30 CHAPTER IE. 



Further, the formula 25) furnishes all the roots of 24). For, if 



a* a + /J 5 OJL+/J, 

 ( a _ ffl )P= ( _ aj ) [modp], 



so that a = i + an integer. Hence there are p n /p = p 71 ~" 1 distinct 

 expressions a p a and hence as many roots ft. Hence 



23,) (^)^_ A ^_(^- 1 + ... + ^-f / 3)^/7[(;^-,) p - (Aa-a*)-/ 3 ! 



the product extending over p n ~ 1 marks f of the 6rjF[j? n ] no two 

 of which differ by an integer. 



42. Consider an irreducible factor x p x /3 of # pw a? 1, 

 where therefore 



Denote by / one root of the equation 



a* re /3 = 0. 



Its remaining roots are I -\- 1, /-]- 2, . . ., / + p 1. 



Then by 23 X ) every root of every /[j), ^ ra ] of the first class 

 is a linear function of I, viz. ; A a? *= I + i, i = integer: 



26) x = (/+ * + 0/A, 



the coefficients I/ A and (* 4- f -)/A being marks of the 6rjP[^ n ]. 



Inversely, every such linear function containing I is the root of 

 an lQ[p, pl 



43. Consider an IQ[p, p n ~\ of class ^t. Its roots belong to the 

 GF[_p n P] and are therefore functions of I of the form 



where the aj belong to the QF[p n ]. By 39, f(I) will be a root of 



27) 2^-tf, 



if be suitable chosen in the GrF[p*~\. But, by 42, 



Hence, by 24, we have for any integer m, 



\_f(I)Y m =f(I^ m -)=f(I+m). 

 Substituting f(I~) in equation 27), X^ being given by 16), we find 



The degree of this equation in J being less than p, it must be an 

 identity. But its first member is the ^ th difference of the polynomial 

 f(I) with respect to the constant difference unity attributed to I. 



