LINEAR GROUP WITH QUADRATIC INVARIANT. 157 



2. Upon applying to f the transformation 



=Di 1; is- & (; = 2,...,m) 



772 I* * j 77% 



we obtain the function 



+ 



Its determinant is 



168. Theorem. A quadratic form f with coefficients in the 

 GF[p n ~\, p > 2, awdf o/ 1 determinant A =4= caw &e reduced by a linear 

 homogeneous substitution belonging to the field to the form 



142) 



(each at =}= 0). 



Since A =j= 0, the coefficients n , 12 , . . ., # lm are not all zero. 

 If 11 = 0, we may suppose that a 12 =|= 0, for example. Applying 

 to f the substitution of determinant 2 A =j= 0, 



I^A^ + ^2, 2 = i fe> 15 & (i = 3, ..., m) 



we obtain a form in which the coefficient of %\ is # 22 + 2Aor 12 . 

 Taking for A any one of the p n 2 marks different from zero and 

 from #22/2^12? the coefficient of J will be not zero. Whether a n 

 be zero or not, we thus obtain a form 



whose determinant A f is not zero by 167. 

 Applying to f the substitution 



. li li~ ^2? If = I/ 



, . . ., ) 



we obtain a form in which the coefficient of | t | 2 is zero, while 

 that |J remains /3 n =(=0. In a similar manner, we can make the 

 coefficients of l^g, . . ., ^ m all zero. In the resulting form 



2, . . . , m 



the coefficients ^ 22 , ^ 23 , . . ., y^ m are not all zero, since the determinant 

 of the transformed form is not zero by 167. 



