ON THE THEORY OP NUMBERS. 129 



vol. liii. p. 142, in which he has established this fundamental proposition on 

 a satisfactory basis.) 



45. Conditions for the Divisibility of the Norm of a Complex Number by 

 a Real Prime*. — Instead of the complex number 



f(«)=a + a 1 a + a. 2 a 2 + +a\_s<A*" a > 



let us now, for a moment, consider the complex number 



1 K'?o) ==c o ? 7o+ c i'?i+ c i !'7 2 + •••• +c«-ii/ e -i, 



which, with its conjugates 



»/ / (Vi)=< 7 o»7 1 +c 1 v 2 +c 2 )73+ .... +c e -i Vo , 



is a function of the periods only, and is therefore a specialized form of the 

 general complex number/(a) ; and let q still denote a real prime, satisfying 

 the congruence g-^=l, mod X. By means of the relation subsisting between 

 the equation-roots w , v^ . . j/e-i, and the congruence-roots u Q , u v . . «<■_], 

 M. Kummer has demonstrated the two following theorems: — 



(i.) "The necessary and sufficient condition that \p(i]) should be divisible 

 by q (i. e. that the coefficients c , c v . .. c e -i should be all separately divi- 

 sible by q) is that the e congruences 



»K M o) = c o M o +c 1 « 1 +c 2 « 2 + .... +c e _n« e _,=0, mod q, 



»K M i) = c o M i +c l u a +e a u 3 + +ce-i u = 0, mod?, 



\p(u e -i)=c u e - 1 +c 1 u +c 2 u i + .... +<?e-i «e_2 = 0, mod q, 



should be simultaneously satisfied." 



(ii.) " The necessary and sufficient condition that the norm of vp(V), taken 



with respect to the periods, i. e. the number ^(ijJiKih) ^(ve~\)> should 



be divisible by q, is that one of the e congruences 



H u o) =0> »K«i) = 0, >K»«-i) =0> mod q, 



should be satisfied." 



These results may be extended to any complex number /(a), by first 

 reducing it to the form 



This is always possible ; for, since the / roots which compose any one 

 period, e. g. y , are the roots of an equation x ( ct )=0 of order/, the coeffi- 

 cients of which are complex integers involving the periods onlyf, we may 

 simply divide/(a) by x( a )> and the remainder will give us the expression 

 of /(«) in the required form. Further, let q now denote a prime apper- 

 taining to the exponent f (not merely satisfying the congruence y/=J,mod A, 

 but also satisfying no congruence of lower index and of the same form). 

 The two preceding theorems are then replaced by the two following, which 

 are analogous to them, and include them. 



The outline of the theory of complex numbers contained in this and the subsequent 

 ticles is chiefly derived from M. Rummer's memoire in Liouville, vol. xvi. p. 411. 

 t Disq. Arith. art. 348. 



articles 



t D 



1860. 



K 



