This book is devoted to quadratic forms over fields of characteristic not equal to two. Besides the basics of the theory, it contains both classical and more recent topics and results, which makes it a good reference book for researchers in many areas of mathematics. The chapter titles are: Foundations, Witt Rings, Quaternion Algebras, Brauer-Wall groups, Clifford Algebras, forms over local and global fields, forms under algebraic and transcendental field extensions, forms over formally real fields and pythagorean fields, Pfister forms and field invariants. In comparison with ‘The Algebraic Theory of Quadratic Forms’ (1972) from the same author, there are two new chapters. These are divided into quite independent sections covering some recent topics in the theory, including forms of low dimension, classification theorems, biquadratic extensions and quadratic invariants of fields.

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. Longer proofs are divided into series of lemmas or steps, which help the reader extract the main idea of the proof. These lemmas are in some cases inserted between the statement and its proof. The reader will also appreciate that definitions are recalled when they are used later in the book. Since the emphasis of the book is not on connections with other parts of mathematics, it is a comprehensive and self-contained introduction to the theory of quadratic forms. Although particular chapters are reworked and two more are added, both the style and the organization of the book has remained the same as that of the above-mentioned book and thus it will appeal to a wide range of readers from students to researchers.

Reviewer:

jhor