On the ■Relatively Abolian Corpora. G9 



this derived corpus is divisible by 3 or not. 



Without losing generahty, we may suppose pi=#=3. Let C be 

 a relatively cubic divisor of IIIi. There are two different derived 

 corpora of C and VI. At least one of these derived corpora is 

 certainly a divisor of Ä"[m] . Let us call it D. If the relative dis- 

 criminant of K[m\ were relatively prime to 3, so would also be that 

 of D. Moreover, since m is odd, this relative discriminant cannot 

 contain the factor 2. Hence D would be a relatively cyclic, cubic 

 corpus different from C, with the relative discriminant, a power 

 of pi, provided that the relative discriminant of K[m\ is relatively 

 prime to 3. 



But, this is evidently impossible, if G be a proper divisor of 

 nil. For, if C does not coincide with IHi, then the relative dis- 

 criminant of C must be a power of ^i, without containing the 

 factQr 2, and consequently it can be proved that D cannot be 

 different from C. '^ 



If (7=111,, then the relative discriminant of C is divisible by 

 2. Compose C with D, and consider the corpus of inertia of a 

 prime ideal contained in p^. This corpus of inertia must be rela- 

 tively cyclic, and its relative discriminant must be a power of 2. 

 Also, since D is not identical with C, this corpus of inertia cannot 

 reduce itself to h{p) (cf. §. M). Therefore it must be relatively 

 cubic. But, we can shew, as follows, that this is again impossible. 



We know that every relatively cyclic, cubic corpus, with 

 respect to A(/'), can be o])tained by adjoining to k{(>) a cube root of a 

 certain integer in h{p). Hence, if there be a relatively cyclic corpus 

 of the third relative degree, whose relative discriminant does not 

 contain any other prime than 2, it must be one of the forms 



In the corpus («), however, the following decomposition holds : 



1 + 25 =(l + ^'2£X/^+ '28)(r + ->^2£) 



1) Takagi : loc. cit., §. 10. 



