MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 47 



Since Z, belongs to the 6rF[p 2n ], we may set ZR*. The equation 38) 



Rt(pn+ 1)= Rl(p n +l ^ 



t(p+ 1) = l(p"+ 1) [mod p* n - 1]. 

 This congruence has p n -\- 1 distinct solutions for t, viz., 



becomes n 



The corresponding values of .Z#= 2/ ^ r/ -j- &| 2 givep w -fl distinct 

 sets of solutions 1? 2 of the given equation. 



65. Theorem. - - The number of sets of solutions ( 1? 2 , . . ., J 2m ) 

 in /*e 6r.F[p ra ], ^? > 2, o^ $e equation 



is a mar/^ =)= m #^ 



^(2m-l)_ V pn(m-l) (ft ^ =(= Q) 



wfwre v is + 1 or 1 according as ( I)" l cricf 2 . . . a* m is a square 

 or a not -square in the field. 



By 64, the theorem is true if m = 1. To prove the theorem 

 by induction, we suppose it true for equations in 2(m 1) variables. 

 The proposed equation is equivalent to the system of two equations 



lll -f 2^2 = If, tf 3 g H ----- h 2 W L= 3C n- 



1. Let x =4= 0. For each of the p n - 2 values of y different from ^ 

 and 0, the first equation has p n A. sets of solutions, while by hypo- 

 thesis the second has jp<2 ) np n ( n ' 2 ), where A = l according 

 as 1 2 is a square or a not -square, and p = + 1 according as 

 ( l) m ~ 1 3 4 . . . 2m is a square or a not- square. For the value y 0, 

 they have respectively 2> n -f(^ 1)^ and p*( 2n ) np n ( m ~ 2 ) sets of 

 solutions. Finally, for >^ = z, they have respectively jp 7 * A and 

 j0(2m-3)_|_ ^^n(m-i)_^(m 2)^ se ^ s ofsolutions. The total number 

 of sets of solutions is therefore 



(p n 



By 61, A^ = i/. Hence the induction is complete. 



2. Let ;c = 0. Separating the two cases 17 ={= and ?? = 0, we 

 find the total number of solutions to be 



(p*- 1)O W - 



-f [p n + 



