130 CHAPTER V. 



Hence, since the product SS~ 1 = l has the determinant unity, we 

 have 



120) a ij \P s + 1 = l. 



From the form of the reciprocal S~ l , it follows that 



121) ^< = -S (M -!,...,) 

 where A,-/ denotes the adjoint of o^ in the determinant 



D= an (i, j = l, . . ., w). 



The value of w, denning the 6rjF[^ n ] to which the coefficients of our 

 substitution S and the quantities A/ were assumed to belong, has 

 played no part in the above formulae. We proceed to prove that 

 our problem can be reduced to a series of similar ones in which 

 n = 2s. Consider the GrF[p* n *\, which includes the G-F[p n ] and 



the G-F[p 2i ]. Raising 119) to the power P 2s _ ^ we have 



if ccij =j= 0. Hence -=- would be the power p s -j- 1 of some quantity 

 in the GF[p 2ns ]. The substitution Tj 



t' t d" 1 OM. Z^ -I. A\ t' - ,,t 



A= bi V^ = ~ - 1 -; ? m ? * \J)i 9j f*; 



transforms cp r into 



in which the coefficients Aj- and A} are equal. Evidently 1} transforms 

 ^ into a substitution with coefficients in the GF\_p* nf \. 



Suppose that the coefficients 13; ]37 . . ., lOTl do not vanish, 

 while iy = for j > m x , in all of the substitutions leaving cp r 

 invariant. Then the group is isomorphic with a group of substitutions 

 in the GF[p 2ns ] leaving invariant 



In the latter substitutions the coefficients u.\j (j > m x ) are all zero. 

 If, among the coefficients 2 y (j> m i)> an J one as ^i^H 8 *^ we 

 transform the invariant <jp[. by T ilt giving the function 



*!=! 



