8 - QUADRATIC FORMS AND APPLICATION TO GEOMETRY

Abstract

This chapter discusses quadratic forms and application to geometry. The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which might be real or complex numbers, rational numbers, or integers. In linear algebra, in analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the p-adic integers Zp. Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in n variables has important applications to algebraic topology.