On the Relatively Abelian Corpora. 47 



But, since cn?Mln?i and cni'dnv can be rationally expressed by 

 riuu and snv respectively, even in the case where ic or v h ix power 

 of 2 (§. 12), we conclude that 



K (sn w) = K (sn u, sn v), Q. E. D. 



§. 14. 



We are now in a position to prove the following important 



theorem. 



Then: can he no other relatlvehj Abelian corpora ivith respect to 

 k{r) than those contained in the dimsion-corpora of the elliptic function 

 sn 'with the singidar modulus 7c=i;K 

 The same thing may also be put in another way as follows : 



(jiven any rclativehj Abelian corpus with respect to h{p), ice can 

 aUraijs find such a corpus which is composed of elementary corpora 

 only and contains the given corpus as a divisor of it. 



It is well known that every Abelian corpus can be decom- 

 posed into cyclic ones, whose degrees are powers of primes. This 

 is true not only for absolutely Abelian corpora, but also for rela- 

 tively Abehan corpora with respect to 1c{p). Hence we shall prove 

 our theorem only for such relatively cyclic corpora. 



Let Ch be a relatively cyclic corpus of relative degree p>\ p 

 being a natural prime, and C\ its divisor of relative degree p. If 

 the relative discriminant of G contain a prime factor in l{f>) rela- 

 tively prime to p, let it be denoted by «. Now, consider the 

 corpus of inertia of « in Ci,. Since the relative discriminant of it 

 must lie relatively prime to !'■, this corpus of inertia cannot contain 

 C'l in it; consequently it must coincide with ^-C/»). On the other 

 hand, the corpus of ramification of i^ is (J,, itself, since ,« is relatively 

 prime to the relative degree / of C,,. Therefore the number /^ is 

 equal to the j/'th power of a prime ideal of the first degree in C,.. 

 If the norm of /^ be denoted by m, we get'^ 



m~l = (m.oà.p''). 



1) Weber ; Algebra II, §. 181. 

 Hilbert : loc. cit., §. 41. 



