22 Art. 5.— T. Takenonchi : 



iiü higher power of it. Hence tlie relative différente of tlie corpus 

 K{;x) and of the number '^j, with respect to K(x'), nuist contain 

 ^1 ^.j to the same power. But, since 



f>- + en 71 



we conckitle that the poAver in question inust l)e the fourtli. 



Therefore the relative différente of Kix) with respect to KQf), 



as well as to k(f'), contains ^i ^2 to the eighth power. Thus, 



we find tliat the relative discriminant of A"(-*') with respect to Z'(.^') 

 is equal to 



4(m-]) 



2~ 3 ^/'.--. 



The corpus /v(.r) is a relatively cyclic corpus of relative degree 

 m— 1. Hence, if d, 1)0 a divisor of ?>? — !, tliere exists certainly a 

 divisor (Unterkörper) of JC(.r), which is of relative degree d, and of 

 course also relatively cyclic. Wo shall denote it l>y d. 



Let 



w-1 = 2"-i3"'+Vï.^^ , 



Avhere j9i, jh, î^i'c distinct natural primes, different from 2 and 



0. Then the corpus Jv(.c) can be looked upon as the result of 

 composition of the relatively cyclic corpora C2'>+^, Csn'+i, C^,^;-., C^i^^.-, 

 , taken all together. 



Lot us now determine the relative discriminant of eacli of 

 these corpora. 



In 62''+!, the number y- l)reaks up into 2''+^ equal prime ideals. 

 Therefore the relative discriminant of G'^+i contains /^- to the power 

 2''+i_l. The same reasoning applies also to otlier corpora Gv+i, 



^i'l''- ? • 



On the other liand, it can be shewn generally that the relative 



discriminant of a relatively cyclic corpus C of relative degree /•)". p 



being a natural prime, cannot contain. the factor 2, unless |9=:2 or 



3. For, if ^9=^2 and 2 enter into the relative discriminant of C, 



then, since 2 is relatively prime to the relative degree p'', the 



corpus of ramification {Verzicelriungslcöriier) of 2 in C must l)e tlie 



