56 CHAPTER V. 



79. Theorem. | w is a SQ[m, p n ~] if, and only if, m be prime 

 to p n 1. 



The theorem follows immediately from the corollary of 63. 

 However, to illustrate a method of proof used below, we will verify 

 that, if m be relatively prime to p n 1, m takes p n distinct values 

 in the GF\p n ~] when does. It is sufficient to prove that from 



j /~\ <-Wi *rYfl 



46) $i = | 2 



follows | t = 2 , provided ^ and | 2 are marks of the field. This being 

 evident if either be the mark zero, we suppose |, 4= 0, ? 2 4 = 0? so that 



47) if-'. -'_ i. 



Raising the members of equations 46) and 47) to the respective 

 powers t and r, chosen ( 7, note) so that tm -f T(j9" 1) = 1, and 

 forming the product of the resulting equations, we find that |,= i 2 . 



80. Theorem. - - For an arbitrary mark a of the GF[j) n ], 



is a SQ[b, p tt ], if p is a prime of the form 5m + 2 and n is odd. 

 To prove that, in the GF[p t>r \, | t = 2 is the only solution of 



(gi-^fe 4 + gfc+ 6?gJ 4- gig -h ^ 



we set 



g 1= =A + ^, * 2 =A t W, 



here 1 ) limiting our proof to the case p>2. Then 16 times the 

 quantity within the braces becomes 



= {(20A 2 + ) 



But 2 ) -f 5 is a quadratic residue of wo odd number of the form 

 bm -f 2 or 5 w 2. Hence ( 62) 5 is a not -square in the 6r.F[5 n ], 

 w being odd and p = 5m + 2. Hence, if the above expression 

 vanishes, we must have 



whence, for p > 2, ^ 2 = 5 A 2 , so that I = p = 0, | x = J 2 . 



1) An analogous proof for j9 = 2 is given in Annals of Math., 1897, pp. 84 85. 

 For an arbitrary prime p, the theorem is a special case of that of 82. 



2) Gauss, Disquisitiones Arithmeticae , Art. 121. 



