Colloquium publications.

INVARIANTS AND NUMBER THEORY.

83

(71) A8=bi(b3-+l)A, b3A=ibb3ai+ * *, a2b3A=blb3ala2+- *,
J = blb2b3 + **, a2J= blb2b3a2 + ',
b3J = bib2b3(aia2 + al + a2) + *., AJ = b1b2b3A +.
These are linearly independent since the first eight do not involve
b1, while all the terms with the factor b1 in the next seven are
given explicitly, likewise all with the factor blb2b3 in the last
four. Hence the 19 functions (71) form a complete set of linearly
independent invariants of F under the group r.
17. Hence, in ~ 14, S1 is a linear combination of the functions
(71). By (672), S + Si is of the form (57) if S2 be denoted by
(56). Now a3be occurs in J, AJ, b3J, a2J, Ag, but in no further
function (71). In the first three, a3bl is multiplied by the linearly
independent functions (62), respectively; in the last two by
b3ala2 and aia2(b3 + 1), whose sum is congruent to the first
function (62). Hence the part of S + S1 involving J,., Ag3
is a linear combination of
(72) (b3 + a2)J = blb2b3aoia2 + b2b3alac2c3,
(73) J + b3J + AP = (b3 + 1) (blb2cla2 - b2A + A).
But b1 occurs in just six of the functions (71) other than the
five just considered. Thus the factor pal of b1 in (57) is a linear
combination of the coefficients of bl in (72), (73), g, a2g, A, a2A,
bsA, a2b3A. Now ai is a factor of the coefficients of b1 in all except
the second, third and fourth, while in these the coefficients are
(b3 + 1)b2o21c2, b3 - a2 + 1, a2b3
and are linearly independent. Hence (73), 3, a20 do not occur
in S + S1. By (57), the latter has no constant term and hence
involves 1, A only in the combination A + 1. This cannot
occur since the total coefficient of as must be of the form pR
and hence vanish for b3 = a2. At the same time we see that
the sum of the constant multipliers of A, a2A, b3A, a2b3A is zero
modulo 2. Hence S + Si is a linear combination of the functions