ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 55 



If we denote the marks of the GF[p n ~\ as follows, 



44) fi , t ui, ji 2 , ..., fv-i; 



the necessary and sufficient conditions that O (|) = ft be solvable in 

 the field for arbitrary /3 are that the marks 



45) 0( k u ), <D(f*i), <t>(> 2 ), .. V 0(^_!) 



be identical with the series 44) apart from their order. In fact, the 

 p n values which 0() takes must all be distinct, since /3 is to have 

 p n distinct values. When the conditions named are satisfied, the 

 series 45) forms a permutation of the series 44), and the quantic 0() 

 is said to represent the substitution 



f 8 ~|_r Po, ^i , -, JV-i ~1 



LO (6)J ^ LO 0*o), o GUI), . . ., o Gv-i)J 



ow $e marks of the GF[p n ~\. For example, 3 represents the sub- 



stitution m_r> i> 2 > 3 > 4 i 



L6 8 J=jo, i, 3, 2, 4j> 



on the marks of the 6rF[5], i. e., the field of integers taken 

 modulo 5. A quantic of degree k with coefficients belonging to the 

 GF\j) n ~\ will be called a substitution quantic SQ[k, jp w ] if it satisfy 

 the above conditions. Its degree Jc will be supposed <_p n in view 

 of the equation ^= | satisfied by every mark of the field. 



77. An arbitrary substitution on the marks of the 6r-F[j?*], 

 ("fo, .v^i .V- 



can be represented by the quantic O (|) given by Lagrange's inter 

 polation formula, 





where 



and F'() denotes the function derived from F(g) by formula 41). 

 Evidently O(|) is an integral function of of degree <p n . 



78. Theorem. Two distinct quantics 0(8) aw^ Y^) belonging to 

 the GF\j> n '\ can not represent the same substitution oti its marks. 



Por > if <D(^) - V(^) (i = 0, 1, . . ., r- 1), 



the equation <t>(J) Y(8) = would have in the field p n distinct 

 roots (ii 9 whereas its degree is less than p n . By 15, it must be 

 an identity in . 



