MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 45 



62. Theorem. - - The not- squares of any 6rF[p*], p > 2, ewe 

 not- squares or squares in the GF[p nm ~\ according as m is odd or even. 



If (5 be a primitive root of the 6rF[p wm ], then Q :EZ <?", where 

 u = (p nm l)/(p n 1), is a primitive root of the GF[p n ~\. Hence 

 the marks =)= of the GF[p n ~\ are given by the formula 



pfEE*"" (t,= l, 2, . ..,JP-1). 



Let U be a not -square in the GF[p n ~], so that v is odd. It will be 

 a not -square or a square in the GF[p nn<r \ according as uv is odd or 

 even, i. e., according as u is odd or even. But 



TO 1 



u = (p nm l)/(p n 1) =p ni = sum of m odd terms. 



Hence u is odd or even according as m is odd or even. 



63. Theorem. If d be the greatest common divisor of m andp n l, 

 there exist exactly (p n \)/d marks =f= in the GF[p n ~\ which are 

 w th powers in the field. 



If ^ 4= be the m th power of some mark v of the field, we find, 

 upon raising ^ = v m to the power (p n l)/d and noting that the 

 power p n 1 of the mark i; 7W / rf =j= is 1, the equation 



37) ^(y- D/<*=1. 



Inversely, there are (p n l}/d roots of 37) in the GF[p n ] by 

 16 and each root is an m th power in the 6r_F[j3 n ]. To prove the 

 last statement, we note first that such a root p is a ^ th power. In 

 fact, the roots of 37) may be exhibited as follows: 



where Q is a primitive root of the GF\jp n ~\. That these roots are 

 distinct is shown by supposing 



Q di - 9 dj (j 5 *' < (P n ~ l)/rf)- 



9 *j( 9 *d -J}- 1) = o [d(i -j) <p- 1]. 



Hence i j = 0. We next prove that ^ = Q di is an w th power. 

 Since m/d is relatively prime to p n 1, we can determine integers I 

 and t satisfying the equation 



t( 



Hence 



Q 



Therefore 



ft = P t = m - 



Corollary. - - Every mark of the 6rjF|jp n ] will be an m ih power 

 in the field if, and only if, d = 1. Extraction of the w th root of an 



