ON THE THEORY OF NUMBERS. 131 



Again, it may be shown that these complex factors of q are primes in the 

 most proper sense of the word : i. e., first, that they are incapable of reso- 

 lution into any two complex factors, unless one of those factors be a complex 

 unit; and secondly, that if any one of them divide the product of two factors, 

 it necessarily divides one or other of the two factors separately. That \p(r) ) 

 possesses the first property is evident, because its norm is a real prime, and 

 that it possesses the second is a consequence of the last theorem of Art. 45. 

 For if iK»/ ) divide/j(a) x/ 2 (a), either f(a) or/ 2 (a), by virtue of that theo- 

 rem, is congruous to zero (mod q) for the substitution n = u ; that is to say, 

 either/ (a) or/ 2 (a) is divisible by ^(>? ). 



Now, if every prime q which appertains to the exponent/ were actually 

 capable of resolution into e complex factors composed of the e periods of 

 /roots, these factors would represent to us all the true primes to be con- 

 sidered in the theory of the residues of Xth powers. And for values of X infe- 

 rior to 11, perhaps to 23, this is, in fact, the case. But for higher values of 

 X, the real primes appertaining to the exponent /divide themselves into two 

 different groups, according as they are or are not susceptible of resolution 

 into e conjugate factors. Let, then, q represent any prime appertaining to the 

 exponent/, whether susceptible or not of this resolution, and let /(a) still 

 denote a complex number which is rendered congruous to zero by the sub- 

 stitution 7/ =w ; /(a) is said by M. Kummer to contain the ideal factor of q 

 appertaining to the substitution rj =u . This definition is admissible, because 

 it is verified, as we have just seen, when q is actually resoluble into e con- 

 jugate factors; and its introduction is justified, as M. Kummer observes, by 

 its utility. To obtain a definition of the multiplicity of an ideal factor, we 

 may employ a complex number ^(r/) possessing the property indicated in 

 the last article. If of the two congruences 



l*Ml n /(«)=0, mod q n , 

 l>Oo)] n+1 /(«)=0, mod y»+», 

 the former be satisfied, and the latter not, /(a) is said to contain n times 

 precisely the ideal factor of q which appertains to the substitution n =u . 



47. Elementary Theorems relating to Ideal Factors. — The following pro- 

 positions are partly restatements (in conformity with the definitions now 

 introduced) of results to which we have already referred, and partly simple 

 corollaries from them. They will serve to show that the elementary proper- 

 ties of ordinary integers may now be transferred to complex numbers. 



(1.) A complex number is divisible by q when it contains all the ideal 

 factors of q. If it contain all of those factors n times, but not all of them 

 « + l times, it is divisible by q n , but not by q n+1 . 



(2.) The norm of a complex number is divisible by q when the complex 

 number contains one of the ideal factors of q. If (counting multiple factors) 

 it contain, in all, h of the ideal factors of q, the norm is divisible by q l f, but 

 by no higher power of q (/denoting the exponent to which q appertains). 



(3.) A product of two or more factors contains the same ideal divisors as 

 its factors taken together. 



(4.) The necessary and sufficient condition that one complex number 

 should be divisible by another is, that the dividend should contain all the 

 ideal factors of the divisor at least as often as the divisor. 



(5.) Two complex number-s which contain the same ideal factors are 

 identical, or else differ only by a unit factor. 



(6.) Every complex number contains a finite number of ideal prime fac- 

 tors. These ideal prime factors (as well as the multiplicity of each of them) 

 are perfectly determinate. 



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