CLASSIFICATION AND DETERMINATION, etc. 21 



By 24 we have in the 6rF[j) TC ] the identity 



Hence, if X 1 be a root of 10), so is every X^ nr . Since <p(X) is an 

 IQ\t> P n ~\> we liave ( 23 ) in tne -*T> w ]> 



Xf ' - X t EE 9 (X,) . * (X,) = [mod F(*)]. 

 Hence, m being a multiple of t, 



XP nm = X, [mod FO)J- 



We next prove that the above t powers of X 1 are distinct modulo 

 F( X ). Indeed, if 



[mod F(a;)] 

 for ft < b < , we would have, upon raising it to the power p n < m a \ 



Xf m EE X t EE Xf (m + 6 - a) [mod .F(a;)], 



so that, by 25, m + ft a would be divisible by m. Hence b = a. 

 Corollary. - - We have in the GrF[p nm ~\ the decomposition 



<p (X) = (X- X,) (X- Xf) . . . (X- Xf - 1) ). 

 In particular, F(x) = has in the G-F[p nm ~\ the distinct roots 



x 9 x^...,xP n(m ~ 1 \ 



32. Theorem. If F(x) le an IQ[m, p n ~\ belonging to the 

 exponent e, every root of F(x) = in the G-F[p nm '] belongs to the 

 exponent e, and inversely. 



We may define the G-F[p nm '\ by means of F(x). In it, all the 

 roots of F(x) = satisfy the equation x e ~L = 0, but do not all 

 satisfy a?/ 1 == for f < e. But, p n being relatively prime to e, a 

 divisor of p nm 1, it follows from the corollary of 31 that the 

 roots of F(x) = in the GrF\j> nm ~] all belong te the same exponent. 

 This common exponent is therefore e. 



In particular, for e =p n 1 9 the roots of F(x) = are primitive 

 roots in the GF[p nm ~]. Such a quantic F(x) will be called a primitive 

 irreducible quantic of degree m in the 6r-F[p n ] and will be referred 

 to as a P IQ [m, p n '\, 



33. Theorem. If e be a prime number, the function 



is irreducible with respect to every prime modulus p which is a primi- 

 tive root of e. 



By hypothesis, p belongs to the exponent e 1 modulo e, so 

 that e is a proper divisor of p* 1 1. Hence, by 30 for n = 1, 



m = e 1, the number of irreducible factors of V is - - = 1. 



