Rational Quadratic Forms and the Local-global Principle

As for most Diophantine equations, quadratic forms were first studied over the integers, meaning that the coefficients aij are integers and only integer values of x1, . . . , xn are allowed to be plugged in. At the end of the 19th century it was realized that by allowing the variables x1, . . . , xn to take rational values, one gets a much more satisfactory theory. (In fact one can study quadratic forms with coefficients and values in any field F . This point of view was developed by Witt in the 1930’s, expanded in the middle years of this century by, among others, Pfister and Milnor, and has in the last decade become especially closely linked to one of the deepest and most abstract branches of contemporary mathematics: “homotopy K-theory.”) However, a wide array of firepower has been constructed over the years to deal with the complications presented by the integral case, culminating recently in some spectacular results. In this handout we will concentrate on what can be done over the rational numbers, and also on what statements about integral quadratic forms can be directly deduced from the theory of rational quadratic forms.