164 REPORT — 1860. 



0j(P m -»=l, mod [>,/(>)]; in the latter, 04O> m -D=— 1, mod [>, /(*)]• 

 And, further, if 6^ and 0. 2 be two primary irreducible modular functions of 



the orders m and n respectively, and if we use the symbols -J and -2 to 

 denote the positive or negative units which satisfy the congruences 6i(P n -' l '> 

 = P-±] , mod (p, 2 ), and 2 4 {l ' m ~ l) = T^l , mod (j>, d,), respectively, these 



two symbols are connected by the law of reciprocity -1 =(— 1)"" \ -~\- 



But the equation (A) admits also of an immediate application to the theory 

 of ordinary congruences with a simple prime modulus. 



In that equation let us assign to 6 the particular value x ; Ave conclude that 

 the function ccp 1 "- 1 — 1, is divisible for the modulus p by/(#), *.«• by every 

 irreducible modular function of order m. Further, if d be a divisor of m, 

 xP m - 1 — 1 is algebraically divisible by xP d ~ l — 1 ; whence it appears that 

 x p m -\ — J j s divisible, for the modulus p, by every function of which the 

 order is a divisor of m. But it is easily shown that xP m ~ l — 1 is not divisible 

 (mod p) by any other modular function, and that it cannot contain any 

 multiple modular factors. Hence we have the indeterminate congruence 



#p»-i_1 ==n/0), mod p, (B) 



in which f(je) denotes any primary and irreducible function, the order of 

 which is a divisor of m, and the sign of multiplication II extends to every 

 value of f{x). This theorem, again, is a generalization of Lagrange's inde- 

 terminate congruence (art. 10). We may infer from it that, when m is >1, 

 the number of primary functions of order m, which are irreducible for the 

 modulus^, is 



i r — m m ~i 



q x , q 2 , . . .denoting the different prime divisors of m. As this expression is 

 always different from zero, it follows that there exist functions of any given 

 order, which are irreducible for the modulus p. 



A congruence F(.r) = 0, mod p, may be considered resolved when we 

 have expressed its left-hand member as a product of irreducible modular 

 factors. The linear factors (if any) then give the real solutions ; the factors 

 of higher orders may be supposed to represent imaginary solutions. We have 

 already observed that even when all the modular factors of F(.r) are linear, 

 we possess no general and direct method by which they can be assigned ; it 

 is hardly necessary to add that the problem of the direct determination of 

 modular factors of higher orders than the first, presents even greater diffi- 

 culties. Nevertheless the congruence (B) enables us to advance one step 

 toward the decomposition of F(.r) into its irreducible factors; for, by means 

 of it, we can separate those divisors of F(.r) which are of the same order, 

 not, indeed, from one another, but from all its other divisors. We may first 

 of all suppose that F(.r) is cleared of its multiple factors, which may be 

 done, as in algebra, by investigating the greatest common divisor of F(.r) 

 and F'(.r) for the modulus p. The greatest common divisor (mod p) of 

 F(#) and ccP~ l — 1 will then give us the product of all the linear modular 

 factors of F(^); let F(.r) be divided (mod p) by that product, and let the 

 quotient be F,(.r); the greatest common divisor (mod p) of F 1 (x) and 

 scp" 1 - 1 — 1 will give us the product of the irreducible quadratic factors of F (a-*); 





