Chap. VIII] NuMBER OF RoOTS OF CONGRUENCES. 233 
N=N* (mod p), where A''*+l is derived from either of Hurwitz's two sums 
for iV+1 by replacing p by P. The same replacement in Hurwitz's expres- 
sion for A — 1 leads to the invariant A* — l, where A* is congruent modulo p 
to the number of distinct sets of solutions in the Galois field of order p""* 
of the equation /(xi, 0:2) =0. 
G. Rados^'^ considered the sets of solutions of 
fix, y) = i\a\i^^ x^'-^+ai'' x''-^+ . . . +05,^^2)2/^"'"'= (mod p) 
for a prime p. Let Ak denote the matrix of D, in (3), with a^ replaced 
by ap\ Let C denote the determinant of order (p — iy obtained from D 
by replacing ak by matrix A^. Then/=0 has a solution other than x^y=0 
if and only if C is divisible by p ; it has exactly r sets of solutions other than 
x=y=0 if and only if C is of rank (p — 1)^— r. 
To obtain theorems including the possible solution x=i/=0, use 
cf>{x, y) = S W^ x^-i+af V-2+ . . . +af.,)y''-^-'^Q (mod p), 
a = 
k=0 
CL2 03 
\ap_i+aoai 
Op-3 Op-2 o,p-\ \ 
ap_2 Op.i+floO 
ap_i+ao Oi 
ap_3 ap_2 / 
and tt/fc derived from a by replacing a^ by al^\ Let y be the determinant 
derived from la| by replacing a^ by matrix a^; and by a matrix whose p"^ 
elements are zeros. Then ^=0 has a set of real solutions if and only if 
7=0 (mod p) ; it has r sets of solutions if and only if y is of rank p^—r. 
*P. B. Schwacha^^ discussed the number of roots of congruences. 
*G. Rados^^ treated higher congruences. 
Theory of Higher Congruences, Galois Imaginaries. 
C. F. Gauss/'' in a posthumous paper, remarked that "the solution of 
congruences is only a part of a much higher investigation, viz., that of the 
factorization of functions modulo p. Even when ^(x) = has no real root, 
^ may be a product of factors of degrees ^2, each of which could be said 
to have imaginary roots. If use had been made of a similar freedom which 
younger mathematicians have permitted themselves, and such imaginary 
roots had been introduced, the following investigation could be greatly 
condensed." As the later work of Serret^^ shows, such imaginaries can be 
*'Arm. Sc. ficole Normale Sup., (3), 27, 1910, 217-231. Math. 6s Term^s firtesito (Report of 
Hungarian Ac), Budapest, 27, 1909, 255-272. 
"Ueber die Existenz und Anzahl der Wurzeln der Kongruenz Sc<x* = (mod w), Progr. Wilher- 
ing, 1911, 30 pp. 
«Math. 6s. Term6s Ertesito, Budapest, 29, 1911, 810-826. 
"Werke, 2, 1863, 212-240. Maser's German translation of Gauss' Disq. Arith., etc., 1889, 
604-629. 
