Part 2 book “Number theory - An introduction to mathematics” has contents: The arithmetic of quadratic forms, the geometry of numbers, the number of prime numbers, a character study, uniform distribution and ergodic theory, elliptic functions, connections with number theory. | VII The Arithmetic of Quadratic Forms We have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask which integers can be represented in the form x 2 + 2y 2 or, more generally, in the form ax 2 + 2bx y + cy 2 , where a, b, c are given integers. The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange’s theorem that every positive integer is a sum of four squares, was continued during the 19th century by Dirichlet, Hermite, . Smith, Minkowski and others. In the 20th century Hasse and Siegel made notable contributions. With Hasse’s work especially it became apparent that the theory is more perspicuous if one allows the variables to be rational numbers, rather than integers. This opened the way to the study of quadratic forms over arbitrary fields, with pioneering contributions by Witt (1937) and Pfister (1965–67). From this vast theory we focus attention on one central result, the Hasse–Minkowski theorem. However, we first study quadratic forms over an arbitrary field in the geometric formulation of Witt. Then, following an interesting approach due to Fr¨ohlich (1967), we study quadratic forms over a Hilbert field. 1 Quadratic Spaces The theory of quadratic spaces is simply another name for the theory of quadratic forms. The advantage of the change in terminology lies in its appeal to geometric intuition. It has in fact led to new results even at quite an elementary level. The new approach had its debut in a paper by Witt (1937) on the arithmetic theory of quadratic forms, but it is appropriate also if one is interested in quadratic forms over the real field or any other field. For the remainder of this chapter we will restrict attention to fields for which 1 + 1 = 0. .
