Introduction To Quadratic Forms

Author by : Onorato Timothy O’Meara
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Publisher by : Springer
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Introduction To Quadratic Forms Over Fields

Author by : Tsit-Yuen Lam
Languange : en
Publisher by : American Mathematical Soc.
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Description : A new version of the author's prize-winning Algebraic Theory of Quadratic Forms (Benjamin, 1973), this book gives a modern and self-contained introduction to the theory of quadratic forms over fields of characteristic not two. Starting with few prerequisites besides linear algebra, the author charts an expert course from Witt's classical theory of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of formally real fields, and structural theorems on Witt rings, to the theory of Pfister forms, function fields, and field invariants. These main developments are seamlessly interwoven with excursions into Brauer-Wall groups, local and global fields, trace forms, Galois theory, and elementary algebraic K-theory, to create a uniquely original treatment of quadratic form theory over fields. Two new chapters totaling more than 100 pages have been added to the earlier incarnation of this book to take into account some of the newer results and more recent viewpoints in the area. As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, field theory, algebraic geometry, algebraic topology, and geometric topology.


Binary Quadratic Forms

Author by : Duncan A. Buell
Languange : en
Publisher by : Springer Science & Business Media
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Description : 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.


Quadratic Forms

Author by : Michael Barot
Languange : en
Publisher by : Springer
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Description : This monograph presents combinatorial and numerical issues on integral quadratic forms as originally obtained in the context of representation theory of algebras and derived categories. Some of these beautiful results remain practically unknown to students and scholars, and are scattered in papers written between 1970 and the present day. Besides the many classical results, the book also encompasses a few new results and generalizations. The material presented will appeal to a wide group of researchers (in representation theory of algebras, Lie theory, number theory and graph theory) and, due to its accessible nature and the many exercises provided, also to undergraduate and graduate students with a solid foundation in linear algebra and some familiarity on graph theory.


The Algebraic And Geometric Theory Of Quadratic Forms

Author by : Richard S. Elman
Languange : en
Publisher by : American Mathematical Soc.
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Description : 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.


Quadratic And Hermitian Forms

Author by : W. Scharlau
Languange : en
Publisher by : Springer Science & Business Media
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Description : For a long time - at least from Fermat to Minkowski - the theory of quadratic forms was a part of number theory. Much of the best work of the great number theorists of the eighteenth and nineteenth century was concerned with problems about quadratic forms. On the basis of their work, Minkowski, Siegel, Hasse, Eichler and many others crea ted the impressive "arithmetic" theory of quadratic forms, which has been the object of the well-known books by Bachmann (1898/1923), Eichler (1952), and O'Meara (1963). Parallel to this development the ideas of abstract algebra and abstract linear algebra introduced by Dedekind, Frobenius, E. Noether and Artin led to today's structural mathematics with its emphasis on classification problems and general structure theorems. On the basis of both - the number theory of quadratic forms and the ideas of modern algebra - Witt opened, in 1937, a new chapter in the theory of quadratic forms. His most fruitful idea was to consider not single "individual" quadratic forms but rather the entity of all forms over a fixed ground field and to construct from this an algebra ic object. This object - the Witt ring - then became the principal object of the entire theory. Thirty years later Pfister demonstrated the significance of this approach by his celebrated structure theorems.


Binary Quadratic Forms

Author by : Johannes Buchmann
Languange : en
Publisher by : Springer Science & Business Media
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Description : 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.


Compositions Of Quadratic Forms

Author by : Daniel B. Shapiro
Languange : en
Publisher by : Walter de Gruyter
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Description : The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics 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 wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany


Arithmetic Of Quadratic Forms

Author by : Yoshiyuki Kitaoka
Languange : en
Publisher by : Cambridge University Press
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Description : Provides an introduction to quadratic forms.


