This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. For some results both classical and geometric proofs are given.Part I includes classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.
Richard Elman is a professor of mathematics at University of California, Los Angeles, CA. Nikita Karpenko is a professor of mathematics at Université Pierre et Marie Curie, Paris, France. Alexander Merkurjev is a professor of mathematics at University of California, Los Angeles, CA.
Introduction Part 1. Classical theory of symmetric bilinear forms and quadratic forms Chapter I. Bilinear Forms Chapter II. Quadratic Forms Chapter III. Forms over Rational Function Fields Chapter IV. Function Fields of Quadrics Chapter V. Bilinear and Quadratic Forms and Algebraic Extensions Chapter VI. u-invariants Chapter VII. Applications of the Milnor Conjecture Chapter VIII. On the Norm Residue Homomorphism of Degree Two Part 2. Algebraic cycles Chapter IX. Homology and Cohomology Chapter X. Chow Groups Chapter XI. Steenrod Operations Chapter XII. Category of Chow Motives Part 3. Quadratic forms and algebraic cycles Chapter XIII. Cycles on Powers of Quadrics Chapter XIV. The Izhboldin Dimension Chapter XV. Application of Steenrod Operations Chapter XVI. The Variety of Maximal Totally Isotropic Subspaces Chapter XVII. Motives of Quadrics Appendices Bibliography Notation Terminology