48 Art. 5.-T. Takcnonehi: 



Hence, iî ]-> ]>e not equal to 2 or 3, we can always find an ele- 

 mentary corpus I, whose relative degree is p' and whose relative 

 discriminant is a power of /^. Let it be called U. Composing J^ 

 with C/n we obtain a corpus of relative degree ^7", where h^n£2h. 

 In this corpus JSQ,, the principal ideal (/^) will be equal to the p"'th 

 power of an ideal, where h£ii'^2//. Then, since JlV/, is itself the 

 corpus of ramification of /^ in EC/,, it follows that tliis corpus must be 

 at least of relative degree p'', and relatively cyclic, with respect to 

 the corpus of inertia C of f^-. But, since it is evident that the 

 Galois' group of EC/, cannot contain a cyclic subgroup whose 

 degree is higher than the ^/'th, the relative degree of EC with 

 respect to C must be exactly equal to p''; whence follows that the 

 relative degree of C is jf''. 



The two corpora E and C have no other common divisor 

 than /•(/'), for their relative discriminants are relatively prime to 

 each other. Hence we get 



CE = CE. (25) 



If 21=2 or 3, the same reasoning can be applied, provided that 

 p' be not the highest power of 2 or 3 contained in m — l. 



If p' be the highest power of 2 or 3 contained in w— 1, we 

 must take an elementary corpus V, YII or III, VI in place of E, 

 according as p= 2 or 3. Then the relative discriminant of ^con- 

 tains the factor 2. It may happen, therefore, that tlie relative 

 discriminant of C is not relatively prime to that of E. Neverthe- 

 less, it i> still valid that C and E cannot have a common divisor 

 other than Z'{/'). For, if there be a common divi^oi'. it must be a 

 relatively cyclic corpus, whose relative discriminant is a power of 

 2. But. it is evident that none of the corpora V, VII, III, VI 

 contains such a divisor. Thus again Ave arrive at (25). 



The relative discriminant of C contains all or a part of the 

 prime factors in the relative dircriminant of 6/„ with the exception 

 of /^. 



If the divisor 6'/, of relative degree ]?, of C still contain a 

 prime f'-' relatively prime to ^7, then apply the above process once 

 more to C to get rid of this factor y'. In this way, repeating 



