232 HlSTOKY OP THE THEORY OF NUMBERS. [CHAP. VIII where the summation extends over the sets of solutions SO of oo+ai+.. .+0r=p-l, a!+2a2+. . .+raf=0 (modp-1). The right member is an invariant modulo p of f(xit x2) with respect to all linear homogeneous transformations on xly x2 with integral coefficients whose determinant is not divisible by p. The final sum in the expression for A — I is congruent to N+l. If r=2, p>2, the invariant is congruent to the power (p — 1)/2 of the discriininant ai2— 4aoa2 of/. *E. Stephan43 investigated the number of roots of linear congruences and systems of congruences. H. Kuhne44 considered f(x) =xm+ . . . +am with no multiple irreducible factor and with am not a multiple of the prime p. For n<m, let g = xn+ +&« have arbitrary coefficients. The resultant R(f, g) is zero modulo p if and only if / and g have a common factor modulo p. Thus the number of all g's of degree n which have no common factor with/ modulo up is pn, where Pn=S|#(/, g)\* (mod pn), w=pn-1(p-l), the summation extending over the pn possible g's. He expressed pfl as a sum of binomial coefficients. For any two binary forms <£, ^ of degrees m, n, it is shown that is invariant modulo pn under linear transformations with integral coefficients of determinant prime to p; Ji is Hurwitz's42 invariant. » M. Cipolla46 used the method of Hurwitz42 to find the sum of the Jfeth powers of the roots of a congruence, and extended the method to show that the number of common roots of /(x)=0, 0(z)=0 (mod p), of degrees r, s, is congruent to —S (?//£,•, where i, j take the values for which 0<^s(p-l), 0<j^r(p-l), i+j=Q (modp), the C's being as with Hurwitz, and similarly The number of roots common to n congruences is given by a sum. L. E. Dickson46 gave a two-fold generalization of Hurwitz's42 formula for the number of integral roots of /(x) = 0 (mod p). The first generalization is to the residue modulo p of the number of roots which are rational in a root of an irreducible congruence of a given degree. A further generalization is obtained by taking the coefficients a» of f(x) to be elements in the Galois field of order pn (cf. Galois62, etc.). Then let N be the number of roots of f(x] =0 which belong to the Galois field of order P = pnm. Then "Jahresber. Staatsoberrealach. Steyer, 34, 1903-4, 3-40. "Archiv Math. Phys., (3), 6, 1904, 174-6. ^Periodico di Mat., 22, 1907, 36-41. -Bull. Arncr. Math. Soc., 14, 1907-8, 313.