ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 59 



we are to prove that 



50) [<K)P= Of + v)riP r -i= r)P r + y^-^ /3 d 



has a solution 17 in the G-F[p n ], for, if ^ be such a solution 

 (necessarily - = 0), then 



will belong to the GF[p n ] and wiU satisfy 0(|) = 0. 



Setting r ( = 1/co in 50) and multiplying by cX, we find 



This has a solution o in the 6rJP[j) w ] for /? arbitrary in the field. 

 Indeed, by 81, corollary, 



represents a substitution on the marks of the GF[p"\, since v//3 rf is 

 not a <2 th power and hence not a (p r 1)*' power in the field. 



Note. - - For p = 3, 5, 7 and partially for # = 11, the author 

 has shown 1 ) that the only SQ[p, p n ] which exist are reducible to 

 the form %(% d v)(P l ^ d 



where d is a divisor of p 1 and v is not a cp h power in the GF[p n \. 



84. Theorem. 2 ) The necessary and sufficient conditions that 

 shall represent a substitution on the marks of the GF[p n ] are: 



1. Every t ih power of 0(1), for t <^p n 2 and prime to p, shall 

 reduce to a degree <p w 2 on applying the equation % pn =* 5; 



2. There shall be one and only one root in the Gf [p n ] 

 of <D(g)-0. 



After the exponents of are reduced below p n , let 



Put for the p n marks ^ of the GrF[p n ~] and add the resulting 

 indentities. We find, on applying 75, 



If O(|) represent a substitution, we must have 



1) Dissertation, Annals of Mathematics, 1897, pp. 101108. 



2) For the case n=l, this theorem is due to Hermite, Comptes Rendus, 

 vol. 57 (1863), pp. 750757. 



