PROOF OF THE EXISTENCE OF THE GF[p*], etc. 17 



It is to be understood throughout the investigation that all our 

 operations upon quantics are performed in the GF[p n \ We may 

 therefore state the results of 23 and 25 as follows: 



An IQlwiup*] is a divisor of x pnr>l x if, and only if, m l be a 

 divisor of m. 



It follows that an irreducible factor of x pnnt x will be of degree m 

 if, and only if, it is a factor of none of the functions 



9) x pnmi x (m^ < w, % a divisor of m). 



After showing that the irreducible factors of any such function are 

 all distinct, it will follow that, if we divide x pnm x by the product 

 of all the distinct irreducible factors of the expressions 9), we obtain 

 a quotient V m>p n which equals the product of all the IQ[m,p n ]. 



For example, if m be prime, the irreducible factors of x pnm x 

 are of degree m or 1. By 13, the product of the distinct linear 

 factors is x pn x. Hence, if m be prime, 



_XP nm -X N _ pnm-pn 



Vm ' p m>p ~~ 



We next prove that the irreducible factors of xP nm x are all 

 distinct. If such a factor be of degree m, it can be used to define 

 the GF[p nm Y)- In this field the equation 



x p nm - x = 



has p nm distinct roots; viz., the marks of the field. Hence no factor 

 can be a multiple factor in this field and therefore not in the in- 

 cluded field the GF[p n ]. If an irreducible factor f be of degree m^ < m, 

 it cannot be a multiple factor. Indeed, m^ must be a divisor of m, 

 and /' must divide x pnmi x in the GF\j> n ~\. By the former case, 

 /' is a simple factor of the expression just given. It remains to prove 

 that f cannot divide the quotient 



It suffices to show that Q and x pn 1 x have no common factor in 

 the GF[p n ~\. Setting 



t /yp wm '_L 1 v pnm I 



y r__ x* i, r= mnm ,_^ 9 

 it suffices to prove that y 1 and 



have no common factor. The condition for a common divisor is 



that r be the mark zero in the field. But r = 1 (mod p). 



1) See 28. 



DlCKSON, Linear Groups. 2 



