gQ Art. 5.— T. Taken ouchi: 



properly chosen in k{py'. Without losing generality, we may 

 suppose that this integer a does not contain such factors that are 

 perfect cubes in k(t^). Then, any prime factor in a, since it be- 

 comes a cube in Äl^^v^«), must necessarily enter into the relative 

 discriminant of this corpus. Hence there can be no other relative- 

 ly cubic corpora, whose relative discriminants are powers of l + ^o, 

 than the following ones: 



E(^f>"(l + 2py), a,, h = 0,1/2 (mod. 3). 



Here the combination « = ^» = should of course be aA'oided ; also it 

 is to be observed that the two pairs of values (a, b) and (— «, —b) 

 give rise to one and the same corpus. It follows therefore that 

 there can be at most four different corpora of this kind, o. E. D. 



Lastly, let us consider the case p—l. From §.12 we see that 

 tlio relative corpus defined by ?^("^) i^ oi relative degree 8, and 

 contains three different relatively quadratic divisors, whose relative 

 discriminants are powers of 2. On the other hand, it is evident 

 that there can be only three relatively quadratic corpora, whose 

 relative discriminants are powers of 2 ; viz. Al^y*), Jv(V'2), Jv{\^i). 

 Our proposition is thus proved. 



The case h=l having been finished, we can now prove, by 

 mathematical induction, that C,;, h>l, is a divisor of a corpus 

 composed of elementary corpora only. 



The theorem is true in the case h=l. Suppose that it is true 

 for all Gz, Jkzu, '?^ being a certain natural number ; and consequent- 

 ly that it is also true for all relatively cyclic corpora of relative 

 degree jV', not necessarily of the kind C. 



Now, let Ml, denote an elementary corpus VIIL IX or X of 

 relative degree p*^, according as p>3, ^9=3 or p=2. Since the 

 divisor C^ of C„ must be identical with a certain J^i contained in JSn, 

 we can find such a relatively cyclic corpus I) that , 



L/n-tiln =^ U 111 „ , 



1) Hubert : loc. cit., §. 101. 



