CLASSIFICATION AND DETERMINATION, etc. 33 



The given quantic being F(x), the roots of F(x) = are 



all belonging to the GF[$ dm '\. If F(x) be reducible in the GF[p d *], 

 the root x will satisfy an IQ[t,p* n ], t<m, of the form 



(IT sAfjT 'rP dn }('Y /rP 2dn } (~X ^p dn ( t ~ 1 ^\ A 



^^v r jjj\^\. Jj )\-^- *' / v**- * / === ^ 



Its constant term must be a mark of the GF[p dn ~], so that 



in virtue of the single relation F(x) = 0. But this requires that tn 

 shall be a multiple of m, and therefore that t be a multiple of m, 

 in contradiction with t < w. In fact, by 23, _F(#) divides in the 

 the function x pkd - x if, and only if, h be a multiple of m. 



48. Theorem. 1 ) - - An IQ^p**] decomposes in the GF[p nv ] into 

 d factors each an IQ Hp p nv , d &em^ ^e greatest common divisor 



of p and v. 



The given quantic being F(x), the roots of F(x) = in the 

 GF[p n ] are 



x, x* n , xP 2n , ..., xP n( - l) [xf=x]. 



They may be separated into d sets each of ji/d roots, 



x" ni , x^ s + '\ x*^ i + i \ . . ., *" [(*-')' + '] 



for i = 0, 1, . . , d 1. A symmetric function of the roots in one 

 set is unaltered upon being raised to the power p nd and therefore 

 belongs to the GF[p nd ~\. The roots of the general set therefore 

 satisfy an equation 



JFi(X) = (X- **0 (X- x'"V + ). - = 0, 



with coefficients belonging to the GF[p nd ~\ and a fortiori to the 

 If 



then 



We next prove that the F { (X) are irreducible in the GF\_p n 

 Suppose, on the contrary, that in the latter field, 



Then . 



F t (X) = f,(X) 9>, 



1) For the case n = 1, this theorem and the corollary of 49 were stated 

 without proof by Pellet, Comptes Eendus, vol. 70 (1870), pp. 328 330. 



DlCKSON, Linear Groups. 3 



