On the Eelatively Abelian Corpora. Q1 



(i) A\'lion r- is not real, put n=-, m^ji. Then 

 p = ^- (mod. 6), 



hence there exist primitive roots for -''. The roots of tlie above 

 equation are 



s"(/'^-^)> ^--0,1,2, ,y-iQ;_i)_i, 



Ï being a primitive root of -'*. 



The corpus obtained by adjoining these roots to h{p) is rela- 

 tively cyclic, the relative degree being p^-iQ^-l). There are two 

 divisors of this corpus, of relative degrees p— 1 and ^V'-^ The first, 

 of relative degree ^?-l, is nothing else but the corpus discussed in 

 §. G and §. 7. The second, of relative degree 79''-^ is also relatively 

 cyclic, and its relative discriminant is a power of -. 



(ii) When ,« is real, put iJ.=q. Then 

 q = —1 (mod. Ö) 



and m=(f. In this case we have to use two integers, say r and r\ 

 which belong to the exponents q^'-\rf—l) and q^'-'^ respectively, in 

 order to obtain all the roots in tlie form 



^ '// //= 0,1,2, ,^"-^-1. 



The corpus obtained by adjoining these roots to /r(/>) is rela- 

 tively Abelian, the relative degree being q-^''~'^\q- — \^. The divisor 

 of this corpus of relative degree y'-l is nothing else Init the corpus 

 discussed in §. and §. 7. The other divisor of relative degree 

 q-^^"^^ is relatively Abelian, and its relative discriminant contains 

 no other prime than q. Since the Galois' group of this latter 

 divisor is of the form 



sH', ./.,,9 = 0,1,2, ,q"-'~\, 



it contains q"\q + l) difïerent cyclic subgroups of relative degree (J"'. 

 Namely, if we use the notation (.t) to denote a cyclic group con- 

 sisting of all the powers of a substitution x, then these subgroups are 



(4 M, {.sei , {sti'-'-^i 



(0, (.s't), (/'{), , (.s^V'-^-i)/). 



