Quadratic And Hermitian Forms
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Quadratic And Hermitian Forms
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Author : W. Scharlau
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Quadratic And Hermitian Forms written by W. Scharlau and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
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.
Quadratic And Hermitian Forms
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Author : McMaster University
language : en
Publisher: American Mathematical Soc.
Release Date : 1984
Quadratic And Hermitian Forms written by McMaster University and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Mathematics categories.
Contains the proceedings of the 1983 Seminar on Quadratic and Hermitian Forms held at McMaster University, July 1983. Between 1945 and 1965, most of the work in quadratic (and hermitian) forms took place in arithmetic theory (M Eichler, M Kneser, O T O'Meara).
Quadratic And Hermitian Forms Over Rings
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Author : Max-Albert Knus
language : en
Publisher:
Release Date : 1991-04-05
Quadratic And Hermitian Forms Over Rings written by Max-Albert Knus and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-04-05 with categories.
This book presents the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial properties of the theory. It is not an encyclopedic survey. It stresses the algebraic aspects of the theory and avoids - within reason - overlapping with other books on quadratic forms (like those of Lam, Milnor-Husemoller and Scharlau). One important tool is descent theory with the corresponding cohomological machinery. It is used to define the classical invariants of quadratic forms, but also for the study of Azmaya algebras, which are fundamental in the theory of Clifford algebras. Clifford algebras are applied, in particular, to treat in detail quadratic forms of low rank and their spinor groups. Another important tool is algebraic K-theory, which plays the role that linear algebra plays in the case of forms over fields. The book contains complete proofs of the stability, cancellation and splitting theorems in the linear and in the unitary case. These results are applied to polynomial rings to give quadratic analogues of the theorem of Quillen and Suslin on projective modules. Another, more geometric, application is to Witt groups of regular rings and Witt groups of real curves and surfaces.
Quadratic And Hermitian Forms
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Author : Winfried Scharlau
language : en
Publisher:
Release Date : 1985
Quadratic And Hermitian Forms written by Winfried Scharlau and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with Forms, Quadratic categories.
Quadratic And Hermitian Forms Over Rings
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Author : Max-Albert Knus
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Quadratic And Hermitian Forms Over Rings written by Max-Albert Knus and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
From its birth (in Babylon?) till 1936 the theory of quadratic forms dealt almost exclusively with forms over the real field, the complex field or the ring of integers. Only as late as 1937 were the foundations of a theory over an arbitrary field laid. This was in a famous paper by Ernst Witt. Still too early, apparently, because it took another 25 years for the ideas of Witt to be pursued, notably by Albrecht Pfister, and expanded into a full branch of algebra. Around 1960 the development of algebraic topology and algebraic K-theory led to the study of quadratic forms over commutative rings and hermitian forms over rings with involutions. Not surprisingly, in this more general setting, algebraic K-theory plays the role that linear algebra plays in the case of fields. This book exposes the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial aspects of the theory. The advantage of doing so is not only aesthetical: on the one hand, some classical proofs gain in simplicity and transparency, the most notable examples being the results on low-dimensional spinor groups; on the other hand new results are obtained, which went unnoticed even for fields, as in the case of involutions on 16-dimensional central simple algebras. The first chapter gives an introduction to the basic definitions and properties of hermitian forms which are used throughout the book.
Quadratic Forms And Their Applications
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Author : Eva Bayer-Fluckiger
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Quadratic Forms And Their Applications written by Eva Bayer-Fluckiger and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
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.
Matrices And Quadratic Forms
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Author : James M. Bower
language : en
Publisher: Hodder Education Publishers
Release Date : 2000
Matrices And Quadratic Forms written by James M. Bower and has been published by Hodder Education Publishers this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Algebras, Linear categories.
This is a lively and accessible introduction to matrices and quadratic forms for students in linear algebra. Examples and exercises are used as teaching aids and ideas for investigation and project work help to place the subject in context. The inclusion of historical contexts, real-life situations and the discussion of links with other areas of mathematics will greatly enhance student motivation, making this the perfect classroom tool.
Integral Quadratic Forms And Lattices
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Author : Myung-Hwan Kim
language : en
Publisher: American Mathematical Soc.
Release Date : 1999
Integral Quadratic Forms And Lattices written by Myung-Hwan Kim and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Mathematics categories.
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.
Quadratic Forms Linear Algebraic Groups And Cohomology
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Author : Skip Garibaldi
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-07-16
Quadratic Forms Linear Algebraic Groups And Cohomology written by Skip Garibaldi and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-16 with Mathematics categories.
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.
Quaternion Algebras
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Author : John Voight
language : en
Publisher: Springer Nature
Release Date : 2021-06-28
Quaternion Algebras written by John Voight and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-28 with Mathematics categories.
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.