On the Relatively Abelian Corpora. 27 



system of bases in general. Supposing that h is not divisible b^^p, 

 the onlj^ thing we can conclude is as follows: 

 Determine an integer (^ from 



1) = -''(> (mod, p'^+^). 



Then we distinguish two cases according as the congruence 



(>-{-x^-^ = (mod. p) 

 has a solution or not. 



If it has no solution, the /(/ numbers given above represent a 

 system of bases. 



If it has a solution, say x = .r,„ then / rational integers a, 

 {1=1,-2, ,/), such that 



a\ , = a^ç^-\-a.,ç.,+ + «^ ç. (mod . p) 



can 1)6 uniquely determined witli respect to mod. j/. Hereljy, 

 Avithout losing generality, we may suppose «i^ 0. (mod. ^?). Then 

 determine an integer c,,, such that 



for all rational integral values of c^, except when 



r,, = C.J = c~= ■ = Cf= (mod. 2J)- 



Now, it can be shewn that all the elements of 3( can be re- 

 presented ))y means of they</ + l numbers: 



l + ^i~\ a = 1,2, , fZ + A— 1, excludiDg m.iiltiples ofp, 



i = l,2. ,f, 



and l + ç„-'+^ 



But, these /(/+1 numbers do not in general represent a system of 

 bases. In fact, tliey represent a system of bases only when o.i=p, 

 where ^a-i denotes the exponent to which the number 1 + ^^-'-' belongs 

 with respect to mod. p". 



Let us now confine ourselves to the corpus /{/'). f denoting a 

 primitive cube root of unity: 



^ _ -l + v^H 



