Colloquium publications.

INVARIANTS AND NUMBER THEORY.

31

In place of the fourth and third we may evidently use
X = (1 -A12)A2, = A1A3 + AoA2- A12A22 = t + Ao2 + X2.
Here o is the discriminant of F3 for p = 3. By ~ 13 there are
11 classes of forms F2. Hence, by ~ 8, there are 3.11 classes of
forms F3. Thus there are exactly 33 linearly independent
seminvariants of F3. Since
A1X A1- O0, aX AoX2, ((T- + Ao2) = 0,
1,(X + Ao) = 0, (1 - A12) AoX,
modulo 3, any polynomial in the seminvariants AO, Al, a, X,,
of the fundamental system is congruent to a linear function of
(44) AoiAlj, Aoirk, AoiAlak, AoiXk, Aoi0k (i, j=0, 1, 2; k=1, 2).
Hence these 33 functions form a complete set of linearly independent seminvariants of F3. The seminvariants
P = 1 - A12 X2 = (1 A2)(1 - A22),
3
(45) Io = (1- Ao2)(P - 2) = (1 - A2),
i=O
E = AoA(a - a2) + Aol = AoA3(AoA2- A1A3+A12-A22)
are seen to be invariants as follows.* The weights of the terms of
each are all even or all odd. Moreover, under the substitution
(AoA3)(A1A2), induced upon the coefficients of F3 by the
interchange of x and y, the functions a, P and Jo are unaltered,
while E is changed in sign. Hence a, P, Io are absolute invariants, while E is an invariant of index unity. We now have
7 linearly independent invariants
(46) Io, E, E2, a', (2, p, 1.
Noting that
(47) E2 = Ao02J + Ao2(- - a-2 + X2) - AoX,
* Or by general theorems, Transactions of the American Mathematical
Society, vol. 8 (1907), pp. 206-207. Note that E is the eliminant of F3 0,
= x, y3 = y (mod 3).