On the Relatively Abelian Corpora. Kl 



ami sucli tliiit its relative degree is less than ]f.'^ Therefore D, and 

 consequently also C„, must be contained in a corpus composed of 

 elementary corpora only. 



PART II. 



§. 10. 



Let ji^nip) be a class invariant, m being a natural number. By 

 adjoining J(wo) to lc{r\ we obtain a relative corpus called order- 

 corpus (Onlnuiigs-korper). Order-corpora are often called class-corjiora 

 (Jvlassenkörjm^). But, to avoid confusion, we shall never use the 

 ^vord class-corpus in this sense. As for the définition of class-corpus 

 used in the following, the reader is referred to Weber's Lchrhuch 

 der Algebra, Vol. Ill, §. 104. 



Tlie order-corpus K{j{mp)) is relatively Abelian witJi respect 

 to /•(;'), and its relative degree is equal to the number of classes of 

 the order [w] {Ordnung mit dem Fährer m). Hence, if the decom^ 

 position of m into distinct natural primes be 



))l — nh\ ph. f)h 



•'- 1 -'- '2 -'- i 



the relative degree is 



h ^ Lii pin~\(p -^i ll 



y, 



3 ^^ V' \3 

 It is known tliat the irreducible equation of the Ath degree, 

 wliich gives J{m;>) as a root, becomes reducible, when we adjoin 



( — 1) '-^ pi, i =\,1, , supposmg that p^^'I, 



a/— 1' if m = (mod. 4), 



V—l , V'2 , if m = (mod. 8), 



to tile fundamental corpus k{p)r' But, no further reduction of the 

 equation can take place, whatever roots of unit}^ may be adjoined 



1) Takagi : loc. cit., §. 9. 

 Hilbert : loc. cit., §. 103. 



2) Fueter : Der Klasseukörper dor quadratischen Körper etc. (Dissertation • Güttiu'i-en 



1903). ' '' ' 



Weber : Algebra 111., §. 13S. 



