176 



CHAPTER VII. 



As shown above, T-i corresponds to C 6 C G . The product 

 seen to be the second compound of 



-i s 



X 







-x ^ 



-x 











c 





 

 x 



But x belongs to the GF[p n ~\, in which 1 is a not -square, if and 

 only if Y (!+(>) is a not -square, which occurs if and only if Of; 4 



is not of the form 2' 4- Hence O 1 contains Ofj4(7 5 (7 6 if Of; 4 is not 

 & $2, 4? hut not in the contrary case. As shown above, O f contains 

 every Ql\ 4 . To prove that 0' contains all the generators of 0_!_i(6,^) n ), 

 it evidently remains only to prove that 0' contains all even substi- 

 tutions on i, (jg, g s , | 4 , 5 , and, if # TC =3, also Fi j2 , 6 . 



Expressing the linear substitution (i^is) i n the indices Y^-, we get 



4-fa-/3 2 ) -ka-0-a/ft -4( 







1 

 1 

 



tfi y(-/ 



This substitution is the second compound of 



x y 



8 W 



161 > -TT Z 



-Y X 



having determinant unity, where 



In order that 161) shall belong to the G-F\_p n ~\, it is necessary and 

 sufficient that x be a mark =[= of the field. We proceed to prove 

 that, for every set of solutions in the field of a 2 + /3 2 = 1 , the 

 expression 



