Colloquium publications.

22

THE MADISON COLLOQUIUM.

For p prime to n, a fundamental system of seminvariants of F,
is given by Ao, 02, **, an together with a fundamental system of
the particular form of order n - 1
F_, = PoFn/y
(23)
(2PoAlxn-l+PoA2x -2y+... +PoAnyn-l (mod p),
where Po = 1 - AoP-1.
Indeed, Ao, a2, * *, ao completely characterize the classes of
forms Fn with Ao = 0. Since yFn-' -= F, identically, when
A0 = 0, the classes of forms Fn with Ao = 0 are completely
characterized by the seminvariants of the fundamental system
for Fnl'.
For example, Ao and PoA1 form a fundamental system of
modular seminvariants of Aox + Aly (since these characterize
the classes represented by Aox and A1y). The corresponding
functions for
Fl' = PoA1x + PoA2y
are PoA1 and
{1 - (PoA1)P-1}PoA2- (1 - AP-1)PoA2 = P1A2 (mod p).
Hence the theorem shows that, if p > 2,
(24) Ao, 22 = 4AoA2 - A12, PoA1, P1A2
form a fundamental system of modular seminvariants of F2. For
f2, these are
(24') 2ao, S2 = aoa2- a2, Poal, Pla2.
8. Order a Multiple of the Modulus.-Next, let n = pq. By
Fermat's theorem, x - xyp- and hence
(25) A = Ao(x - xyP-1)
is unaltered modulo p by any transformation (1). Hence if, for
each value of the seminvariant Ao, we separate the forms
(26) F,_ -- (Fn -
y