On the Eelatively Abelian Corpora. 45 



The relative différente of At///,) is therefore 

 Accordingly the relative discriminant of K\ii,) is 



0(3/(— '*)- - . 



Now, the corpu. At.V„). /'>2. is relatively Abelian, and ils 

 Galois' group is ot the tonn 



s'C', « = U, 1,.;, ■••■,- 



;, = o,i,2, ,v-'-\. 



In this Galois' group, there are 2'-= cyclic suhgroups of order 2'-, 



^'^''" (.,), (rf). Uf'^. ■ (st"'-=-'l- 



Besides them, there are also 2'- cyclic subgroups of order 2'-=, viz. 



if), im, (s't), (s"-p'-'-«0- 



Therefore the corpus A'(/a) contains as its divisors 2'-= relatively 

 yclic corpora (we shall call them A) of relatno clegree_2 . and 

 also ■->"- relatively cyclic coi-pora (B) of relative degree 2 



Similarly, the corpus A%,.«) ^vill contain 2'- relatively cychc 



aivrso (C) of ;elative degree 2'-, an.l 2^ U» of relative degree 2 . 



'n^e relation between C an.l A is such that each one of C contains 



^ attain one of J, and convesely, each one of A is contained m 



certain two of C. Similar relation exists between 1) an.l A. 



Hence we see that, by the division of perio.ls by powers ot -, 



^™°'*''" 2..-. + o'.-. = 3.2'- 



different relatively cyclic corpora in all, whose relative degrees are 



all 2»-', and whose relative .liscriminants are powers ot -. 



§. U!. 



We have considered all the division-corpora arising from the 

 division ot periods ot sn with the singular modulus x=lohy powers 

 of primes in iO»); and analysed them into relatively- cyclic compo- 

 nents. Let us here tabulate these component corpora. 



