Quadratic Forms

Abstract

Quadratic forms constitute a large domain of research with roots in classical mathematics and remarkable developments in the last decades. This chapter discusses the Milnor Conjecture and its solution by Voevodsky. The classification results for quadratic forms and Witt rings are presented in the chapter. While classification of quadratic forms up to equivalence is an obvious central problem of the theory, its solution does not suffice to determine the structure of the Witt ring. The classification of Witt rings over a field F is considered the ultimate goal of the theory. The chapter discusses the selected applications of function fields of quadratic forms. This technique brings to quadratic form theory the methods of algebraic geometry, which proved to be extremely important and successful. The properties of particular quadratic forms, the sums of squares, are discussed in the chapter. The chapter describes the level of a field or ring, the composition of sums of squares, and the Pythagoras number of fields and rings. The most natural generalization of the original Witt ring, the Witt ring of a commutative ring, is discussed in the chapter.