ON THE THEORY OF NUMBERS. 371 



7-1 



( — 1) 2 



If »i=3, mod 4, the value of ^ — > — (art. 125) corresponding to one of the 



two forms (A, B, C), (C, — B, A) is + */n, and that corresponding to the 



v-i 



( — 1) 2 

 other — s/n; if n=l, mod 4, the values of - — —^ — corresponding to those 



two forms are both\/« or both — n/n, according as the generic character of 



/-i /-i 



the two forms is ( — 1) 2 = + 1, or (— 1) 2 = — 1; in either case, therefore, 

 the decomposition of ^(n, x) into two factors, can be effected by comparing 



y-l 



it with the equations Xi(^ n > ^)=°» Xi(~ ^"> d ')=®> ^ Xi ( — W — »** ) =0 



is the equation satisfied by the multipliers appertaining to the transformations 

 of order n. 



But M. Kronecker has shown that ^(n, x) admits of a more profound de- 

 composition, when n is a composite number. In fact, if v is the number of 

 the primes p 1 p a lh • • ■ dividing n, ^ x {n, x) can be resolved into 2" factors, 

 each of the order —^ h(n), and containing no irrationalities but i/pa >/p 2 , 



*/p 3 , ... If n=3, mod 4, the order of each factor is precisely equal to the 

 number of classes in a properly primitive genus of det.— n; and the roots of 

 each factor are the values of 8 (w) corresponding to forms of a determinate 

 genus. If n= 1, mod 4, instead of the equation ^(n, x)=0, we consider the 

 equation, of which the roots are the h(n) quantities x(l—x) ; this equation 

 can, as before, be resolved into 2" generic factors, each corresponding to a 

 determinate genus. In either case the factors only differ from one another 

 in respect of the signs of the radicals *Jp l} *J p 2 , . . . , and these signs are de- 

 termined for each factor by the characters of the corresponding genus with 

 respect to the primes p lt p 2 , . . . For an account of the method by which this 

 resolution into generic factors is obtained, we must refer to the original note 

 of M. Kronecker (Monatsberichte, June 26, 1862, p. 370). 



In one of Abel's memoirs ((Euvres Completes, vol. i. p. 272) we find the 

 theorem that the modules which admit of complex multiplication are capable 

 of expression by radicals. At an earlier period Abel had doubted of the truth of 

 this result, and he has given no indication of the method by which he obtained 

 it. JM. Kronecker' s researches on complex multiplication were originally sug- 

 gested by Abel's theorem ; and they have led him to a complete demonstration 

 of it. Let n=3, mod 4, and let 8 ( w i)> 8 (w 2 ), ... be the roots of one of the 

 generic equations into which ^(x, n)=0 is resolved, the imaginaries 

 w u w 2 , . . . being determined by the equations 



A x + 28^ + 0^=0, A 2 +2B 2 u, 2 +C 2 wt=0, ... 



so that the forms (A u B„ C^, (A 3 , B 2 , C 2 ) belong to the same genus. If the 

 determinant — n is regular (art. 117), it can be proved by the theory of com- 

 position, that it is possible so to select the representative forms (A u B 1} C 2 ), 

 (A 2 , B 2 , C 2 ) as to satisfy the equations b).,=6w 1 , w 3 = dm 2 , . . ., denoting an 

 uneven number prime to n, or indeed a prime not dividing n. We may 

 thus represent the roots of the generic equation by the expressions ^ 8 (<i>i)> 

 ^(flwi), ^(O 2 ^) . . . But it can also be proved (by a comparison of 

 the equations relating to the transformations of the orders n and 0) that 



2c2 



