58 Art. 5.— T. Takenouchi : 



a and h being rational integers, then taking the conjugate number, 

 Ave get 



«'= a + hrr = 1 (mod. m). 



Therefore 



7?(«) = aa' = 1 1 



'mod. m). 



I 



a — o!— hiji — o-) = 



From tlie latter congruence, it follows that h must Ije divisible by 

 m. Hence we see that all the numbers of Ä arc included in i> ; 

 syml^olically we may write 



B ^ A, 



the sign > standing for "includes, as a proper divisor." There- 

 fore it follows tliat 



Let //' be the relative degree of P\m) with respect to Txif). In 

 general, H' is equal to '^^(f{m)^\ m being regarded as an integer in 

 }c{f>), not as a natural number. Hence, if 



m = ;)''! 9)''-' p^'i , 



1 ;.' ■'- i 



wdicre ])i, p-2, , Ph are distinct natural primes, then 



the meanings of r and s are as in §. 10. The only exceptional case 

 is when w«=2, in which case we have 



Hence the equality 



P\m) = K\_m'] 



holds only in the following cases: 



(i) î' = l, s = 0; i.e. m =//', 2) being an odd prime, 

 (ii) r = 0, s = l\ i.e. m=^, 

 (iii) 7» = 2. 

 NoAV, since the class-in\'ariant ji^co) can always be rationally 



1) Weber ; Algebra III, §. 154. 



