Volume 2 of two - also available in a set of both volumes.

During the 1980s, profound connections were discovered relating modern algebraic geometry and algebraic $K$-theory to arithmetic problems. The term ``arithmetic algebraic geometry'' was coined during that period and is now used to denote an entire branch of modern number theory. These same developments in algebraic geometry and $K$-theory greatly influenced research on the arithmetic of fields in general, and the algebraic theory of quadratic forms and the theory of finite-dimensional division algebras in particular. This book contains papers presented at an AMS Summer Research Institute held in July 1992 at the University of California, Santa Barbara. The purpose of the conference was to provide a broad overview of the tools from algebraic geometry and $K$-theory that have proved to be the most powerful in solving problems in the theory of quadratic forms and division algebras. In addition, the conference provided a venue for exposition of recent research. A substantial portion of the lectures of the major conference speakers--Colliot-Thelene, Merkurjev, Raskind, Saltman, Suslin, Swan--are reproduced in the expository articles in this book.

Developments in Mathematics is a book series devoted to all areas of mathematics, pure and applied. The series emphasizes research monographs describing the latest advances. Edited volumes that focus on areas that have seen dramatic progress, or are of special interest, are encouraged as well.

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.

An introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra.