Colloquium publications.

INVARIANTS AND NUMBER THEORY.

47

is involved in the assumption that a certain cj is not divisible by p.
First, the remaining c; are divisible by p. For if also ci q 0,
let kgArirqi be the term of Pi of highest degree in A. Since
yo and r are of degrees p and np, and of weights - 2 and 0
(mod p - 1), 7oiPi is of degree pi + 2ri + sinp and of weight
2i + 2ri (mod p- 1). But p- 1 (mod n). Hence
i + 2ri j+ 2rj, 2i + 2ri 2j + 2rj (mod n),
so that i — j (mod n). But i and j are positive integers < n.
Hence i = j. Multiplying our invariant by a suitably chosen
integer, we have the invariant
(39) yjPj(A, r) + E yo0i(a, A, r), Pj = ATrs +...
i=O
Now - (c - ka)b-1 is the term of highest degree in b in 7k.
Hence
(40) = - b-l- *', r = abn(p-l) +..,
(41) a = n{- (c - ka)} (- c) + (- a)~ (mod p),
where k ranges over the non-residues of p, the last following
from (34) for x = c/a. Since o7 and r are of even weights,
only even powers of b enter (39). Hence an invariant (39) is
symmetrical in a and c. We shall prove that this is not the
case for the terms of highest degree in b. For 7ojPy this term is
(42) (- c)jorb, 3 = j(p - 1) + 2r + sn(p - 1).
Let CiaeiAfirgi be one of the terms of 4i in which the exponent of
b is a maximum. Then in 7yoii the highest power of b occurs
in the terms
(43) Ciaei(- c)iagb, f = 2f + gin(p - 1) + i(p - 1).
Since the weight and degree is the same as for (42),
2i + fi 2j + 3 (mod p- 1),
(44)e+ + n+ = +s+
ei q- i q- gin - fii = j q- sn - fl.