Quadratic Forms over Formally Real Fields

Abstract

The very first theorem discovered in the algebraic theory of quadratic forms was the law of inertia of Jacobi and Sylvester. This theorem is concerned with quadratic forms over the field of real numbers. In the present chapter we will be interested in various generalizations of this result, and more generally in the connections between the theory of quadratic forms and the theory of ordered fields. Our ground field will be a formally real field, that is one in which — 1 cannot be expressed as a sum of squares. The theory of these fields was developed by Artin and Schreier in a series of now classical papers. Today it is a part of basic algebra. In the years since about 1970 it has been discovered that a substantial part of this theory can be developed in a simple and elegant manner in the framework of the theory of quadratic forms.