3Q Art. 5.— T. Takeuouchi : 



S. 



Let lis again iissume tliat /^ is a piimary iiitogcr relatively 

 prime to 3, and m its norm. It was shewn in §. 4 that 



sn ait = — '—^ , X = sni(, 



7.1 



where </\ = A,,(x), /, = D,,{x). 



Ô 



Substituting x-^-^ in place of .r, we get 



x^ 



where '/". is a rational integral function of x of degree wr— 1, wliicli 

 must he divisible 1)}^ ^j. If we put 



then (I'.^ is necessarily of the form 



<J>., = Sx" + a.x'--+ + rto, A'"--" + + n, 



where £ is an algebraic unity and «o, , «o;., are all integers 



in /•(/') divisible by y-. 



It can 1)e shewn ])y matliematical induction, tliat in general 



where <^„ is of degree m''~'^(/n—l), and its coefficients possess the 

 same property as those of ç\,. 



Hence, if we put 



this equation is irreducible, and gives as its roots the values of sn^^ 

 corresponding to the /^-''-division of jwriods. Tliat tlie discriminant 

 uf this equasion can contain only two distinct prime factors 2 and 

 /•Î has already been remarked at the end of n^. o. 



To find the Galois' group of tliis equation, we have to dis- 

 tinguish the following two cases. 



