36 CHAPTER IE. 



Corollary. Every PIQ[m, p n ] is given by the formula 



F t (x) = (x- E 1 } (x - E ( p n ) . . . (x - R'p n(m -\ 



where t is an integer relatively prime to p nm 1. 

 Evidently F t = F tp n = F ip 2 n = . . - 



51. The determination of a primitive root in the GF[p nm ~\ is 

 one of the most important as well as most difficult problems in the 

 theory. Special methods of procedure are illustrated in 54 57. 

 We may determine simultaneously all the PIQ\m, p n ~\ and therefore 

 all the primitive roots of the GF[p nm ~\ by the following method of 

 undetermined coefficients. 



The roots of F t (x) = are the t ih powers of the roots of F l (x) = 0. 

 Hence the equations 



are equivalent in the GrF[p nm ~\. Since t is prime to p nm 1, we may 

 determine t' by the congruence 



tt' = l (modp nm 1). 



Hence F \x ' ) = and F 1 (#*') = are equivalent equations in virtue 

 of xP nm = x. By 30, the product of all the PIQ[m, p n ] is given 

 thus: nm _ l 



n 

 . ^ } = 



where q^, q 2 , . . . denote the distinct prime factors of p nm 1. 



are equivalent if t and t' each run through the integers less than and 

 relatively prime to p nm 1, which give distinct functions F(x). 

 Giving F l (x) the undetermined form 



F^(x) = x m + ax m ~ l + ~bx m ~* H ---- , 

 and forming the product of the <$>(p nm 1) distinct quantics F 



the result may be identified with the above fractional expression in x, 

 giving a series of conditions for the coefficients a, b, . . . The 

 examples which follow will serve to make clear the method. 



52. For p n = 3, m = 2, we have p nm 1 = 2 3 . The integers less 

 than and prime to 2 3 are 1, 3, 5, 7. But 



