46 CHAPTER IV. 



arbitrary mark of the G-F[p n ~\ is possible if, and only if, m be 

 relatively prime to p n 1. With this condition satisfied, there exists 

 but one m th root of each mark. 



Number of solutions of certain quadratic equations in a Galois 

 Field, 6467. 



64. Theorem. 1 ) - - If v = 4- 1 or 1 according as a^ is a 

 square or a not-square in the GrF[p n ~\, p > 2, the equation belonging 

 to tte field, l6 j + ak g_ x (0l + , 3 +0), 



has p n v or p n -\- (p n \)v sets of solutions according as % =f= or K = 0. 

 Setting KJ^^^, the equation becomes 



i? 2 -f- fl^tf = !. 

 1. If a^g^ ^ 2 ; a square =|= in the 6rF(j> w ], we set 



7? + Ai 2 =0, ??-A 2 =(>, 

 whence l 



n = y (9 + *)> 2 = YT ((> - 4 

 The equation becomes nfi __ H 



fj O I* ^ /v. 



If ^ =f= 0, we can give to any one of the p n 1 marks =|= in 

 the 6r-F[j> n ], when the corresponding value of Q is determined by 

 the equation. There are in this case p n 1 sets of solutions | t , | g 

 in the field of the given equation. 



If ?c = 0, there are evidently 1 -f 2(p n 1) sets of solutions. 

 2. If ffjtfg be a not -square in the GF\j) n ~\, the equation 



g> 2 = ajflfg 



is irreducible in the field. If one root be i, the other is i^ n r - * 

 by the corollary of 31. We therefore have the identity 



We are thus led to determine the number of roots in the 

 of the equation in the unknown Z = TI -f it%, 



38) ^*+ 1 = 1 x. 



If ^ == 0, we have ^ = and hence a single set of solutions 



fc-cvt-o. 



If 40, let JR be a primitive root of the ,F[> 2w ]. We ma J 

 set ^ ^ = R k , whence 



so that 7c(p n 1) is divisible by p 2w 1, the exponent to which E 

 belongs. We may therefore set / = l(p n + 1), being an integer. 



1) The theorems of 64 67 are immediate generalizations ofNos. 197 200 

 of Jordan's Traite des substitutions. 



