On the Eelatively Abelian Corpora. 23 



corpus C itself, while the coi'pus of inertia {Tnigheitshöiyer), whose 

 relative «liscri minant cannot contain 2, must necessarily l)e a proper 

 divisor of C. Hence the relative degree of the corjms of ramification 

 with respect to the corpus of inertia must he //', h'^h^l. Then, 

 if / he the degree of a prime ideal contained in 2 in the", corpus 6' 

 we get'* 



4^-1 = (mod./). 



But, since/nuist ])e a power of ^?, it follows from this congruence 

 that 



4-^-1 = 4-1 = (mod. p), 



wliich is imp()ssil»le, unless ]j='d. 



Therefore, among the corpora Ave are considering, only the 

 two, viz. t'^h+i and CV'+i, rnay contain '1 in their relative discrimi- 

 nants. The relative discriminants of other corpora CpK , Cph,, 



are equal to //J'i'''-i, /^?'v"-i, respectively. 



Tlie corpus Jv(ff), i.e. Ck-J, has the relative discriminant 

 relatively prime to 2. Hence the corpora C2'' and Csv, l)oth heing 

 divisors of dtri, must also have the relative discriminants relativelv 

 jmme to 2. Their relative discriminants are therefore /^^''~^ and 

 ^ß'>'-i respectively. 



As for the excei)tional corpora CV'+i and Cs'-'+i, we have to 

 consider them a little further. If we put 



m' = 9]^ 'pi'ipl^ ^ 



the relative discriminant of the corpus C,,, is relatively prime to 2. 

 Composing C,„. with (3''' or Cs'i'+i, we ohtain K{y^) or 7v"(//) re- 

 spectively. Hence, the relative cl'ifferente of CV+i with respect to 

 63''' must contain as many ideal factors of 2 as the relative différ- 

 ente of K{y) with respect to K^jf) does. Therefore the relative 

 discriminant of 63" +1 is S^-^"' //'''+'-i. 



Next, to investigate the relative discriminant of Ck'^+i, let us 

 consider the corpus -Ä"(~), where 



-^7)- 



1) Wtbtr: Algebra II, §. 181. 

 Hillxrt: loc. cit.. Ç. 11. 



