LINEAR HOMOGENEOUS GROUP IN THE GF[2n] etc. 199 



will belong to the GF\2 n ~\. It transforms into 



187) 6ii, + 6564+-"+6-36-+6i-i+6-i6+ 



d being such a mark that the equation 



188) 2 + 6 + * = 



is irreducible in the .F[2 n ]. It follows from 188) that 



Hence 188) has a root | in the GF[2 re ] if and only if 



The left member being its own square in the 6rF[2 n ] and hence 

 either or 1, it follows that 188) is irreducible in that field if and 

 only if 



189) d f <52+<5 4 + ...+ d 2n - 1 =l. 



Applying to the quadratic form 187) the transformation 



Si-i im-i+a&ii, 65 60 (* i,...,w; t + w i) 



the constant 8 is replaced by 



which is therefore a root of 189). Giving to A all possible values 

 in the GF[2 n ], we obtain the 2"- 1 roots of 189). Indeed, if in 

 the 



we must have A x = A or A -f 1. Hence all irreducible quadratic forms 

 in two variables of the 6rF[2 n ] can be transformed linearly into each 

 other. Applying, finally, the transformation 



^ 



187) becomes F^. 



200. Changing the notation used in exhibiting F, the canonical 

 quadratic form for an odd number 2m + 1 of indices may be written 



The conditions upon the coefficients of the substitution S: 



