38 CHAPTER III. 



54. The eight PIQ[, 3] are the factors of 



<m (a 80 - 1) (X - 1) _ 32 _ 24 , T 16_ T 8 , 1 



(a"-l)(a!"-l) = 



It suffices, however , in view of 50, to determine a primitive root $ 

 of the OF[3 4 ]. To get an /$[4,3], we employ the theorem of 37 

 for A = 4, i = 2, p = 2, I = 1, giving the decomposition 



x*+ 1 = 77 (# 4 x 2 - 1) (mod 3). 

 Hence a root i of the irreducible congruence 

 x*-x 2 1 = (mod 3) 



belongs to the exponent 16. If then we find a mark 6 belonging to 

 the exponent 5, Q = i<5 will, by 14, be a primitive root of # 80 = 1. 

 We readily verify that the fifth power of ?' 2 * is congruent to unity 

 modulo 3. To find the irreducible congruence satisfied by the primi- 

 tive root p = i (?' 2 i), we form its powers, 



0*- if t'q: i _ 1, pS- qp **- ; 1, () 4 =qi ^_ ?: 2 T ;_ hL 



Eliminating the powers of i, we have 



P 4 s 3 + ? 2 + ( - 1 = (mod 3). 



The product of the two PI$[4,3] thus reached is Q+ Q*+ ? 4 -h 1. 

 Since the expression 31) contains only exponents which are multiples 

 of 4, we would expect the new factor p 8 6 + 4 + 1. In fact, the 

 product of these two quantics of degree 8 gives p 16 -f- p 12 p 4 -f 1, 

 which divides 31) giving the quotient 



pl_ p l* + p*+ 1 = ( ? + p4+ p 2 + 1) (,8 + p 4_ p2+ !) 



We therefore have two new PIQ[4, 3] given by the decomposition 



e 8 - (> 6 + ^> 4 + 1 = (? 4 + <> 3 - 1) ( ? 4 - (> 3 - 1). 



Since 8 + (> 4 p 2 + 1 is derived from $ 8 Q*+ Q*-\- 1 upon replacing 

 Q by in the latter and multiplying by p 8 , we find 



Q*+ e 4 + (> 2 + 1 = n ((> 4 P 3 - (> 2 =F e - 1), 



(> 8 +(> 4 -(> 2 +l = //(9 4 + (>-!). 

 Hence the eight P/C[4 ; 3] are 



55. To obtain a primitive root p of the &F[5 4 ], we define the 

 latter by means of a root i of the irreducible congruence 



x*=2 (mod 5). 



Indeed, by 35, x* 3 3 is an /$[4,5] belonging to the exponent 16. 

 Since 5 4 1 =16 3 13, we seek marks belonging to the exponents 3 



