68 CHAPTER V. 



The conditions for the identity >' = $Z are readily seen to be 



60) 



Since the left members of 59) and 60) are the powers p n and p 2n , 

 respectively , of the left member of 58) , we must have pp w = g. Hence 

 the totality of substitutions 57) for which the expression 



is a mark of the GF[p n ~\ form a group leaving Z relatively invariant. 



94. Consider next the substitutions 57) which multiply the 

 function 



by a parameter p ? where Z) is a mark =f= in the GrF[p 3n ]. 



To form the function T' into which 57) transforms Y 9 we 

 note that 



Denoting by W the product of the expression on the right by D and 

 forming the sum T = W+ Wv n + W^ n , we find that the conditions 

 for the identity Y' = $Y are the following six relations: 



where, for brevity, 



61) T = DA 3 A? 4- D*AfAf "+ D^ n A 3 A 



62) f 



In particular, it follows that $?"= Q. Hence those substitutions 57) 

 whose coefficients A 1 , A, A 3 make r = and give to the function f/D 

 a value belonging to the G-F\_p n ~] form a group with the relative in- 

 variant Y. 



The method may be readily extended to determine for general m 

 the substitutions 54) which leave relatively invariant the following 

 function TO _i 



(D in the GF[p* m l[), 

 where s may be any integer < m, except perhaps m/2. 



