On the Relatively Abelian Corpora. 49 



this process a number of times, we sliall arrive at a corpus C^, of 

 relative degree ^/ (ji^h), the relative discriminant of whose divisor 

 Öl, of relative degree p, docs no longer contain a prime relatively 

 prime to p. 



§. 15. 



It was proved by Hubert" that every cyclic corpus, whose 

 degree is an odd prime ?«, and whose discriminant does not contain 

 any other prime than u, must necessarily be a kreishörpcr. Fol- 

 lowing his example, we can now prove tliat the corpus 6\ of rela- 

 tive degree p is nothing else but an elementary corpus VIII, IX or 

 X, according as/7>-3, p=3 or^) = 2. 



In the case p>'), the proof can be effected without any 

 difficulty by Hilbert's method"'. As for the method, the reader is 

 referred to his original work, as it will occupy too much space to 

 reproduce it here. 



Next we have to piove that the relatively cyclic corpora of 

 relative degree 3, whose relative discriminants are powers of 1 + 2/*, 

 are exhausted l)y the four relativel}^ culiic ones of the elementary 

 corpus IX. 



From what has been shewn in n^. lU and §. 11, it follows that 

 the relatively cyclic corpus obtained by adjoining 



(1 + 2/0^ 



to /'{/') is of relative degree 0, and that this corpus contains four 

 different relatively cubic divisors, Avhose relative discriminants are 

 powers of 1 + 2/'. 



But, on the other hand, it can be shewn that there are no 

 more relatively cubic corpora, whose relative discriminants are 

 powers of l + 2/>, tlian the above four corpora. For, since the 

 fundamental corpus lie) contains the primitive cuIjo roots of unit3^ 

 every relatively cyclic corpus of the third degree can be obtained 

 from k{l>) by adjoining to it a cube root of a certain integer, say «, 



1) Hilbert: loc. cit., §. 103. 



2) Also cf. Takagi : loc. cit., §, 13 and §. 14. 



