62 CHAPTER V. 



87. If 0(|) = a S* + ai^-M---- be a [&, P n ], then will also 

 ), obtained by forming the compound substitution, 



+ * - 



(fc/J 



if a = 1. We may dispose of T>, /3, $ to simplify Oi(|). We take 

 y = tfo"" 1 ? and, in case Jc is prime to p, we choose /5 so that 



) 



Finally, we take d= yO(/3). The quantic O^l), in which the 



coefficient of (;* is unity, the constant term zero, and, when k is not 



a multiple of j), the coefficient of I*" 1 is zero, will be called the 

 reduced form of 0() for the GF\j>*\. 



88. To illustrate the use of the theorem of 84, we apply it to 

 determine all reduced SQ\3, p n ]. For p =J= 3, the reduced cubic in 

 the GrF[p n ] is | 3 + aj. The sub -case ^) w = 3m + 1 must be rejected, 

 since the m ih power of 3 -f J contains the power | 8 = | p *~ 1 with 

 coefficient unity and hence =j= 0. For the sub- case ^> w = 3m + 2, the 

 condition given by the power m -f 1 is (m + 1) a = 0. But if m + 1 

 be divisible by^>, then would also 3m -f 3 =p n -\- 1. Hence must a = 0. 

 The resulting form | 3 is a SQ[3, p n = 3m + 2] by 79. 



There remains the case p n = 3", when the reduced cubic is 



Raising it to the power 3 n ~i_f 3 W ~ 2 H ------ \- 3 + 1 =(3 n l)/2, we 



find (mod 3), 



fc 2.8-. 3- fc 3- /fe3 



g + 2 j-.-Cr-H^r 



The highest exponent of | in this product is < 2(3 W 1). The 

 coefficient of 



g8 w 1= 12(3^ irS 71 2 H ----- 1-3 + 1) 



is evidently aj ^ x . Hence must a.\ = 0. Applying then 



the corollary of 81, the resulting form 3 -}- 2 g is a 5Q[3, 3 W ] if, 

 and only if, either 2 = or else -~ 2 is a not -square in the 



89. To treat a more characteristic example, we seek the SQ[5,p n '] y 

 when p n is of the form 5m -f 3. The reduced quintic is 



The power m+1 gives (m + l)/3 as the coefficient of |5+2= |f w -i. 

 If m -f 1 = (mod p), then 5m -f 5 =p n -\- 2 = and therefore # = 2. 



