PROOF OF THE EXISTENCE OF THE GF[p], etc. 15 



22. Theorem. - - A function E(x) belonging to the G-F[j) n ] can 

 be decomposed into factors belonging to and irreducible in the GF[p n ~\ 

 in a single way. 



For if E(x)-f 1 f,...f t -F 1 F,...F t , 



where fi(x) and F t (x) are irreducible, F must by 21 divide one 

 of the factors /J, and, since the latter are irreducible, be identical 

 (apart from a factor independent of x) with one of them, say f v 

 Proceeding similarly with the equality 



/2/3 fh = F 2 F S . . . F k , 



we may suppose f% = F 2 , etc. In particular, h = ~k. 



23. Theorem. - - Every function F(x) of degree m belonging to 

 and irreducible in the G-F[p n ~\ divides 



x^ m - x. 



Upon dividing any function E(x) belonging to the GF\j) n ~\ by F(x\ 

 we obtain a residue of the form 



the a's being marks of the GF[p ri \. We denote the p nm distinct 

 residues of the above form by 



5) X, (*- 0,1,... ,/ m -l), 



and, in particular, by X$ the residue zero. Consider the products by a 

 fixed residue Xj=f= -Xo^ 



6) XjXt (*-0,l,...,i-"-l). 



By the theorem of 21, the products 6) are all distinct and different 

 from X . Hence the residues obtained on dividing them by F(x) 

 must coincide apart from their order with the residues 5). Forming 

 the products of the residues not zero in each series, 



i [mod 

 t=i t=i 



Since nX,-=|=0, we have by 21, 



Taking for X,- the particular residue x 9 the proof of the theorem 

 follows. 



24. Theorem. - - If f(x) belongs to the G-F[p*], we have, for 

 every integer t, the following identity in the field: 



