ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 57 



81. Theorem. - - The quantic belonging to the 6rjF f [j) wm ] 7 



will represent a substitution on its marks if, and only if, X = is the 

 only solution in the field of H'(X) == 0. 



Indeed , the necessary and sufficient condition is that 



= 



shall require X 1 = X 2 , or X x X 2 = 0. 



Corollary. X l>nr AX*"* represents a substitution on the marks 

 of the GF[p' im \ if, and only if, either A = or else A is not the 

 power p nr ~ p" s of a mark of the field. 



82. Theorem. If k be an odd integer relatively prime to p 2n 1, 

 and if a be an arbitrary mark of the GF\j) n ~\, the quantic 



represents a substitution on the marks of the GF[p n ~]. 

 We are to prove that the equation 



48) <i\.(i, ) - ft 



has a solution | in the GF[p n '\, ft heing an arbitrary mark of the 

 field. By Waring's formula, ^(?, a) is the sum of the & th powers 

 of the roots of the quadratic 



?? 2 -^- = 0. 

 Hence, in virtue of the equation 



| = n a/?}, 



we have the identity /i . N . , , N7 



0*(g, a) = 7/- (0j)*. 



The equation 48) thus becomes 



^ fc __ ^y^ *. o. 



Setting F = iy*, this becomes 



49) P ! -/3F-a*=0. 



According as 49) is reducible or irreducible in the 6r.F[j) n ], it is 

 solvable in the GrF[p n ~\ or in the GF[p 2n ], and therefore always 

 solvable in the larger field. Call its roots Y t and Y 2 . Since k is 

 prime to p in 1, we can determine uniquely ( 79 or 63) two 

 marks r tl and % belonging to the GF[p**\ such that 



