On the Relatively Abelian Corpora. 37 



ïlie rc4ative différente of w- with respect to K^jj^,.^ is 



ylÀyj:-i-yi:-^{y,-yk-i) 

 = (i+2.>)r,_,[(i+2;.)^,_,r;-i] ^f^. 



Therefore tlie relative différente of AX'//.) with respect to K{ii;,.^ 

 must be equal to {l + ^f>)^,:-\- Consequently, that of K(i/-2n+-2) witli 

 respect to k(p) is 



whence we conclude that the relative discriminant of K{y-zn+'i) is 

 equal to 



(1 + '2.y)-"- 3'' + 3 + 3-+ + 3-" 



(4» +3)3-'' -3 



As in case (ii) of ^. 0, it can be shewn that the corpus K()/-2„u.i) 

 contains as its divisors 4. o"'^ different relatively cyclic corpora of 

 relative degree 3", whose relative discriminants are powers of 1-1-2^6». 



Next, let us consider the corpus 



which is relatively cubic with respect to Z{;'). Since the discrimi- 

 nant of the equation 



x^-^ = 



is —2'. 3", the relative discriminant of C cannot contain any other 

 prime than 2 and l + 2;>. If we put 



l+2/> 



^ = v2, ■'/ = "rfrr' 



then 



i+.2,„ = -^yL. 



1 — X 



Therefore, since 1—x is an algebraic unity, we get 



2 = x^, 

 1 + 2;,^ y ', .//--^1 + ar. 



