54 CHAPTER V. 



Hence if 41) be true for f(x) = with the roots ffi, . . . 3 a m , a like 

 formula is true for the equation F(x) = with the roots 1? . . ., a m , 



Forming the sums 



m m in 



"V/Y ^ "V -/Y -1 ^V/Ya-) 

 we derive the new identities, 



> si****-***-*-* 



Corollary. -- If/'(#) = have a double root a, the right member 

 of 41) must vanish for # = . 



75. Theorem. - - // t be a positive integer and , % ; . . ., u p i 

 denote the ma/rks of the GF[p n ~], then 



4~> 1-1 



In fact, the marks / are the roots in the G-F\j) u ~\ of the 



equation 



X P )I - x = 0. 



Applying to the latter the identities 40), we find 



CHAPTER V. 



ANALYTIC REPRESENTATION OF SUBSTITUTIONS 

 ON THE MARKS OF A GALOIS FIELD. 



76. Consider the problem to find every quantic <$>() belonging 

 to the GF [p n ] such that the equation <t> (i;) = ft has a root in the 

 field whatever mark of the field ft may be. For example, 



8 = ft (mod 5) 

 is solvable for every integer ft, since we have 