Quadratic Forms And Hecke Operators

Author by : Anatolij N. Andrianov
Languange : en
Publisher by : Springer
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Description : The numerous explicit formulae of the classical theory of quadratic forms revealed remarkable multiplicative properties of the numbers of integral representations of integers by positive definite integral quadratic forms. These properties were explained by the original theory of Hecke operators. As regards the integral representations of quadratic forms in more than one variable by quadratic forms, no multiplicative properties were known at that time, and so there was nothing to explain. However, the idea of Hecke operators was so natural and attractive that soon attempts were made to cultivate it in the neighbouring field of modular forms of several variables. The approach has proved to be fruitful; in particular, a number of multiplicative properties of integral representations of quadratic forms by quadratic forms were eventually discovered. By now the theory has reached a certain maturity, and the time has come to give an up-to-date report in a concise form, in order to provide a solid ground for further development. The purpose of this book is to present in the form of a self-contained text-book the contemporary state of the theory of Hecke operators on the spaces of hoi om orphic modular forms of integral weight (the Siegel modular forms) for congruence subgroups of integral symplectic groups. The book can also be used for an initial study of modular forms of one or several variables and theta-series of positive definite integral quadratic forms.


Rational Quadratic Forms

Author by : J. W. S. Cassels
Languange : en
Publisher by : Courier Dover Publications
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Description : 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.


Quaternary Quadratic Forms

Author by : Gordon L. Nipp
Languange : en
Publisher by : Springer Science & Business Media
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Description : This book of tables includes a reduced representative of each class of. integral positive definite primitive quaternary quadratic forms through discriminant 1732. The classes are grouped into genera; also included are Hasse symbols, the number of automorphs and the level of each such form, and the mass of each genus. An appendix lists p-adic densities and p-adic Jordan splittings for each genus in the tables for p = 2 and for each odd prime p dividing the discriminant. The book is divided into several sections. The first, an introductory section, contains background material, an explanation of the techniques used to generate the information contained in the tables, a description of the format of the tables, some instructions for computer use, examples, and references. The next section contains a printed version of the tables through discriminant 500, included to allow the reader to peruse at least this much without the inconvenience of making his/her own hard copy via the computer. Because of their special interest, we include tables of discriminants 729 and 1729 at the end of this section. Limitations of space preclude publication of more than this in printed form. A printed appendix through discriminant 500 and for discriminants 729 and 1729 follows. The complete tables and appendix through discriminant 1732 are compressed onto the accompanying 3.5 inch disk, formatted for use in a PC-compatible computer and ready for research use particularly when uploaded to a mainframe. Documentation is included in the Introduction.


The Geometry Of Positive Quadratic Forms

Author by : Sergeĭ Sergeevich Ryshkov
Languange : en
Publisher by : American Mathematical Soc.
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Description : Papers and articles about quadratic forms.


Basic Quadratic Forms

Author by : Larry J. Gerstein
Languange : en
Publisher by : American Mathematical Soc.
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Description : The arithmetic theory of quadratic forms is a rich branch of number theory that has had important applications to several areas of pure mathematics - particularly group theory and topology - as well as to cryptography and coding theory. This book is a self-contained introduction to quadratic forms that is based on graduate courses the author has taught many times. It leads the reader from foundation material up to topics of current research interest - with special attention to the theory over the integers and over polynomial rings in one variable over a field - and requires only a basic background in linear and abstract algebra as a prerequisite. Whenever possible, concrete constructions are chosen over more abstract arguments. The book includes many exercises and explicit examples, and it is appropriate as a textbook for graduate courses or for independent study. To facilitate further study, a guide to the extensive literature on quadratic forms is provided.


Integral Quadratic Forms And Lattices

Author by : International Conference on Integral Quadratic Forms and Lattices
Languange : en
Publisher by : American Mathematical Soc.
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Description : This volume presents the proceedings of an international conference held at Seoul National University (Korea). Talks covered recent developments in diverse areas related to the theory of integral quadratic forms and hermitian forms, local densities, linear relations and congruences of theta series, zeta functions of prehomogeneous vector spaces, lattices with maximal finite matrix groups, globally irreducible lattices, Mordell-Weil lattices, and more. Articles in the volume represent expository lectures by leading experts on recent developments in the field.The book offers a comprehensive introduction to the current state of knowledge in the arithmetic theory of quadratic forms and provides active directions of research with new results. Topics addressed in the volume emphasize connections with related fields, such as group theory, arithmetic geometry, analytic number theory, and modular forms. The book is an excellent introductory guide for students as well as a rich reference source for researchers.


