On the Relatively Abelian Corpora. 2T 



then these three numbers are conjugate to one another witli respect 

 to Jy(y"); and 



consequenth' 



(^') = {n = {n = \\v. 



Now, the ralative différente of ? with respect to Jy(i/^) is 



{^ — ■?')(l^ — ^") = (en u + en oiL)(cn u — cii o-u) 



_ (en u dn u + l)(on"^i — dn u) 

 cnuànu 



Hence, observing that both en?/, and i\nit are algebraic unities, and 



also that 



cmiàniL + l = ç"cn-u—frç- + 2.o~, 



ciihi — dn /<; = o|' d nit + ^-'— '20- en ?^ — 2 dn ii, 



we see that tlie différente in question contains ViV-z to the 



second power. 



Next, to investigate the relative différente of K{pc) with respect 



to K{/). put 



_ l + en?i + sn;;. 

 .r + CTiu 



and let ^' Ije conjugate to f] witli respect to K(^x^)j then 



, l+en?i — sn?^ 

 ;7- + en a 

 and 



-^ , ,/_ '2(1 + en, a) 

 .)- + en ?t 



. . , _ 'icnuj l + en «,) 



which shew that ^ and ■'^' are integers in AX'O- Also, since ■ 

 1 + en ?i = 2 + ;v + {o- + en ?/), 



the number 1 + cn?* is relatively prime to 2. We conclude there- 

 fore that each of the ideals p,, p,., must break up into two equal 



prime ideals in K{:c). Let us put 



\\\h = (^:^. r, 



then the integers ^ and '^/ are l)oth divisible by -^^ ^,, , Ijut by 



