28 -^rt. 5.— T. Takenouchi : 



(i) When p^l (mod. G), ;? can be decomposed into two 

 different prime factors of the first degree in /■(,'>). Hence 



and consequently r=l. There exist primitive roots of p". 



(ii) When p=— 1 (mod. G), j^ is itself a prime number of tlie 

 second degree in k(/>), and 



d = l, f='2, /. ■ = 1. 



If n = l, then r=l. There exist primitive roots of p. 

 , If n>l, then r=2. The invariants of 31 are ^9"-^ p"-^ 

 Putting 



Ç being a primitive root of p, we obtain a system of leases of 51^^; 

 viz,, 



l+p, l + ^p. 



As for the root of 33, we may take such a power of ç as pre- 

 scribed in §. 1 of my paper just cited. 



(iii) The natural prime 3 is associated with the square of a 

 prime in /{;>), viz. 



Hence 



and thus we meet with a case where d is divisible by p— 1. 



If 71=1 or 2, then r=l. Consequently there exist jnimitive 

 roots of l + 2;7 and (i+2//)-, e.g. —1 and —o respectively. 



If n = 3, then r = 2; the invariants of 3t Ijeing o, 3. Putting 



--.-1, e,= l, 



we obtain a system of bases uf 3C : 



p, l — 3p. 



If ;^>3, then ;-=3. If we put 



^ = —(1 + 2..:/), ç^ = fr, ç^,= //-, 



1) Strictly speaking, we obtain a system of numbers representing a system of bases of 

 ?(; for, the elements of 31 are not numbers, but the classes of numbers. But, for the sake of 

 simplicity, we shall herafter use this abridged term. 



