66 



CHAPTER V. 



92. Theorem. The necessary and sufficient condition that the 

 transformation 



m 



54) X'-pAtZF* (X,Ai in the 



shall represent a substitution on the marks of the GF[p nm ~\ is 

 A l A 2 ...Am 



Ap n A_P n Ap n 



+ 0. 



p n( 



p n(m 1) p n(m 1) 



I . . . A. m _ i 



We seek the condition under which 54) is solvable for X. 

 Raising 54) to the powers 1, p n , p 2n , . . ., p n ( m ~V and reducing the 



powers of X by 



ZP WW X, 



we obtain the following m equations (written with detached coefficients) : 



Z' 



A, 

 At 



Af 



n(m 



p n(m L) n(m 

 - -Am 



Am -A-i . . . -ft-m 2 -^-m 1 



The solution of this system of equations in X, X pn , . . . gives 

 55) - ^ ^ 



n(m-l) n(m-l) 



n(m-l) n(m _ 



2 <&* 



56) AX* 



,n (m i) 



An 



.A{' 



j _ 2 



n (m- 



The condition A =j= is necessary , since otherwise there would exist 

 a relation between the powers of X' with exponents < p nm . To 

 prove that the condition A =J= is sufficient , we need only verify 

 that the X given by 55) satisfies the relations 56) for i = 1, 2, . . ., w 1. 

 Observing that J.^ n = ^l in the field, we find the following relations 

 upon raising 55) to the power p n ( m ^ and moving the first i rows 

 below the last m i rows: 



