20 Art. 5.— T. Takenonchi : 



Now, the numljer a breaks up into two factors in Jy{->'')^ as 

 follows : 



a = (1— /y)^(?^)sn-M 



= (1 — 10) — sn-?/ 



= (en 2( + ,o-)(cii « — .(/). 



Since the difference between these two factots is 2^o-, both of them 



iiiust be divisible by the same power of PiPz Hence we find 



that 



Consequent^ the relative différente of A"(a:'), i.e. of ^(^), with 

 respect to the corpus K{;if), must contain the same power of Pi p^ ... 

 as the relative différente of the number cnw+//^ with respect to the 

 corpus K{if) does.^-* 



Under our present supposition that p- is primary, the following 

 relation holds for a variable ii : 



cn/i« = cn?/jR(sn-w), 



where it(snV) denotes a certain rational function of sn^^"' From 

 this relation, putting 



we obtain 



Hence, if we put 



1) Hilbert: Theorie der algebraischen. Zahlkorper (Jahreshericht dor Deutschen 

 Mathematiker-Vereinigung, IV), §. 126. 



2) Weber: Elliptische Funktionen und algebraische Zahlen (1st edition), §. 117. 



