Colloquium publications.

INVARIANTS AND NUMBER THEORY.

5

transformation
Xi = a 2I y x- =- alml % m -- (i 2, * * *x i - i1)
of determinant unity. Hence there exists a linear transformation with complex coefficients of determinant unity which
replaces qm by
(4) Xz2 + + X21I + DXm2 X12 +.. + Xr2,
according as r = m or r < m. In the first case, the final coefficient is D since the determinant (2) of a form qm equals that
of the form derived from qm by any linear transformation of determinant unity. Hence all quadratic forms (1) may be separated
into the classes
(5) Cm.D, Cr (D + O r=0,1,..., m- 1),
where, for a particular number D 4= 0, the class Cm, D is composed
of all forms qm of determinant D, each being transformable into
(41); while, for 0 < r < m, the class Cr is composed of all forms
of 'ank r, each being transformable into (42); and, finally, the
class Co is composed of the single form with every coefficient
zero. In the last case, the determinant D is said to be of rank
zero. Using also the fact that the rank of the determinant of a
quadratic form is not altered by linear transformation, we conclude that two quadratic forms are transformable into each other
by linear transformations of determinant unity if and only if they
belong to the same class (5).
2. Single-valued Invariants of qm.-Using the term function
in Dirichlet's sense of correspondence, we shall say that a singlevalued function 0 of the undetermined coefficients Oij of the
general quadratic form qm is an invariant of qm if 0 has the same
value for all sets /3.j,.j, * * * of coefficients of forms qi, q,..*
belonging to the same class.* The values V,, vD Vr of 0 for the
various classes (5) are in general different. For example, the
determinant D is an invariant; likewise the single-valued func* Briefly, if 4 has the same value for all forms in any class.