52 -'^'"t« ^—'i-'- Takenouclii : 



to the fundamental corpus'^ Hence, if a primitive 7nih root of 

 miity, in terms of which all the square roots above given can l)e 

 rationally expressed, be adjoined to the order-corpus JC(jivif>)), 

 then the resultant corpus must be of relative degree 



Avhere r denotes the number of distinct odd primes in w, and 



.9 = 0, if VI ^0 (mod. 4), 



5 = 1, if 7n = 4: (mod. 8), 



s = 2, if 7)1=0 (mod. 8). 



This corpus of the ITth relative degree is relatively Abelian with 

 respect to Ii{p). We shall denote it by the symbol Ä"[wi]. 



It is known that tlie corpus Ä^[7?i] is the class-corpus corre- 

 sponding to the group of numbers «, which belong to tlie order 

 [??^] . and satisfy the condition 



Ilia) = 1 (mod. 77i), 



all the prime factors of ni l^eing taken in the excludent.-' Such a 

 group of numbers is called a straJd by E. Fueter^*. Hence the 

 corpus Ä"[w?] may well be called a strahl-corpus. 



Weber concluded tliat his so-called division-corpus, i.e. the 

 corpus obtained by adjoining to k(p) the numbers m and >S|-^I, 

 where 



ß being an integer in k{('). can always l;)e looked upon as a divisor 

 of a certain s^7'a/?/-corpus**. 



It seems to me, however, that there are two defects in his 

 proof. The one is the misconjecture that his division-corpus for 



1) Weber: Algebra III, §. liJ8. 



2) Weber : Algebra III, §. 16S. 



3) Fneter : Crelle's Journal, Vols. 130, 132 ; Math. Ann., Vol. 75. 



4) Weber : Algebra III, §. 169. 



