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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Knebusch, Manfred. Specialization of quadratic and symmetric bilinear forms. London ; New York: Springer, ©2010 (DLC) 2010931863 (OCoLC)646114290 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Manfred Knebusch |

ISBN: | 9781848822429 1848822421 |

OCLC Number: | 701718760 |

Description: | 1 online resource (xiv, 192 pages) : illustrations. |

Contents: | Cover13; -- Preface -- Contents -- 1 Fundamentals of Specialization Theory -- 1.1 Introduction: on the Problem of Specialization of Quadratic and Bilinear Forms -- 1.2 An Elementary Treatise on Symmetric Bilinear Forms -- 1.3 Specialization of Symmetric Bilinear Forms -- 1.4 Generic Splitting in Characteristic 2 -- 1.5 An Elementary Treatise on Quadratic Modules -- 1.6 Quadratic Modules over Valuation Rings -- 1.7 Weak Specialization -- 1.8 Good Reduction -- 2 Generic Splitting Theory -- 2.1 Generic Splitting of Regular Quadratic Forms -- 2.2 Separable Splitting -- 2.3 Fair Reduction and Weak Obedience -- 2.4 Unified Theory of Generic Splitting -- 2.5 Regular Generic Splitting Towers and Base Extension -- 2.6 Generic Splitting Towers of a Specialized Form -- 3 Some Applications -- 3.1 Subforms which have Bad Reduction -- 3.2 Some Forms of Height 1 -- 3.3 The Subform Theorem -- 3.4 Milnor's Exact Sequence -- 3.5 A Norm Theorem -- 3.6 Strongly Multiplicative Forms -- 3.7 Divisibility by Pfister Forms -- 3.8 Pfister Neighbours and Excellent Forms -- 3.9 Regular Forms of Height 1 -- 3.10 Some Open Problems in Characteristic 2 -- 3.11 Leading Form and Degree Function -- 3.12 The Companion Form of an Odd-dimensional Regular Form -- 3.13 Definability of the Leading Form over the Base Field -- 4 Specialization with Respect to Quadratic Places -- 4.1 Quadratic Places; Specialization of Bilinear Forms -- 4.2 Almost Good Reduction with Respect to Extensions of Quadratic Places -- 4.3 Realization of Quadratic Places; Generic Splitting of Specialized Forms in Characteristic 2 -- 4.4 Stably Conservative Reduction of Quadratic Forms -- 4.5 Generic Splitting of Stably Conservative Specialized Quadratic Forms -- References -- Index. |

Series Title: | Algebras and applications, 11. |

Responsibility: | Manfred Knebusch ; translated by Thomas Unger. |

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### Abstract:

## Reviews

*Editorial reviews*

Publisher Synopsis

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 fur 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 has been given such a comprehensive treatment." (Detlev Hoffmann, Zentralblatt MATH, Vol. 1203, 2011) Read more...

*User-contributed reviews*

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## Similar Items

### Related Subjects:(5)

- Forms, Quadratic.
- Bilinear forms.
- Algebraic fields.
- Mathematics.
- MATHEMATICS -- Algebra -- Elementary.