JOURNAL OF THE COLLECrE OF SCIEN'CE, TOXYO IMl'EF;[A.r, UX^VERSU'Y. 



VOL. XXXVir. ARTICLE 5. 



On the Relatively Abelian Corpora w ith respect to the 

 Corpus defined by a Primitive Cube Root of Unity. 



By 



Tanzo Takenouchi, Ui'jah-n.'i/ii, 

 Professor of Matliematics in the Eighth High School. 



It wa^^ conjectured by Kronecker that all the relatively 

 Abelian corpora {KOiyer) with respect to an imaginary quadratic 

 corpus are probably exhausted by those which arise from the 

 ecjuations of transformation of elliptic functions with singular 

 moduli. Prof. T. Takagi^-* investigated this problem in the 

 remarkably interesting special case, in which the fundamental 

 corpus is defined by the imaginary unit i, and proved that the 

 relatively Abelian corpora with respect to k{_i) are completely 

 exhausted b}^ tlie division-corpora {TeilungshJriJer) of the function 

 sn with the singular modulus j<=l. Following his example, I am 

 going to treat of another interesting special case in whicli the 

 fundamental corpus is h{l'), (' denoting a primitive cube root of 

 unity. 



The present paper consists of two parts. In the first part it 

 will be proved that the relatively Abelian corpora with respect to 

 h{l') are completely exhausted by the division-corpora of the func- 

 tion sn Avitli the singular modulus ;£ = /,". 



Now, if CO be a quadratic number whose imaginary part is 

 positive, and m a natural number, then the invariant j(mco) is 

 called a class-invariant. By adjoining the class-invariant jimoi) 

 and a primitive ??^th root of unity to the quadratic corpus Ic(oj), we 

 obtain a corpus, to whicli I shall (§. 10) give the name strahl- 

 coiyus. It is known that the strahl-QOY\\\\ii is relatively Abelian 



1) Takagi : Journal of the College of Science, Tokyo Imperial University, Vol. XIX, Art. 

 5 : Proceedings of the Tckyo Mathematico-Physical Society, 2ncl Ser., Vol. VII, Xo. 21. 



