CHAPTER V - Genera and semi-equivalence

Summary

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.