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.