Arithmetic of Quadratic Forms
Author: Yoshiyuki Kitaoka
Publisher: Cambridge University Press
Published: 1999-04-29
Total Pages: 292
ISBN-13: 9780521649964
DOWNLOAD EBOOK →Provides an introduction to quadratic forms.
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Arithmetic of Quadratic Forms
Author: Goro Shimura
Publisher: Springer Science & Business Media
Published: 2010-08-09
Total Pages: 245
ISBN-13: 1441917322
DOWNLOAD EBOOK →This book is divided into two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. There are two principal topics: classification of quadratic forms and quadratic Diophantine equations. The second topic is a new framework which contains the investigation of Gauss on the sums of three squares as a special case. To make the book concise, the author proves some basic theorems in number theory only in some special cases. However, the book is self-contained when the base field is the rational number field, and the main theorems are stated with an arbitrary number field as the base field. So the reader familiar with class field theory will be able to learn the arithmetic theory of quadratic forms with no further references.
Introduction to Quadratic Forms
Author: Onorato Timothy O’Meara
Publisher: Springer
Published: 2013-12-01
Total Pages: 354
ISBN-13: 366241922X
DOWNLOAD EBOOK →Basic Quadratic Forms
Author: Larry J. Gerstein
Publisher: American Mathematical Soc.
Published: 2008
Total Pages: 274
ISBN-13: 0821844652
DOWNLOAD EBOOK →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.
The Arithmetic Theory of Quadratic Forms
Author: Burton W Jones
Publisher: American Mathematical Soc.
Published: 1950-12-31
Total Pages: 212
ISBN-13: 1614440107
DOWNLOAD EBOOK →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.
Rational Quadratic Forms
Author: J. W. S. Cassels
Publisher: Courier Dover Publications
Published: 2008-08-08
Total Pages: 429
ISBN-13: 0486466701
DOWNLOAD EBOOK →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.
Quadratic and Hermitian Forms
Author: W. Scharlau
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 431
ISBN-13: 3642699715
DOWNLOAD EBOOK →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.
The Sensual (quadratic) Form
Author: John Horton Conway
Publisher: American Mathematical Soc.
Published: 1997-12-31
Total Pages: 152
ISBN-13: 1470448424
DOWNLOAD EBOOK →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.
A Course in Arithmetic
Author: J-P. Serre
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 126
ISBN-13: 1468498843
DOWNLOAD EBOOK →This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapters contain some preliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols. Chapter V applies the preceding results to integral quadratic forms of discriminant ± I. These forms occur in various questions: modular functions, differential topology, finite groups. The second part (Chapters VI and VII) uses "analytic" methods (holomor phic functions). Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. 2.2). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. The two parts correspond to lectures given in 1962 and 1964 to second year students at the Ecole Normale Superieure. A redaction of these lectures in the form of duplicated notes, was made by J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They were very useful to me; I extend here my gratitude to their authors.
Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups
Author: Alexander J. Hahn
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 296
ISBN-13: 146846311X
DOWNLOAD EBOOK →Quadratic Algebras, Clifford Algebras, and Arithmetic Forms introduces mathematicians to the large and dynamic area of algebras and forms over commutative rings. The book begins very elementary and progresses gradually in its degree of difficulty. Topics include the connection between quadratic algebras, Clifford algebras and quadratic forms, Brauer groups, the matrix theory of Clifford algebras over fields, Witt groups of quadratic and symmetric bilinear forms. Some of the new results included by the author concern the representation of Clifford algebras, the structure of Arf algebra in the free case, connections between the group of isomorphic classes of finitely generated projectives of rank one and arithmetic results about the quadratic Witt group.
