On the Relatively Abelian Corpora. 27 



Since it is evident, liowever, that the discriminant of the 

 latter equation must be a divisor of that of the equation Af.{x) = 0, 

 the following conclusion is valid: 



The discrlmmanf of the equation of f^-diviston of jxrlods of sn 

 cannot contain an odd 2'>ri7nc, which is relatively j^i'imc to /-«, /^ Ijeing 

 an odd integer in h{i>). 

 Hence it follows, exactl3^ as in the case of primary /^, that 



tJie discriminant of the corres'ponding division- equation for the 

 function ^Tj cannot contain apriine, which is relativeh/ privî<^ to 3 and. /^. 



§. 6. 



Again, let /^ be a primary integer relatively prime to 3, and ?/? 

 its norm. By adjoining a root x of the equation 



Ä^(x) = 



to k(f'), a corpus is determined, which we shall denote by A"(a-), 

 instead of k(f', x), for the sake of simplicity. This corpus is of the 

 (w— l)th relative degree and relatively cyclic, when we take k(f>) as 

 the fundamental corpus. In the following, when the fundamental 

 corpus is not explicitly mentioned, it should always be understood 

 to Ije k(p). 



Since the discriminant D of the above equation contains only 

 two distinct primes, viz. 2 and /^, the relative discriminant of the 

 corpus Jy^(x) contains no other prime than these two. 



It is well-known that all the roots 



sn(;-^ — ), ; = 0,1,2, •••,m-'2, 



are associated with one another, and 



Hence (sn— ) is a prime principal ideal {Ifauptideal) in K{;x)^ and [^ 



is associated with its (m— l)th power. Therefore the relative dis- 

 criminant of K{x) contains i« to the (wi— 2)th power. 

 As for the prime 2, we proceed as follows : 



