130 REPORT 1860. 



(i.) " The necessary and sufficient condition that f(a) should be divisible 

 by q, is that the congruences 



^ (w*;)=0, ^(KijsO, $f-x (ttjfc)=MO, mod q, 



should be simultaneously satisfied for every value of k." 



(ii.) " And the condition that the norm of /(a) should be divisible by q, 

 is that the same congruences should be satisfied for some one value of k." 



When the congruences >// (ma.)=0, ^(mj^sO, .... i/y-i(M/fc)— °> mo ^ 9' 

 are simultaneously satisfied, /(«) is said to be congruous to zero {inod q),for 

 the substitution r) =Uk- These /congruences may be replaced by a single 

 congruence in either of two different ways. Thus, if we denote by F(?/ ) the 

 complex number involving the periods only which we obtain by multiplying 

 together the/ complex numbers 



f(a),f(ay e ),f(ay 2e ),....f(ay (f - 1)e ), 



it may be proved that the single congruence F(i<j-)=0, mod q, is precisely 

 equivalent to the/ congruences 



Or, again, if we denote by ^(»? ) a complex number congruous to zero for 

 every one of the substitutions j/ =i< 1 , 7 f =u !i , .... rj =u e -i, but not con- 

 gruous to zero for the substitution v =u (such complex numbers, involving 

 the periods only, can in every case be assigned)*, it is readily seen that the 

 same/ congruences are comprehended in the single formula 



¥(»;e-i)/(a)=0, mod q. 



The utility of this latter mode of expressing the/ congruences will appear in 

 the sequel: the formula F(wi)=0, mod q, is of importance, because it 

 supplies an immediate demonstration of the important proposition, that "if 

 a product of two factors be congruous to zero for the substitution ?j =w;t, 

 one or other of the factors must be congruous to zero for that substitution." 

 46. Definition of Ideal Prime Factors. — To develope the consequences of 

 the preceding theorems, let us consider a prime number q appertaining to 

 the exponent/; and let us first suppose that it is capable of being expressed 

 as the norm (taken with respect to the periods) of a complex number 4>(r) ), 

 which contains the periods of/ terms only; so that 



9=^( r lo) «K>h) ^(Ve-i)' 



If the substitution of u in \p render ^(11^=0, mod q, we may distinguish 

 the e factors of q by meaus of the substitutions which respectively render 

 them congruous to zero; so that, for example, vi(?/ e — ft) is the factor apper- 

 taining to the substitution j/ =wj.. 



We thus obtain the theorem that if/(a) be congruous to zero, mod q, for 

 any substitution ri =u , /(a) is divisible by the factor of q appertaining to 

 that substitution. For if *//(jy ) be that factor of q, 



H%)~ 9 



but f(a)\p(t] ^^(tIz) ...i//(jj(,_i) is congruous to zero, mod q, for every one of 

 the substitutions j? =m , j? =Mj> ... v — u e-i ! '* * s consequently divisible 

 by q; i.e. /(a) is divisible by ^(>? ). A useful particular case of this theo- 

 rem is that uu— »?,fc=0, mod ^(jj ), *f *K M o)=°> m °d 9- 



* Crelle, vol. liii. p. 145. The number ¥(?;) of this memoir possesses the property in 

 question. 



