Quadratic and Hermitian Forms

  • Winfried Scharlau

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 270)

Table of contents

  1. Front Matter
    Pages I-X
  2. Winfried Scharlau
    Pages 1-27
  3. Winfried Scharlau
    Pages 28-105
  4. Winfried Scharlau
    Pages 106-141
  5. Winfried Scharlau
    Pages 142-165
  6. Winfried Scharlau
    Pages 166-200
  7. Winfried Scharlau
    Pages 235-279
  8. Winfried Scharlau
    Pages 280-325
  9. Winfried Scharlau
    Pages 326-345
  10. Winfried Scharlau
    Pages 346-390
  11. Back Matter
    Pages 391-421

About this book

Introduction

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.

Keywords

Abstract algebra Impress arithmetic boundary element method classification development finite field form linear algebra mathematics number theory object prime number quadratic form time

Authors and affiliations

  • Winfried Scharlau
    • 1
  1. 1.Mathematisches InstitutUniversität MünsterMünsterGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-69971-9
  • Copyright Information Springer-Verlag Berlin Heidelberg 1985
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-69973-3
  • Online ISBN 978-3-642-69971-9
  • Series Print ISSN 0072-7830
  • About this book