134 CHAPTER V. 



is the number of substitutions affecting one index only, we have 



Q p P P 



m,p,s 'm,p, s'm 1, p, s r J,p, s 



To evaluate P w?jM , the number of sets of solutions of 



we note that, for the P n _i 5j p vS sets of values of %, . . ., rj n which make 



n 



NI^ 77 J* 8 -M = 1^ the corresponding value of r^ is zero*, while, for each 



i=2 



of the _pM-i)_ p n __ ljJM sets of values in the GF[p* s ] for which 

 that sum =(= 1, there exist p s -f 1 values in the GF[p* s ~\ for 7? r Indeed, 



belongs to the GF[p s ] and is therefore the power p s -\-l of a mark 

 in the GF[p 2s ]. Hence we have 



Since Pi, P ,s = p SJ r 1, we find by mathematical induction that 



For another proof of this result, we consider only the case p > 2. 

 Then if v be a not- square in the 6rF[jj*] ; the 6r_F[j 2 *] may be 

 defined by means of the irreducible equation 



we have 

 Hence 



By 65, this quadratic equation has j? s(2n ~ 1) ( l)^ 5 ^"^ sets of 

 solutions 17 ...,, f$ 19 . . ., ($. n in the 6rjF[_p a ]. Hence Q^^, equals 



146. Theorem. - - //" n , 12 , . . ., i ?n be any system of solutions 

 in the GF[p 2s ] of the equation 125), there exists a substitution S in 

 the group G m ,p,s which replaces ^ by 



