CLASSIFICATION AND DETERMINATION etc. 39 



and 13. We verify at once that 2i 2 -\- 2 belongs to the exponent 3. 

 To find the most general mark rj which belongs to the exponent 13, 

 we simplify the calculations by first determining the marks 



% = ai s -\- bi 2 + ci + d 



of the jF[5 4 ] for which rff = 1. Then either (+ T^) 18 or (- yj** 

 equals unity. Now 



+ d 



= ai*-\- bi 2 ci -f d. 

 The condition r^=\ thus gives 



(6* 8 +0 8 -(ai s +ci) 8 =l. 



Reducing by i* = 2, we obtain the conditions , modulo 5, 

 _ 2a 2 - c 2 + 2bd = 0, - 4ac -f 2& 2 -f d 2 EE 1. 

 For a = 0, the only solutions are seen to be 



6 8 =1, f? 2 = 1, c*=l; &^c = 0, f? = l. 



Hence -i 2 +ci2 (e = 1, 2, 3 or 4), or else the negative of this 

 expression, belongs to the exponent 13. We may verify that ** 2 -M-f 3 

 belongs to the exponent 13. We may therefore take 



q = -(2**+ 2) (t s -f i + 3) = 3^ 3 + 2^ 2 + 4. 

 Then 



2= _ ^3_ ^2_ t - _ 1 3= ^D_ 2*8+ i + 1 = - + 2. 



Hence we obtain the following PJ[4, 5] satisfied by the primitive 

 root ?> 9 4 - p 3 - p - 2 = (mod 5). 



This quartic can be decomposed into the two PIQ\2, 5 2 ], 



But p 3 = 4f s + 3?: 2 + 4, 85 ==2 8 + 2t a + 4, p 185 = 8 + 3*+ 4. 

 Hence ^ - ? ) (^ - p 25 ) = ^ 2 - a?(- 8 + 3) -f 3 8 + 4, 



56. The determination of primitive roots in the 6r.F[5 6 ] and in 

 the 6rF[5 3 ] may be made to depend upon the congruence 



32) x+ x*+ x*+ x*+ x+ x 2 + 1 = (mod 5), 



which, by 33, is irreducible. The root x belongs to the exponent 7. 

 The general mark of the 6rF[5*] may be expressed in the form 



5 



x 1 (each d an integer). 



