The Algebraic and Geometric Theory of Quadratic Forms

The contents of this book is divided into three parts. The first part treats classical theory of symmetric bilinear and quadratic forms, including a study of forms under field extensions and construction of basic algebraic invariants of quadratic forms (e.g. the Witt ring, the u-invariant and norm residue homomorphisms). In the second part of the book, basic parts of algebraic geometry are developed (including Chow groups of algebraic cycles modulo rational equivalences on an algebraic scheme). Methods of the second chapter are then applied in the last part to a description of algebraic cycles on quadrics and their powers. Some basic combinatorial objects associated to quadrics (e.g. shell triangles and diagrams of cycles) are also introduced, leading to a simplified visualization of algebraic cycles and operation on them. The book is self-contained; almost no prerequisites are needed. Its point of view is characterized by a neat interplay of geometric approach (given by properties of a quadric and its function field) and algebraic approach (given by properties of quadratic maps over a field).

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