CLASSIFICATION AND DETERMINATION, etc. 31 



Since it reduces to the constant 6 =j= 0, the degree of /"(/) is 

 exactly ft 1 ). Hence 



We have therefore proved that the roots of every IQ\]p,p n ] of class ft 

 are integral functions of I of degree ft. 



44. We can readily obtain a formula including all 

 In the above expression f(I) 9 let the ct, be arbitrary such, however, 

 that f(I) does not reduce to . To set up the equation of which 

 f(P) is a root, consider the p equations 



PUW-Q-Q (i- 0,1,... ,1-i} 



Reducing the exponents of I below p by using the identity 



we obtain the series of equations 



2 J 2 + ---- h Op-1/'- 1 ^ 0, 



- g 4- ,-!)!+ i/ 2 -f r Hh ^- 2 ^- 1 = 0, 



Eliminating J, J 1 , . . ., J^" 1 from these ^9 equations, we reach the 

 required irreducible quantic 



Setting a /u _|_i= a / ,_|_2= = cc p i== and giving to a , i, . . ., a/ui 

 all possible values in the GF[p n ] and to a^ every value =f= 0, we 

 obtain p n t'(j) n 1) irreducible quantics of class ft. Since f(I-\~m) 

 leads to the same determinant as /"(/), if w be an integer, the number 

 of distinct IQ[p,p n ~\ of class ft is pwifjp* 1), a result also follow- 

 ing from 39. 



For ft = 1, we find that 



so that we may derive a new proof of formula SSj). 



1) Boole, Calculus of Finite Differences, p. 5 and p. 19, formula 3). 



