2 Art. 1.— T. Takenouchi: 



the following holds. Namely, C being given, when A or B is 

 given, then the other is uniquely determined. Hence these 

 reduced system of classes form an Ahelian group, which we shall 

 call 3». 



Since 9JÎ is Ahelian, it contains a system of elements (classes) 



called bases, say A u A 2l , A s , such that each element of 9)ï 



can be represented uniquely in the form 



S=A?A?- Af, «;=0,1,2, , ai -l (!) 



0=3, 2, ,s) 



where a t denotes the order of the element A { . Systems of bases 

 may be constructed in different ways, and the orders of the bases 

 of course vary according to different systems of bases. But, if we 

 decompose them into powers of prime factors, say 



--p-'Y 1 ,«.=?!' , ,a s =p'" s q" 



then these powers f l \ f m \ ,p v,s , <f\ q 1h , , cf , , 



as a whole remain conserved independently of the choice of bases. 

 Following H. Weber, ]) we call these powers the invariants of 

 group 3JÎ. 



The object of the present paper is the determination of the 

 number of the invariants of 9JÎ. At the same time, we shall also 

 find the invariants themselves and a specimen of a system of bases 

 as far as possible. 2) 



I take this opportunity to express my thanks to Professor 

 T. Takagi for his kind suggestions. 



§• 1. 



General Considerations. 



First of all, observe that we may confine ourselves to the case 

 where the modulus is a power of a single prime ideal. For, if 



1) Weber : Lehrbuch der Algebra, Bd. II. §. 12. 



2) For the natural Itorper, the problem is completely treated in the elementary theory of 

 [lumbers. For the quadratic korper, the problem is said to be treated by A. Eanum in Transac- 

 tions of the American Mathematical Society, Vol. 11 which was inaccessible to me. See also 

 Takagi : Journal of the College of Science, Tokyo University, Vol. XIX, Art. 5, pp. 13-15. 



