CHAPTER VIII. LINEAR HOMOGENEOUS GROUP IN THE jF[2] etc. 197 



and consequently also their product belong to the GF[p n ]. But 

 2 accdd belongs to that field only when a or 8 vanishes, so that 

 ad fry = 0. 



If W reduce to the form T Z) x , then a a = K^ dl$ = xr\ = <y = 0, 

 j= = 1, 12 = P = 0. By the latter, ad = a~d. Then 5 = I gives 

 d~a = 1. But ad = ad /ty = 1. Hence a = a", so that % = a 2 , 

 a belonging to the GF[p n ]. 



CHAPTER 



LINEAR HOMOGENEOUS GROUP IN THE GF[2 n ] DEFINED 

 BY A QUADRATIC INVARIANT. 



199. Theorem. If a quadratic form with coefficients in the GF\2 n ~\ 



can not be expressed in the field as a quadratic form in fewer than m 

 linear homogeneous functions of 1; . . ., m , it can be reduced by a linear 

 homogeneous substitution belonging to the field to one of the canonical 

 forms 



JF = 6J 2 + i 8 S 4 + ".+ i m _ 8 g m _i+S l (m odd) 



(meven) 



where A, is zero or is a particular one of the values X for which 



is irreducible in the GF[2 n ~\. 



We first prove that, if m ^> 3, / can be transformed into a 

 quadratic form having a u =0. If every Uij (i,j = 1, . . ., m; i <j) 

 were zero, f would reduce modulo 2 to the form 



i + y% 2 + + 



This being contrary to our hypothesis, we may assume that cr 23 =f= 0, 

 for example. We may also suppose that or 22 =(= 0, since otherwise the 

 transformed of f by (l^) would have cc n = 0. The terms of f which 

 involve 2 may be written thus, 



8S + fefeg^l + ^23^3 + ^24^4 + ' ' ' + 



