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 + 2 y^2 or, more generally, in the formax^2 + 2 bxy+cy^2 ,wherea,b,care 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 cen-
tury by Dirichlet, Hermite, H.J.S. Smith, Minkowski and others. In the 20th century
Hasse and Siegel made notable contributions. With Hasse’s work especially it be-
came 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, theHasse–Minkowski
theorem. However, we first study quadratic forms over an arbitrary field in the geo-
metric formulation of Witt. Then, following an interesting approach due to Fr ̈ohlich
(1967), we study quadratic forms over aHilbert 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 chapterwe will restrict attention to fields for which
1 + 1 =0. Thus the phrase ‘an arbitrary field’will mean ‘an arbitrary field of charac-
teristic=2’. The proofs of many results make essential use of this restriction on the
W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext, 291
DOI: 10.1007/978-0-387-89486-7_7, © Springer Science + Business Media, LLC 2009