Quaternion Orders Quadratic Forms And Shimura Curves

Author by : Montserrat Alsina and Pilar Bayer
Languange : en
Publisher by : American Mathematical Soc.
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Description : 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 multiplication points. 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. Titles in this series are co-published with the Centre de Recherches Mathématiques.


Quadratic Forms Linear Algebraic Groups And Cohomology

Author by : Skip Garibaldi
Languange : en
Publisher by : Springer Science & Business Media
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Description : Developments in Mathematics is a book series devoted to all areas of mathematics, pure and applied. The series emphasizes research monographs describing the latest advances. Edited volumes that focus on areas that have seen dramatic progress, or are of special interest, are encouraged as well.


Singular Quadratic Forms In Perturbation Theory

Author by : Volodymyr Koshmanenko
Languange : en
Publisher by : Springer Science & Business Media
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Description : The notion of singular quadratic form appears in mathematical physics as a tool for the investigation of formal expressions corresponding to perturbations devoid of operator sense. Numerous physical models are based on the use of Hamiltonians containing perturba tion terms with singular properties. Typical examples of such expressions are Schrodin ger operators with O-potentials (-~ + AD) and Hamiltonians in quantum field theory with perturbations given in terms of operators of creation and annihilation (P(


Quadratic Forms And Their Applications

Author by : Eva Bayer-Fluckiger
Languange : en
Publisher by : American Mathematical Soc.
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Description : 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.


Quadratic Forms In Infinite Dimensional Vector Spaces

Author by : H. Gross
Languange : en
Publisher by : Springer Science & Business Media
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Description : For about a decade I have made an effort to study quadratic forms in infinite dimensional vector spaces over arbitrary division rings. Here we present in a systematic fashion half of the results found du ring this period, to wit, the results on denumerably infinite spaces (" NO-forms'''). Certain among the results included here had of course been published at the time when they were found, others appear for the first time (the case, for example, in Chapters IX, X , XII where I in clude results contained in the Ph.D.theses by my students W. Allenspach, L. Brand, U. Schneider, M. Studer). If one wants to give an introduction to the geometric algebra of infinite dimensional quadratic spaces, a discussion of N-dimensional O spaces ideally serves the purpose. First, these spaces show a large number of phenomena typical of infinite dimensional spaces. Second, most proofs can be done by recursion which resembles the familiar pro cedure by induction in the finite dimensional situation. Third, the student acquires a good feeling for the linear algebra in infinite di mensions because it is impossible to camouflage problems by topological expedients (in dimension NO it is easy to see, in a given case, wheth er topological language is appropriate or not).


Quadratic Forms With Applications To Algebraic Geometry And Topology

Author by : Albrecht Pfister
Languange : en
Publisher by : Cambridge University Press
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Description : A gem of a book bringing together 30 years worth of results that are certain to interest anyone whose research touches on quadratic forms.


The Arithmetic Theory Of Quadratic Forms

Author by : Burton W Jones
Languange : en
Publisher by : American Mathematical Soc.
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Description : 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

Author by : Montserrat Alsina
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 66
Total Download : 729
File Size : 54,9 Mb
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Description : 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.


Ternary Quadratic Forms And Norms

Author by : O. Taussky
Languange : en
Publisher by : CRC Press
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Total Read : 21
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Description : This book covers topics including the Redei-Reichardt theorem, automorphs of ternary quadratic forms, facts concerning rational matrices leading to integral ternary forms representing zero, characteristics polynomials of symmetric matrices, and Gauss' theory of ternary quadratic forms.


Ternary Quadratic Forms And Norms

Author by : O. Taussky
Languange : en
Publisher by : CRC Press
Format Available : PDF, ePub, Mobi
Total Read : 93
Total Download : 510
File Size : 41,9 Mb
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Description : This book covers topics including the Redei-Reichardt theorem, automorphs of ternary quadratic forms, facts concerning rational matrices leading to integral ternary forms representing zero, characteristics polynomials of symmetric matrices, and Gauss' theory of ternary quadratic forms.