40 Art. 5. -T. Takenouchi : 



where/, and (ji, are defined l>y the recursion-formulae 

 with tlie initial values 



By similar reasoning as in the preceding section, it can 1)6 

 shewn that the roots of the equation of the 2'''"''th degree 



fu-x-iffju-i =0, c = 0, 1, 2, h > 2, (23) 



are the values of />(?/), where 



./-Xl + 2or(l-2o)'' « = 0,1,2, ,2"-i-l, 



2 b =0,1,2, ,2"-^-l; 



or, in other words, 



,,_ />^-'(^+^/>),, ^ = 1.5, ,4(2"--r)+i, 



^ --^ = 0,2, ,2(2"-i-l). 



As will he shewn hereafter (§. 17), the corpus Ki^iji) is the 

 class-corpus {K lasse nkor per) corresponding to the group of numhers 

 a in /•(,'/), such that 



a = 1 (mod. 2''). 



Therefore the relative degree of J^(>/1) cannot he less than^^ 



V'C^"^ i.e., 2-'. 



(3 



Hence, of course, the relative degree of A"(//a) cannot be less than 

 2'''"'. Thus we see that the equation (23) must he irreducil.de in 

 /{,"). .Vnd it follows, at the same time, that 



K(yf,) = K(y,) ; 



whence it can l)e proved, as in the preceding section, that 



for all values of //. 



1) Weber: Algebra III.. §. 167. 



