Colloquium publications.

38

THE MADISON COLLOQUIUM.

We are now ready to prove that any rational integral invariant
I, with integral coefficients, of the group G is a rational integral
function of L and Q with integral coefficients.
After removing possible factors L and Q, we may assume that
I vanishes for no special point. If I is not a constant, it vanishes
at a point (c, d) and hence at the co distinct points conjugate with
(c, d) under the group G. The invariants*
p+l p(P-1)
(11) q= Q 2, l= L 2
are of degree co. The constant r, determined by
q(c, d) + r ~ l(c, d) 0 (mod p),
is a root of a congruence of a certain degree t with integral coefficients and irreducible modulo p. Now q + rl is a factor of I.
Since q, 1 and I have integral coefficients, I has also the factors
(12) q + TPl, q + rPl, ***, q + TP-l.
For, by Galois's theorem mentioned above,
T P2 ''2-1
are the roots of our irreducible congruence of degree t. Since
the conditions which imply that q + zl shall be a factor of I are
congruences satisfied when z = r, they are satisfied when z = rpk.
Hence if we multiply q + rl by the product of the invariants
(12), we obtain an invariant T with integral coefficients modulo p.
Since L and Q have no common factor, no two of the functions
q + rl and (12) have a common factor. Hence T is a factor of I.
Proceeding in like manner with I/T, we arrive finally at the
truth of the theorem.t
3. Invariants of Smaller Binary Groups.-We shall later need the
theorem that a fundamental system of rational integral invariants
* If p = 2, we omit the divisor 2 in the exponents.
t Proved less simply in Transactions of the American Mathematical Society,
vol. 12 (1911), p. 1. Still simpler is the proof that various coefficients of an
invariant are zero, Quarterly Journal of Mathematics, 1911, p. 158.