4 CHAPTER V. 



Reduced quantic Suitable for p n = 



a 2 



5 -f a 3 -f -g-j; (a arbitrary) 5m + 2 



| 5 + a| 3 + 3 2 | (a = not -square) ... 13 

 5 2a| 3 -f 2 | (a = not -square) . . . 5 W 



That in fact these quantics do represent substitutions on the 

 marks of the corresponding GF[p n ] follows from 79, 80, 81, 83 

 and 86, with the exception possibly of the eleventh and thirteenth 

 forms. To verify 1 ) that the latter two are substitution -quantics, set 



k (i 



Since a is a not -square, we can choose an integer a so that fi~ 2 

 shall be a particular not -square v. But 



fi-^O*!) - ^-f^-^-f 3Gi- a )*g p- 8 | s . 



Since ^, 3 = + 1 (mod 7), we can choose the sign of p = (/v) 1/2 to 

 make the coefficient of | 2 unity. It follows, therefore, from 87 

 that 0(1) and M*() will be substitution -quantics modulo 13 and 7, 

 respectively, for cc an arbitrary not -square, if they be such for a a 

 particular not -square v and for the -j- sign in H^). We take v = b, 

 a non-residue of both 7 and 13. In the notation of 76, these 

 reduced forms represent the substitutions, 



n \ = /0, 1, 2, 3, 4, 5, 6\ 



U 5 + o| 3 + | 2 + 5|j - VO, 5, 2, 3, 1, 6, 4)' 



( g \ = /O, 1, 2, 3, 4, 5, 6, -6, -5, -4, -3, -2, -1\ 



l|+ 5| 8 + 10i; VO, 3, 1, 5, 6, 4, -2, 2, -4, -6, -5, -1, -3/ 



modulo 7 and 13 respectively. 



The Betti-Mathieu Group, 9194. 

 91. It was shown in 81 that the quantic belonging to the GF[p nm } 7 



represents a substitution upon the p nm marks of the field if, and only 

 if, X = is the only solution in the field of the equation 



0. 



1) For a verification by means of the theorem of 84, see the author's 

 paper, American Journal , vol.18, pp.210 218; in particular, 7 and 9. 



