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Specialization of Quadratic and Symmetric Bilinear Forms

  • Book
  • © 2010

Overview

Authors:
  1. Manfred Knebusch

    1. Fak. Naturwissenschaft I, FB Mathematik, Universität Regensburg, Regensburg, Germany

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  • Written by the founder of specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms
  • Comprehensively covers specialization and generic splitting theories
  • Contains a final chapter containing research never before published on specialization with respect to quadratic place

Part of the book series: Algebra and Applications (AA, volume 11)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Fundamentals of Specialization Theory

    • Manfred Knebusch, Thomas Unger
    Pages 1-53

  3. Generic Splitting Theory

    • Manfred Knebusch, Thomas Unger
    Pages 55-90

  4. Some Applications

    • Manfred Knebusch, Thomas Unger
    Pages 91-164

  5. Specialization with Respect to Quadratic Places

    • Manfred Knebusch, Thomas Unger
    Pages 165-183

  6. Back Matter

    Pages 185-192
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Keywords

About this book

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).

Reviews

From the reviews:

“In the present book Knebusch develops a theory of specialization covering all the situations occurring by combining the cases when the characteristic of the field L is equal or unequal to 2 with the cases when the forms are either symmetric bilinear or quadratic and have a prescribed type of reduction under the place. … The book is a welcome addition to the literature on quadratic forms supporting the tendency of presenting results for forms over all fields independently of the field characteristic.” (K. Szymiczek, Mathematical Reviews, Issue 2011 h)

“The author developed a specialization theory and a subsequent generic splitting theory of quadratic and symmetric bilinear forms over fields. … This book comprehensively covers the theory without the restriction of the characteristic. … offers results on specialization with respect to quadratic places providing a look on further research. A book for the specialist!” (H. Mitsch, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

“This is an important monograph in which the author has done an excellent job putting together in one place many important results on specialization of quadratic and bilinear forms and generic splitting of quadratic forms that were previously dispersed in several research articles. It belongs on the shelf of any mathematician interested on the algebraic, geometric and arithmetic aspects of quadratic forms. The book under review joins the class of the ever expanding literature on quadratic forms … .” (Feline Zaldivar, The Mathematical Association of America, January, 2011)

“The present book represents the state of the art in the theory of specialization of quadratic forms, containing a substantial amount of previously unpublished new results. It is a welcome addition to the literature as this is the first monograph in which this original and important approach to the algebraic theory of quadratic forms hasbeen given such a comprehensive treatment.” (Detlev Hoffmann, Zentralblatt MATH, Vol. 1203, 2011)

Authors and Affiliations

  • Fak. Naturwissenschaft I, FB Mathematik, Universität Regensburg, Regensburg, Germany

    Manfred Knebusch

Bibliographic Information

  • Book Title: Specialization of Quadratic and Symmetric Bilinear Forms

  • Authors: Manfred Knebusch

  • Series Title: Algebra and Applications

  • DOI: https://doi.org/10.1007/978-1-84882-242-9

  • Publisher: Springer London

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag London Limited 2010

  • Hardcover ISBN: 978-1-84882-241-2Published: 14 September 2010

  • Softcover ISBN: 978-1-4471-2586-0Published: 09 November 2012

  • eBook ISBN: 978-1-84882-242-9Published: 22 January 2011

  • Series ISSN: 1572-5553

  • Series E-ISSN: 2192-2950

  • Edition Number: 1

  • Number of Pages: XIV, 192

  • Topics: Algebra

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