On the Relatively Abelian Corpora. 67 



Abelian corpus, whose relative degree is evidently twicy that of 

 Jvlm]. Therefore 



YJ^[m] := \JC > K[i)i], 



which shews tliat Vi is not a divisor of K[ui\. 



Thus we are led to the following important conclusions : 



Relatlvchj Ahcllan covpora wlfli, ra^pcct to h{i>) are not exhausted 

 Inj strald-coiyora. 



If l^ he a)i odd 'prime In lc{p)^ m its norm, and if 2'' he the highest 

 power of 2 contained in m—1, then tJte relatively cyclic corpus of rela- 

 tive degree 2'' contained in the division- corpus SÇf-'-) can never he con- 

 tained in a strahl-corjius. 



Now, in §. 14 and §. 15, we have given a method of finding 

 a corpus composed of elementary corpora only, such that any given 

 relatively Al:)elian corpus may he a divisor of it. In that method, 

 V and VII are used only in the case when the relative degree of 

 the given corpus is even. Hence we see that 



relatively Ahelian corjyora of odd relative degree with, respect to 

 kÇ") arc exhausted hy strahl-corpora. 



I '1-2. 



According to Fueter, " all the prime factors of tlie relative 

 discriminant of K\_m\ must be contained in m; and conversely, all 

 the |)rime factors of m must, in general, be contained in the rela- 

 tive discriminant of A"[m]. The only exceptional case is when m 

 is divisible by 3, Init not by 3"; in this case, tlie relative discrimi- 

 nant of 7l'[7;^] does not contain the factor 3. 



But, I think, this is not quite correct. In the first place, the 

 most obvious exceptional case m = -l is not noticed in the above 

 statement. Secondly, his so-called exceptional case is not neces- 

 sarily exceptional. For example, take the value m=6, tlien 



A^t^) =^A-3 + 3V_3) =j{^~ri). 

 But, we know that 



where x'—?>x- — '^x — l — Op i.e. x =1 + ^2^ + ^4- 



1) Pneter : Math. Ann., Vol. 75. 



2) Weber : Alg-ebra III, p. 722. 



