36 Art. 5. -T. Takenouclii : 



But it follows from (17) tliat z,._-^ cannot contain any ideal factoi 

 relatively prime to 3. Hence it must also be the case with D, and 

 consequentl}^ also witli the relative discriminant of the corpus 

 K{:lli^. Therefore tlie relative discriminants of the corpus of de- 

 composition (^ZeTlegvmjslcorper) and of the corpus of inertia of the 

 number l + 2/> in K{yi) must be an algebraic unity. Hence both of 

 these corpora coincide with /{/'). It follows therefore that l + 2,o is 

 equal to the 3^"th. power of a prime ideal of the first degree in AX///.). 

 That this prime ideal is a principal ideal can be shewn as 

 follows. 

 Put . _ l + 2,> z. -3 4 o,, , o 



and let its conjugate numbers with respect to K{i/i.^ be denoted by 

 Zl and cl. Then, observing that, if ij,„ y',, y",, denote the three 

 roots of (20), 



Ave ol)tain 



2(l + '2o). 



./.-/. I ~ /.■ - /.■ T^ -ï /.■ ~ :■ 



2//.-1 

 1 . 



- /. - 1 



' ■ ' Ul 



Since //;,_i is an algebraic unity, C/, must be an integer, provided 

 that r;_i is an integer. But, this is really the case with 



for 



C:w^+:^r^+:^r:; = 2(i+2/.n 



Hence C/. is certainly an integer. Its relative norm taken with 

 respect to AX.V/,_i) is, as shewn above, associated with ^/._i. Hence 

 we infer that the relative norm of C/, taken with respect to k(f>) 

 must be associated with that of C3. Thus we get 



(i+2o) = (r,„,,)^" 



