers of Lars Ahlfors, in which the author himself supplied afterthoughts for each paper or group of papers. These were tough acts to follow, but I hope that some readers will find at least some of my afterthoughts interesting. Second, I was permitted to add eight previously unpublished items. My model here, to a certain extent, was the charming little book, A Mathematician's Miscel lany by J. E. Littlewood. In picking these eight I had quite a selection to make -from fourteen loose-leaf notebooks of such writings. Here again I hope that at least some will be found to be of interest.

Quadratic Forms and Their Applications

Author : Eva Bayer-Fluckiger
Publisher : American Mathematical Soc.
ISBN 13 : 0821827790
Total Pages : 330 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Quadratic Forms and Their Applications by : Eva Bayer-Fluckiger

Download or read book Quadratic Forms and Their Applications written by Eva Bayer-Fluckiger and published by American Mathematical Soc.. This book was released on 2000 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Primes of the Form X2 + Ny2

Author : David A. Cox
Publisher : Wiley-Interscience
ISBN 13 :
Total Pages : 372 pages
Book Rating : 4.:/5 (321 download)

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Book Synopsis Primes of the Form X2 + Ny2 by : David A. Cox

Download or read book Primes of the Form X2 + Ny2 written by David A. Cox and published by Wiley-Interscience. This book was released on 1989 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 class 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

Author : John Horton Conway
Publisher : American Mathematical Soc.
ISBN 13 : 1470448424
Total Pages : 167 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis The Sensual (quadratic) Form by : John Horton Conway

Download or read book The Sensual (quadratic) Form written by John Horton Conway and published by American Mathematical Soc.. This book was released on 1997-12-31 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Algebraic Theory of Quadratic Numbers

Author : Mak Trifković
Publisher : Springer Science & Business Media
ISBN 13 : 1461477174
Total Pages : 206 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Algebraic Theory of Quadratic Numbers by : Mak Trifković

Download or read book Algebraic Theory of Quadratic Numbers written by Mak Trifković and published by Springer Science & Business Media. This book was released on 2013-09-14 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: By focusing on quadratic numbers, this advanced undergraduate or master’s level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes. The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory.

An Illustrated Theory of Numbers

Author : Martin H. Weissman
Publisher : American Mathematical Soc.
ISBN 13 : 1470463717
Total Pages : 341 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis An Illustrated Theory of Numbers by : Martin H. Weissman

Download or read book An Illustrated Theory of Numbers written by Martin H. Weissman and published by American Mathematical Soc.. This book was released on 2020-09-15 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae

Author : Catherine Goldstein
Publisher : Springer Science & Business Media
ISBN 13 : 3540347208
Total Pages : 579 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae by : Catherine Goldstein

Download or read book The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae written by Catherine Goldstein and published by Springer Science & Business Media. This book was released on 2007-02-03 with total page 579 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since its publication, C.F. Gauss's Disquisitiones Arithmeticae (1801) has acquired an almost mythical reputation, standing as an ideal of exposition in notation, problems and methods; as a model of organisation and theory building; and as a source of mathematical inspiration. Eighteen authors - mathematicians, historians, philosophers - have collaborated in this volume to assess the impact of the Disquisitiones, in the two centuries since its publication.

Compositions of Quadratic Forms

Author : Daniel B. Shapiro
Publisher : Walter de Gruyter
ISBN 13 : 3110824833
Total Pages : 433 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Compositions of Quadratic Forms by : Daniel B. Shapiro

Download or read book Compositions of Quadratic Forms written by Daniel B. Shapiro and published by Walter de Gruyter. This book was released on 2011-06-24 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

The Class Number of Binary Quadratic Forms ...

Author : George Hoffman Cresse
Publisher :
ISBN 13 :
Total Pages : 117 pages
Book Rating : 4.L/5 ( download)

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Book Synopsis The Class Number of Binary Quadratic Forms ... by : George Hoffman Cresse

Download or read book The Class Number of Binary Quadratic Forms ... written by George Hoffman Cresse and published by . This book was released on 1923 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Combinatorial and Additive Number Theory III

Author : Melvyn B. Nathanson
Publisher : Springer Nature
ISBN 13 : 3030311066
Total Pages : 237 pages
Book Rating : 4.0/5 (33 download)

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Book Synopsis Combinatorial and Additive Number Theory III by : Melvyn B. Nathanson

Download or read book Combinatorial and Additive Number Theory III written by Melvyn B. Nathanson and published by Springer Nature. This book was released on 2019-12-10 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on talks from the 2017 and 2018 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 17 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, commutative algebra and discrete geometry, and applications of logic and nonstandard analysis to number theory. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Classical Theory of Algebraic Numbers

Author : Paulo Ribenboim
Publisher : Springer Science & Business Media
ISBN 13 : 0387216901
Total Pages : 676 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Classical Theory of Algebraic Numbers by : Paulo Ribenboim

Download or read book Classical Theory of Algebraic Numbers written by Paulo Ribenboim and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 676 pages. Available in PDF, EPUB and Kindle. Book excerpt: The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics.

Lectures on Number Theory

Author : Peter Gustav Lejeune Dirichlet
Publisher : American Mathematical Soc.
ISBN 13 : 0821820176
Total Pages : 297 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Lectures on Number Theory by : Peter Gustav Lejeune Dirichlet

Download or read book Lectures on Number Theory written by Peter Gustav Lejeune Dirichlet and published by American Mathematical Soc.. This book was released on 1999 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions.