240 History of the Theory of Numbers. [Chap, viii 
R. Dedekind"^ developed the subject of higher congruences by the 
methods of elementary number theory without the use of algebraic prin- 
ciples. As by Gauss^° he developed the theory' of the g. c. d. of functions 
modulo p, a prime, and their unique factorization into prime (or irreducible) 
functions, apart from integral factors. Two functions A and B are called 
congruent modulis p, M, \i A—B is divisible by the function AI modulo p. 
We may add or multiply such congruences. If the g. c. d. of A and B is 
of degree d, Aij=B (mod p, M) has p'^ incongruent roots y{x) modulis p, M. 
Let <t){M) denote the number of functions which are prime to M modulo 
p and are incongruent modulis p, M. Let ii be the degree of M. A pri- 
mary function of degree a is one in which the coefficient of a:" is = 1 (mod p). 
If D ranges over the incongruent primary divisors of M, then 20(Z))=p''. 
If M and A^ are relatively prime modulo p, then 4){MN) =4){M)<t){N). If 
A is a prime function of degree a, 0(A'') =p*'(l -~ Vp")- If Af is a product 
of powers of incongruent primary prime functions a,. . ., p, 
*W=p-(i-l)...(i-l). 
If F is prime to M modulo p, F'^^-^^^= 1 (mod p, M), which is the generaliza- 
tion of Fermat's theorem. Hence if A is prime to M, the above Unear con- 
gruence has the solution y=BA'^~^. 
If P is a prime function of degree tt, a congruence of degree n modulis p, 
P has at most n incongruent roots. Also 
(3) y''-'-l^Il{y-F)^(mod p, P), 
identically in y, where F ranges over a complete set of functions incongruent 
moduhs p, P and not divisible by P. In particular, l+nF=0 (mod p, P), 
the generalization of Wilson's theorem. 
There are (/)(p'— 1) primitive roots modulis p, P. Hence we may em- 
ploy indices in the usual manner, and obtain the condition for solutions 
of ?/"=A (mod p, P), where A is not divisible by P. In particular, A 
is a quadratic residue or non-residue of P according as 
^(p'-i)/2^_^^ or -1 (mod p, P). 
His extension of the quadratic reciprocity law will be cited under that topic. 
A function A belongs to the exponent p with respect to the prime func- 
tion P of degree tt if p is the least positive integer for which A^''=A (mod 
p, P). Evidently p is a divisor of tt. Let N{p) be the number of incon- 
gruent functions which belong to an exponent p which divides w. Then 
p^='ZN'{d), where d ranges over the divisors of p. By the principle of 
inversion (Ch. XIX), 
isr(p) =p''-2p''/''+2:p''/''*-2p''/'''^+ . . ., 
where a, 6, . . . are the distinct primes di\'iding p. Since the quotient of 
this sum by its last term is not divisible by p, we have A^(p)>0. 
"Jour, fiir Math., 54, 1857, 1-26. 
