162 CHAPTER VII. 



174. Theorem. If a 17 ff 2 , a 3 ~be any set of solutions in the 

 GF[p n ~],p>2, of the 



, 



there exists a substitution S, derived from the generators of % 173 which 

 leave If + If + ^|| invariant, such that S replaces |j_ % ^1^1 + ^2 + ^3^3- 

 The proposition follows if 1 a 2 or 1 af be a square =j= in 

 the GrF[ n ]. For example, if 1 | = r 2 , then 



so that we may take 



The proposition will be true for a 19 & 2 , a s if true for the quantities 

 where { = i9-2^ i ~ cr^ + a a (, < = 8 , 



so that we have 



148) I i +i i + 7i l -! + J + 7!-i. 



In fact, if the group contains a substitution /S' which replaces | x by 

 a ii+ 2&+ ^3^3? i* w iH contain the product S=Ol]^ a S' which 

 replaces ^ by aji + 2 fe + ^3^3- 



175. Consider first the case in which 1 is a not -square in the 

 GF[p n ~\. By 64, there are p n + 1 sets of solutions p, ^ in the field 

 of the equation p 2 +a 2 =l. Not more than two of these sets of 

 solutions give the same value to 



Indeed, upon eliminating tf, we obtain a quadratic for p. Hence a' 2 

 takes at least y(p n H- 1) distinct values. But, by 67, there are 

 exactly y(p w 3) distinct marks y =j= for which rf 1 is a square 1 ), 

 so that 1 if is a not- square. Hence there exist at least two values 

 of tfg f r which 1 Kg 2 is a square or zero. If it be a square, our 

 theorem follows from the previous section. There remains the case 

 ^ 2 = 1, for which, by 148), 



If fi=l, we have { = a$ = 0, since 1 is a not -square, and the 

 required substitution is S*=Cfyt*. If ^ be a not-square, we may 

 take ft = 1 , so that 



1) Zero is not reckoned as a square. 



