ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 61 



85. Theorem. 1 ) - - If r be less than and prime to p n 1 and s 

 be a divisor of p n 1, and if /'(*) be a rational integral function 

 of s with coefficients belonging to the G-F[p n ] such that /"((*) = has 

 no root in the field, then 



represents a substitution on ttie p n marks of the field. 



The conditions of the theorem of 84 for a substitution quantic 

 are all satisfied by the given quantic. In fact, upon raising it to 

 any power , not divisible by s, we obtain a set of terms whose 

 exponents are of the form ms -f Ir and therefore not divisible by s 

 and consequently not by p n 1. If, however, we take the power 



l = ts<p n -l, 



we get the result t, lr , since the power p n 1 of /*((*) ^= is unity in 

 the field. But Ir is not divisible by p n 1. 



Condition 2 is satisfied by our quantic, since it vanishes in the 

 field only when 5 = 0. 



86. As examples under the preceding theorem, we note first r 

 if r be prime to p n 1 [Compare 79]. Next, if p > 2, 



represents a substitution on the marks of the GF[p n ~\ if x be any 

 mark in the field except +1, 1, 0. For the remaining p n 3 

 marks T, the quantics 52) coincide in pairs. We note the following 

 special substitution quantics 52): 



n = 1, p = 7: 4 3 and 5 2g 8 (Hermite), 



For w = l, ^ = 7, v 3 = 1, the theorem of 85 gives the 

 quantics 2 3 _ 



6(6- v )= 2 (| 5 -f 2 

 which together give the following SQ[b, 7] of 80: 

 6 5 + a S 8 + 3 a 2 ^ ( = arbitrary). 



1) For w z= 1 , this theorem is due to Rogers , Proc. Lond. Math. Soc., 

 vol. 22 (1890), pp. 210 218. 



