Rational Quadratic Forms

Abstract

Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x² + 2y², which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a _ij x _i x _j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a _ij x _i x _j = a. This leads to the problem of classification of quadratic forms over $\mathbb{Q}$, which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.