34 CHAPTER HI. 



the coefficients of /v_j_i(X) being the power p n of the corresponding 

 ones of //(X), those of fa being the power p n of those of /y_i. The 

 coefficients of the product fofi .. .fs\ are consequently unchanged 

 when we replace the coefficients of f by their (p n ) ih powers and are 

 therefore unaltered upon being raised to the power p'\ Hence that 

 product belongs to the GF[jp n ~\, so that F(x) would be reducible in 

 that field, contrary to hypothesis. 



Since the degree ft/d of JR(X), an IQ[pfi,p nd ], is relatively 

 prime to v/8, F { (X) is irreducible in the GF[p* v ] by 47. 



49. Theorem. //' F() be an IQ[m,p n ~\ in which the coefficient 



a Q j |/n 1 ^ sw fo faofi fa ffe GF[pn] 



a _|_ a,p-\- a p ~-\ -f- a pU ~~ 1 =)= 0, 

 then F(&- 5) is an IQ[mp,p n l 



If x be one root of JP(|) = 0, its roots are 



By the hypothesis concerning the coefficient 



a = - x xP n - r pn(m ~ 1] 



tX/ <AJ ii> j 



we have 



/yj _j_ wPJL. /Y*P _j_ . . . I ^P^ m I f\ 



Hence, by 40, p g x is irreducible in the GF[p nnt ~\. The 

 same holds for each of the quantics 



X, = f- | - *"' (* = 0,1,..., m- 1). 

 Consider the function belonging to the GF[p*\ 9 



By 22, it has in the GF[$ nm ~\ no irreducible' factors other than 

 the X { . Hence if F(%P |) have a factor /"() belonging to and 

 irreducible in the GF[p n ~\, /"(I) must be in the 6rjP[> wm ] a product 

 of the X,-, 



an identity in virtue of JP(rc) = 0. Replacing # by ^^ n , another root 

 of F(x) = 0, and therefore X> by X f+1 (* < m) and X w by X , we 

 obtain from the above identity, 



/"(I) =X r .|_i 



Hence /'(|) contains every factor X,- and therefore coincides with 

 I)- The latter function is therefore irreducible in the 



