44 -^'^t. 5. — T. Takcnonehi : 



Hence the relative discriminant of /vX///.) nmst consist of i<leals, 

 which are all divisors of 2. 



It follows therefore that tlio relative discriniinant of K{}/i) 

 with respect to /■(/') cannot contain any other prime than 2. 

 Hence the principal ideal (2) must l)e equal to the 2''''Hli power of 

 of a prime ideal of the first degree in Jv(j//,). 



Tliat this is a principal ideal can l)e shewn as follows. Put 



then, as shewn in §. 3, '/, is a divisor of 2. The norm of ^/, taken 

 with respect to A"(//,,_i) is, 1)y (24), 



44 



(4z/;.-i) 



n"- ) = ^^ = —2 • 



(-2?/i-,)" 



whence we conclude that 



(2) = U,,y , Q. E.D. 



Let us next consider the relative différente of C/, with respect 

 to Jy(i/u-i), supposing that A>2. If we indicate the conjugates of 

 any num1)er l>y placing accents upon it, the dlfflrentc of C/, is 



4- 



1 1 \/ 1 1 \/ I 



u'h u'i: J^ Vk uT J\ yt y'l 



j^-J yl+'^f+^yL^y:: 



yû-iyk 



Therefore the différente is divisil)le hy 2'' CS, hut l)y no ihgher 

 power of C/,. 



In the corpus AX'/O? ^■'^' A^/), the numl)er (2) is decomposed 

 as follows : 



2 = (V3+ixvä-i) ^ {V3+iy. 



Hence the relative différente of A"(/yO is equal to 2. 



