CHAP, viii] NUMBER OF BOOTS OF CONGRUENCES. 233 N=N* (mod p), where N*+l is derived from either of Hurwitz's two sums for N+ 1 by replacing p by P. The same replacement in Hurwitz's expression for A — 1 leads to the invariant A*— 1, where A* is congruent modulo p to the number of distinct sets of solutions hi the Galois field of order pnm of the equation /(a?!, z2) -0. G. Rados47 considered the sets of solutions of , y) =s x»-2+a(» **-3+ . . . +a£!a)ir*-a=0 (mod p) for a prune p. Let Ak denote the matrix of D, in (3), with a{ replaced by a,ik\ Let C denote the determinant of order (p— I)2 obtained from D by replacing ak by matrix Ak. Then/=0 has a solution other than z=y=0 if and only if C is divisible by p; it has exactly r sets of solutions other than x==i/=0 if and only if C is of rank (p— I)2— r. To obtain theorems including the possible solution rc=2/=0, use <l>(x, y) = 2 W} *"-1+a{*)o;'>-2+ . . . +a<,*21)2/*-fc-1=0 (mod p), QI . . . ap_3 ap_2 «P-i \ 02 ... ap_2 Op-i+aoO I 0 and ak derived from a by replacing a* by af\ Let 7 be the determinant derived from |a| by replacing ak by matrix ak and 0 by a matrix whose p2 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 7 is of rank p2—r. *P. B. Schwacha48 discussed the number of roots of congruences. *G. Rados49 treated higher congruences. THEORY or HIGHER CONGRUENCES, GALOIS IMAGINARIES. C. F. Gauss,60 hi 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 £(z)=0 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 Serret74 shows, such imaginaries can be "Ann. Sc. ficole Normale Sup., (3), 27,1910, 217-231. Math. 6s Termds firtesitd (Report of Hungarian Ac.), Budapest, 27, 1909, 255-272. "Ueber die Existenz und Anzahl der Wurzeln der Kongruenz 2^=0 (mod m), Progr. Wilher- ing, 1911, 30pp. "Math. &. Termed Ertesito, Budapest, 29, 1911, 810-826. "Werke, 2, 1863, 212-240. Maser'u German translation of Gauss' Disq. Arith., etc., 1889, 604-629.