On the Eelatively Abelian Corpora. 55 



then the relative corpus determined by r(z/)-\ i.e. by ^-(w)', i^ the 

 class-corpus corresponding to the' group A of numbers «, such that 



a = ±1, ±f>, ±f>- (mod. //), 



Avhere tlie excludcnt consists of all the prime factors of i'-. 



Tlie relative degree of K{r{uf) is not higher than '^^, where 

 m is the norm of /A Hence, in proving the above statement, no 

 generality will be lost, if we suppose that m>l. Under this sup- 

 position it can easily be verified that the group A does not contain 

 a divisor of 3, 



As shewn in §.3, if 



/i=^7r'" or (1 + 2/0"'", 



Tz' being a prime in Tiip), then the reciprocal of z{n) is an algebraic 

 integer. In all other cases z{u) is itself an algebraic integer. At 

 any rate, it follows from (27) or (28) that, if t be a primitive integer 

 of the corpus 7vXr(?/?), and p any prime ideal factor of - in the 

 same corpus, we get 



t = P (mod. v), ('29) 



supposing that ~ belongs to the group A. If - be a prime, which 

 is relatively prime to the cxcludent, but does not belong to A, 

 then the congruence (20) does not hold. For, since the discrimi- 

 nant of the equation for r{uf is not divisible by p (§. 5), we get 



z{t(?rP = ziziif ^ Ti^if (mod. p). 



Hence we see that tlie congruence (20) holds good, when and 

 only when p is a prime ideal factor of -, where - is a prime integer 

 of the first degree in ^-(,0) belonging to the group A. But, this is 

 the very condition, necessary and sufticient, in order that the 

 corpus K(j{u>f') may be the class-corpus corresponding to the group 

 A. Our statement is thus proved. 



Now, in general, if G be any group of numl)ers in an imagi- 

 nary quardratic corpus, and E the group of algel)raic unities in the 

 same corpus, then the class-corpus corresponding. to G is identical 

 with that Avhich corresponds to EG'\ Hence, for the sake of 



1) Weber : Algebra III, §. 167. 



