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Specialization of Quadratic and Symmetric Bilinear Forms

Manfred Knebusch

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ISBN 10: 144712586X / ISBN 13: 9781447125860
Published by Springer
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192 pages. Dimensions: 9.2in. x 6.1in. x 0.5in.A Mathematician Said Who Can Quote Me a Theorem thats True For the ones that I Know Are Simply not So, When the Characteristic is Two! This pretty limerick rst came to my ears in May 1998 during a talk by T. Y. Lam 1 on eld invariants from the theory of quadratic forms. It ispoetic exaggeration alloweda suitable motto for this monograph. What is it about At the beginning of the seventies I drew up a specialization theoryofquadraticandsymmetricbilinear formsover elds32. Let : K L be a place. Then one can assign a form ()toaform over K in a meaningful way if has good reduction with respect to (see1. 1). The basic idea is to simply apply the place to the coecients of, which must therefore be in the valuation ring of. The specialization theory of that time was satisfactory as long as the eld L, and therefore also K, had characteristic 2. It served me in the rst place as the foundation for a theory of generic splitting of quadratic forms 33, 34. After a very modest beginning, this theory is now in full bloom. It became important for the understanding of quadratic forms over elds, as can be seen from the book 26of IzhboldinKahnKarpenkoVishik for instance. One should note that there exists a theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic groups, parallel to the theory of generic splitting of quadratic forms (see 29 and the literature cited there). This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Bookseller Inventory # 9781447125860

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Title: Specialization of Quadratic and Symmetric ...

Publisher: Springer

Binding: Paperback

Book Condition:New

Book Type: Paperback

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A Mathematician Said Who Can Quote Me a Theorem that’s True? For the ones that I Know Are Simply not So, When the Characteristic is Two! This pretty limerick ?rst came to my ears in May 1998 during a talk by T.Y. Lam 1 on ?eld invariants from the theory of quadratic forms. It is―poetic exaggeration allowed―a suitable motto for this monograph. What is it about? At the beginning of the seventies I drew up a specialization theoryofquadraticandsymmetricbilinear formsover ?elds[32].Let? : K? L?? be a place. Then one can assign a form? (?)toaform? over K in a meaningful way ? if? has “good reduction” with respect to? (see§1.1). The basic idea is to simply apply the place? to the coe?cients of?, which must therefore be in the valuation ring of?. The specialization theory of that time was satisfactory as long as the ?eld L, and therefore also K, had characteristic 2. It served me in the ?rst place as the foundation for a theory of generic splitting of quadratic forms [33], [34]. After a very modest beginning, this theory is now in full bloom. It became important for the understanding of quadratic forms over ?elds, as can be seen from the book [26]of Izhboldin–Kahn–Karpenko–Vishik for instance. One should note that there exists a theoryof(partial)genericsplittingofcentralsimplealgebrasandreductivealgebraic groups, parallel to the theory of generic splitting of quadratic forms (see [29] and the literature cited there).

From the Back Cover:

The specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms were invented by the author in the mid-1970's. They came to fruition in the ensuing decades and have become an integral part of the geometric methods in quadratic form theory. This book comprehensively covers the specialization and generic splitting theories. These theories, originally developed mainly for fields of characteristic different from 2, are explored here without this restriction. In this book, a quadratic form φ over a field of characteristic 2 is allowed to have a big quasilinear part QL(φ) (defined as the restriction of φ to the radical of the bilinear form associated to φ), while in most of the literature QL(φ) is assumed to have dimension at most 1. Of course, in nature, quadratic forms with a big quasilinear part abound. In addition to chapters on specialization theory, generic splitting theory and their applications, the book's final chapter contains research never before published on specialization with respect to quadratic places and will provide the reader with a glimpse towards the future.

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