The arithmetic theory of quadratic forms may be said to have begun with Fermat in 1654 who showed, among other things, that every prime of the form 8n + 1 is representable in the form x2 + 2y2 for x and y integers. Gauss was the first systematically to deal with quadratic forms and from that time, names associated with quadratic forms were most of the names in mathematics, with Dirichlet playing a leading role. H. J. S. Smith, in the latter part of the nineteenth century and Minkowski, in the first part of this, made notable and systematic contributions to the theory. In modern times the theory has been made much more elegant and complete by the works of Hasse, who used p-adic numbers to derive and express results of great generality, and Siegel whose analytic methods superseded much of the laborious classical theory. Contributors have been L. E. Dickson, E. T. Bell, Gordon Pall, A. E. Ross, the author and others. Exhaustive references up to 1921 are given in the third volume of L. E. Dickson's History of the Theory of Numbers.

The purpose of this monograph is to present the central ideas of the theory in self-contained form, assuming only knowledge of the fundamentals of matric theory and the theory of numbers. Pertinent concepts of p-adic numbers and quadratic ideals are introduced.