28 CHAPTER III. 



Determination and classification^ of the IQ[jP>p*]j 39 46. 

 39. Consider for positive integers p the auxiliary quantics 



16) X u 



where C^ k denotes the number of combinations of p things k at a 

 time. Since C p r^ k is a multiple of p, if < & < p r y we have 



17) X p r=xP npr -x (mod p). 



Hence , by 26 7 the product of all the /<?[#*,#"] is given by 



18) V p s, p n= 



We derive a simple expression for the quotient 18) as follows. 

 From 



we deduce at once the congruence 



19) X^EEXf-X, (modi.). 

 Multiplying together the congruences (for i = 1, 2, . . ., v) 



X. + , = X* + ,. _ ,- X. + , _ ! (mod i,), 

 and dividing the resulting formula by the product 



X U _j_ 1 X U _j_ 2 X M ^- y _ 1 , 



we find 



u -j- f 1 



20) X u + v = X u fJ(Xf~i-l) (modp). 



Taking w =p-i, w _j_ v= ^ we fi n( j f rom jg^ an( i 20) the result 



21) rp,r 



Further, if v lf v^ 7 . . ., v p n_^ denote the marks 4= of the GF[p n ], 

 we have 



Since X f i/ 7 - is of degree p ni in a?, it must decompose in the 

 into p ni ~ s factors each an IQ\jf,p*]. 



1) For the case n 1, Serret, Jownal de Mathematiques , 1873, p. 301; 

 Algebre, II, ch. IV. For general w, Dickson, Bw?Z. J.wer. Ma^. ^oc., 1897 

 pp. 384 - 389. 



