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Introduction to Quadratic Forms over Fields
T.Y. Lam, University of California, Berkeley, CA

Graduate Studies in Mathematics
2005; 550 pp; hardcover
Volume: 67
ISBN-10: 0-8218-1095-2
ISBN-13: 978-0-8218-1095-8
List Price: US$87
Member Price: US$69.60
Order Code: GSM/67
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See also:

Basic Quadratic Forms - Larry J Gerstein

This new version of the author's prizewinning book, Algebraic Theory of Quadratic Forms (W. A. Benjamin, Inc., 1973), gives a modern and self-contained introduction to the theory of quadratic forms over fields of characteristic different from two. Starting with few prerequisites beyond linear algebra, the author charts an expert course from Witt's classical theory of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of formally real fields, and structural theorems on Witt rings, to the theory of Pfister forms, function fields, and field invariants. These main developments are seamlessly interwoven with excursions into Brauer-Wall groups, local and global fields, trace forms, Galois theory, and elementary algebraic K-theory, to create a uniquely original treatment of quadratic form theory over fields. Two new chapters totaling more than 100 pages have been added to the earlier incarnation of this book to take into account some of the newer results and more recent viewpoints in the area.

As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.


Graduate students and research mathematicians interested in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.


"The book reads very well; notions and statements are supported by examples involving cases over both finite and infinite fields. At the end of every chapter, there are a number of exercises that could be useful, especially for teachers using the book as a basis for their course."

-- EMS Newsletter

"(This) book is a wonderful achievement. Its genesis is recounted with charm and warmth in the preface. The author's lucid style and expository skill, his judicious choice of topics and their impeccable layout, not to mention the beautiful typesetting make the book a joy to read or just to browse. it will be a must for anybody working in quadratic forms or on topics related to or using quadratic forms, be it for learning the theory of quadratic forms over fields from its foundations, or be it as a reference."

-- Zentralblatt Math

Table of Contents

  • Foundations
  • Introduction to Witt rings
  • Quaternion algebras and their norm forms
  • The Brauer-Wall group
  • Clifford algebras
  • Local fields and global fields
  • Quadratic forms under algebraic extensions
  • Formally real fields, real-closed fields, and pythagorean fields
  • Quadratic forms under transcendental extensions
  • Pfister forms and function fields
  • Field invariants
  • Special topics in quadratic forms
  • Special topics on invariants
  • Bibliography
  • Index
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