232 History of the Theory of Numbers. [Chap, viii 
where the summation extends over the sets of solutions ^ of 
ao+ai+- • .+a, = p — 1, ai + 2a2+. . .+ra^=0 (mod p-1). 
The right member is an invariant modulo p oi f{xi, x^ with respect to all 
linear homogeneous transformations on Xi, Xo with integral coefficients 
whose determinant is not divisible by p. The final sum in the expression 
for .4 — 1 is congruent to A^+1. If r = 2, p>2, the invariant is congruent 
to the power (p — 1)/2 of the discriminant a^—^aQa^ of/. 
*E. Stephan^^ investigated the number of roots of linear congruences 
and systems of congruences. 
H. Kuhne^ considered J{x)=x"'-\- . . .-\-a„ with no multiple irreducible 
factor and with a^ not a multiple of the prime p. For n<m, let ^ = x'*+ . . . 
+ 6„ 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 ^'s of degree n which have no common factor with / modulo p is p„, 
where 
P,.^^\R{J, gYr (mod p':), co = p''-np-l), 
the summation extending over the p" possible ^'s. He expressed p„ as 
a sum of binomial coefficients. For any two binary forms 4>, \p of degrees 
w, n, it is shown that 
is invariant modulo p" under linear transformations with integral coeffi- 
cients of determinant prime to p ; Ji is Hurwitz's^" invariant. 
M. Cipolla^^ used the method of Hurwitz^^ to find the sum of the kth 
powers of the roots of a congruence, and extended the method to show that 
the number of common roots of /(x) = 0, g{x) = (mod p), of degrees r, s, is 
congruent to —XCjKi, where i, j take the values for which 
0<i^s{p-l), 0<;^r(p-l), i+j=0 (mod p), 
the Cs being as with Hurwitz, and similarly 
g{xy-' = Ko+K^x+.... 
The number of roots common to n congruences is given by a sum. 
L. E. Dickson^^ gave a two-fold generalization of Hurwitz's^^ formula for 
the number of integral roots of f{x) = (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 generaliza- 
tion is obtained by taking the coefficients Oi of f{x) to be elements in the 
Galois field of order p'* (cf. Galois^^ etc.). Then let N be the number of 
roots of f{x) = which belong to the Galois field of order P = p"'". Then 
"Jahresber. Staatsoberrealsch. Steyer, 34, 1903-4, 3-40. 
"Archiv Math. Phys., (3), 6, 1904, 174-6. 
«Periodico di Mat., 22, 1907, 36-41. 
«BuU. Amer. Math. Soc, 14, 1907-8, 313. 
