On the Eelativelj Abelian Corpora. 59 



expressed by j^nw), n 1 )eing any natural number, we get 



P\m) ^ Jv[7;il ^ A'[/>Ji] K\_pl^ Kip]"] (30) 



Avhere . -^^^C^jfO = P\pl'')^ ^i general, 



the only exception being 



P''(2") > 7v[-2"], 7/>2. 



§. 19. 



Before proceeding further, let us insert here a few preliminary 

 considerations. 



I. Let Ci, Co, , Q, be absolutely or relatively cyclic 



corpora of degrees |/'i, p^-, , ^/"* respectively, p being a prime. 



We suppose that these corpora are independent of one another, 

 i.e., we suppose that none of these corpora has common di^'isors 

 with the corpus composed of all the others. 



If we compose these cyclic corpora all together, we get an 



Abelian corpus of degree j??''i+"- + +"''-, which we" shall call A. 



The Galois' group of ^1 must be of the form 



s? s;' s:' 5^ c, = o, 1,2, ,pai-^i. 



{i - 1,2, ,//) 



Here >Si denotes a substitution, by which all the numbers in Ci are 

 changed into their conjugate numbers, while those of other C's 

 remain unchanged. We may express this fact by saying that the 

 cor})us 



CiC.2 C',_i C(.Li Cj, 



IkIoikjs to the cjx-lic group 



S'^ , r,= 0,l,2, ,pn-K 



Now, let C- be the divisor of G of degi'ee p'^'"^, then the 

 Abelian cordons 



c[a c; d 



belongs to the groujD of the form 



>sr^f^ ^f ^':\ si=sr" ^-^1^0^...^,^ 



c,^0,\,2,.-,p-\, 



