ON THE THEORY OF NUMBERS. 165 



and by continuing this process, we shall obtain the partial resolution of T(x) 

 to which we have referred. 



71. Imaginary Solutions of a Congruence. — We have said that the non- 

 linear modular factors of F(.r)=0, mod p, may be considered to represent 

 imaginary solutions. These imaginary solutions can be actually exhibited, 

 if we allow ourselves to assign to x certain complex values. The following 

 proposition, which shows in what manner this may be effected, is due to 

 Galois : — 



"If f(x) represent an irreducible modular function of order m, the con- 

 gruence 



F(fl)=0, mod !>/<»]. 



is completely resoluble when F(x) is an irreducible modular function of 

 order in, or of any order the index of which is a divisor of in." 



To establish this theorem, write for x in equation (B); we find Op™- 1 — 1 

 =11 F(0), mod />, the sign of multiplication n extending to every irreducible 

 modular function having m or a divisor of m for the index of its order. 

 But the congruence 0P m - 1 ==l, mod [jo, /(#)], admits of as many roots as 

 it has dimensions; therefore also every divisor of flp'"- 1 — ], and, in particular, 

 the function F(0) considered as a congruence for the same compound modu- 

 lus, admits of as many roots as it has dimensions. 



Let the order of the congruence F(0)=O, mod [jo, /(#)], be S, and let 

 any one of its roots be represented by r ; it may be shown that all its roots 

 are represented by the terms of the series r, rP, rP 2 , . . . rP s ~ l . For, if 

 F(r)=0, mod [>,/(»], we have also F(rP)=[F(r)]/'=0, mod [p,f(x)j, 

 and similarly F(7-/> 2 ) = [F(r)]^=0, mod p; so that r, rP, rP 2 , . . . rP s ~ x are 

 all roots of F(0)=O, mod Ljp, /(»)]. It remains to show that these $ func- 

 tions are all incongruous, mod [jo, /(#)]. If possible let rP k+k '^rP k ', 

 mod lp,f(x)'], h and k' being less than I; we have, raising each side of this 

 congruence to the power p s ~ k ', r p s+k =rP s , mod ip,f(x)~], i.e. rP k =r, or 

 r p k -i = l, mod [/»,/(*)], observing that rP*=r, mod [jo,/(»], because 

 r p s ~ l — l i s divisible by F(V) for the modulus p. We conclude, therefore, 

 that r is a root, mod \_p, /(#)], of some irreducible modular divisor of the 

 function Qp k -i — 1, i. e. of some irreducible function of an order lower than d, 

 because k is less than 2; r is therefore a root, mod [£>,/(#)], of two different 

 irreducible modular functions, which is impossible. 



If, therefore, we suppose x to represent, not an indeterminate quantity, 

 but a root of the equation f(x)=0, we may enunciate Galois' theorem as 

 follows : — 



" Every irreducible congruence of order in is completely resoluble in com- 

 plex numbers composed with roots of any equation which is irreducible for 

 the modulus jo, and which has m or a multiple of in for the index of its order. 



"Anil all its roots may be expressed as the powers of any one of them." 



72. Congruences having Powers of Primes for their Modules.— \t remains 

 for us to advert to the theory of congruences with composite modules — a sub- 

 ject to which (if we except the case of binomial congruences) it would seem 

 that the attention of arithmeticians has not been much directed. We shall 

 suppose, first, that (he modulus is a power of a prime number. 



The theorem of Lagrange (art. If), and the more general proposition of 

 art. 69, in which it is (as we have seen) included, cannot be extended to 

 congruences having powers of primes for their modules. 



Let the proposed congruence be F(a-) = 0, modjo" 1 ; and let us suppose 

 (what is here a restriction in the generality of the problem) that the coeffi. 



