Introduction. If f and g are forms with integer coefficients and n and m variables respectively (n > m) the results of the last chapter give us methods of finding whether or not some form in the genus of f represents g integrally. Corollary 44a or 44b may be used to show that the existence of such a representation depends on the solvability of the congruence f ≡ g (mod 8 | g | P) where P is the product of the odd primes in |g|·|f|, or on the existence of representations in R(p) for all p dividing 2|f||g|. When there is only one class in the genus of f, the same criteria serve to determine the existence of representations of g by the form f. However, when there is more than one class in the genus except for certain very special cases and asymptotic results there are no known criteria for existence of representations.

When it comes to determining the number of representations of g by f, known results, except for those in section 38 below, depend on analytic theory which is beyond the scope of this book. However we shall describe such conclusions.

For the case m = n two fundamental problems arise. First, the question of equivalence cannot in general be elegantly resolved in the ring of rational integers. Faced with such a problem one would first test for semi-equivalence by methods of the previous chapter; then employ a reduced form such as is described in theorem 23.

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of Hasse, who used p-adic numbers to derive and express results of great generality, and Siegel whose analytic methods superseded much of the laborious classical theory.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.

Congruence. When two forms with rational coefficients may be taken into each other by linear transformations with rational elements, the forms are frequently called rationally equivalent. But, to be consistent with our terminology, we shall call two such forms rationally congruent or congruent in the field of rational numbers and reserve the term “equivalent” for transformations with coefficients in a ring. We shall see that there is an intimate connection between the fundamental results of this chapter and those of the previous chapter.

Since the rational numbers form a field we have shown in theorem 1 that every form is rationally congruent to a diagonal form. As in the last chapter, we can specialize this still further; for (r/s)x2, where r and s are integers, becomes rsy2 if x is replaced by sy and any square factor of a coefficient may be absorbed into the variable. Hence we have

Theorem 22. Every form with rational coefficients is rationally congruent to a diagonal form whose coefficients are square-free integers (that is, integers with no square factors except 1).

Equivalence and reduced forms. So far in this book we have considered transformations whose elements are in the same field as the coefficients of the form.

Definitions. Reasoning by analogy from the results of chapter III one might deduce that the function of this chapter would be to prove that if a form f with integral coefficients represents a number N in R(p) for all p and in the field of reals, then there would be integer values of the variables of f for which f = N. One might also suppose that a similar result would hold for equivalence in R(p) and in the ring of integers. But, while it is true that equivalence (or representation) in the ring of integers implies equivalence in R(p) for all p, yet the converse statement is not true as is shown, for instance, by the fact that 8/5, 1/5 is a solution of f = x2 + 11y2 = 3 in the field of reals, in R(2), R(3) and R(11). Thus f represents 3 in all R(p), from corollary 14 and theorem 34, but f = 3 has no solution for integer values of x and y. However, two things do follow from the fact that f represents 3 in all R(p). First, for any integer q, there is a solution of f = 3 in rational numbers with denominators prime to q. Second, there is a form g with integer coefficients such that g = 3 has an integral solution and such that for every integer q there is a transformation which takes f into g and whose elements are rational numbers with denominators prime to q.

Introduction. Most of the results in this chapter are classical—some dating back to the time of Gauss and earlier—and can be derived independently of the previous general theory. But viewing the binary forms as special cases of our previous results illuminates the general theory on the one hand and economizes labor on the other. Furthermore certain problems, such as the determination of all automorphs, inaccessible in the general case, can be completely solved for binary forms.

Since much of the theory of binary forms was developed in advance of the general theory there is a wide divergence in the use of the term “determinant” as applied to a form. Gauss wrote the binary form a s f = ax2 + 2bxy + cy2 and defined the determinant of f to be b2 − ac. Kronecker preferred f = ax2 + bxy + cy2 and called b2 − 4ac its determinant. These expressions or their negatives have been variously referred to as the “discriminant” of the form. The confusion of terminology is so great that, in reading the literature, one must take great care to inform himself of the meaning of the author. We shall in this book make a clean break with tradition and define the determinant of a binary form just as it was denned for forms in more variables. That is, the determinant of ax2 + 2b0xy + cy2 shall be ac − b02 and that of ax2 + bxy + cy2 shall be ac − b2/4.