Arithmetic of quadratic forms / Yoshiyuki KitaokaType de document : MonographieCollection : Cambridge tracts in mathematics, 106Langue : anglais.Pays : Grande Bretagne.Éditeur : Cambridge : Cambridge University Press, 1993Description : 1 vol. (x-268 p.) ; 24 cmISBN : 9780521404754.ISSN : 0950-6284.Bibliographie : Bibliogr. p. -267. Index.Sujet MSC : 11E12, Forms and linear algebraic groups, Quadratic forms over global rings and fields
11E08, Forms and linear algebraic groups, Quadratic forms over local rings and fields
11E10, Forms and linear algebraic groups, Forms over real fields
11E88, Forms and linear algebraic groups, Quadratic spaces; Clifford algebrasEn-ligne : Zentralblatt | MathScinet | Edition 1999
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"The purpose of this book is to introduce the reader to the arithmetic of quadratic forms'', says the author in the preface. This is, however, not the only goal pursued in this book. It also covers some fairly advanced and technical parts of the arithmetic theory of quadratic forms that are not readily available in other textbooks on the subject and serves as a welcome reference for these. (Some of the material in this category, especially Witt's theorem over rings, quantitative versions of Hensel's lemma, and the theorem of Minkowski and Siegel in the positive definite case can, however, be found in M. Kneser's lecture notes ("Quadratische Formen'', Univ. Göttingen).) ... The book contains a number of exercises and an appendix of notes and problems which contains quite a few important results and open problems that could have been included in the main text as well.
The book treats neither the theory over number fields or their integers (apart from the theory of scalar extensions mentioned above) nor the computational or algorithmic aspects of the theory (which recently have had quite a rapid development). For the former the reader should consult the book of O'Meara, while for the latter the book of Conway and Sloane is presently the best reference.
Who should read this book? The first four chapters along with Sections 5.1–5.3 and 6.1 give a good introduction to the basics of the subject (occupying about 90 pages). The rest is more for seriously interested people, and the casual reader will perhaps find it difficult to locate important results without going through all the details. This is especially true if one does not know what precisely one is looking for. Nevertheless I think that this is an excellent book. It is very carefully written; even extremely technical material is so well organized that it becomes accessible and it contains a lot of material that is otherwise scattered in the literature. It will certainly be indispensable for anybody working in the field, and its more elementary parts give a good introduction to the basics. (MathSciNet)
Bibliogr. p. -267. Index