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^ + ly'^ or, more
generally, in the form ax^ + Ihxy + cy^, where a,h,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, H.J.S. 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 tlie way to the study of quadratic forms over arbitrary fields,
with pioneering contributions by Witt A937) and Pfister A965-67).
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 Frohlich A967), we study
quadratic forms over a Hilbert field.