Quadratic Forms and Ordered Fields

Abstract

Most readers will have met inner products before. Here we take up the subject in a more general form and look at the properties of quadratic forms over a general field and its group of isometries (Sections 8.1-8.3). With each quadratic form a certain algebra is associated, the Clifford algebra, and with the set of all forms on a field the Witt group is associated; these form the subject of Section 8.4 and Section 8.5 respectively, with a further development, the Witt ring of a field, in Section 8.9. In Section 8.10 we take a brief look at symplectic groups and in Section 8.11 we consider quadratic forms in characteristic 2. We also briefly discuss the related topic of ordered fields in Section 8.6, leading to a construction of the real numbers in Section 8.7 and formally real fields in Section 8.8. Some of the later topics are included for completeness, but do not really have a place in a basic account; thus at a first reading the later parts of Sections 8.7-8.11 can be omitted.