24 Art. 5.— T. Takenouchi : 



Since 



the corpus K{z) must either be relatively quadratic with respect to 

 J^df) or coincide with it. But, if Jv(z) coincide with K{;if), the 

 corpus AX'/), i.e. 7v(.r), must contain l)Oth :; and sm^^. ïlien. by 

 the relation 



z — — ^ , 



K{lf) must also contain snw ; which is of course impossible. Hence 

 K{z) must be relatively cjuaclraric with respect to J^(jf), and con- 

 sequently K(x) is relatively cubic with resj^ect to IC(zy 



The principal ideal (2) is equal to the sixth power of an ideal 

 in KÇx). Hence, in AXt;), (2) is equal to the square of an ideal, say 



(2) = (Q.q, f. 



If we regard q^ q,, as an ideal in Ä"(a-), it follows that 



QiQ. =i%'^-2 f- 



Now, in general, if a prime ideal p in a certain corpus k is 

 equal to the cth power of an ideal $ in another corpus JC, which 

 contains k as a divisor, and if e be divisiljle Ijy p, then the relative 

 différente of Jv with respect to k contain >p at least to the eth 

 power. ^^ Hence, in the present case, there lative different e of K{z) 



with respect to At//) must contain at least {c\i(\» )'. On the 



other hand, if the différente in question contain (qi t]2 )^ the 



relative différente of K{x) witli respect to K{;}f) must be divisible 



at least by ('l^i ^o )', Avhich is contradictory to the result 



obtained in the preceding section. Therefore the relative différente 

 of K{z) contains Qi q.^ exactly to the second power. 



Composing Cs^'+i or €& with Cm" , where 



we obtain K{z) or K{}/') respectively. Tlius, here tlie corpus K{z) 

 playing the part of K{if), \\Q can shew just as in tlie case of Gv+i, 

 that the relative discriminant of Ck'^+i is equal to 22''"*"^ /^^'^^^-i. 



1) Hilbert: loc. cit , §. 12. 

 AVeber: Algebra II, §. 174. 



