16 CHAPTER II. 



Let f(x) = c + dx + c. 2 x 2 + - + c k x k , 



where the e's belong to the G-F[p n ~\, so that 



7) ef-c, (-0,l,...,t). 



Raising f(x) to the power |j and noting that the multinomial coeffi- 

 cients of the product terms (viz., those not p ih powers) are multiples 

 of p, we have the algebraic identity , 



We obtain by induction the formula 



W*)f - c1+ cfV'+ - + <$'x tp +p Q s (x). 

 Applying 7), we obtain in the G-F[p n ~\ the identity: 



Our theorem now follows by a simple induction. 



25. Theorem. - - A function F(x) of degree m belonging to and 

 irreducible in the GF[p n ~\ divides (in the field) the function 



only when the integer t is a multiple of m. 



Let t = sm -f fj where < r < m. By the theorem of 23, 



we have 



x p nt - x =(xP nsm )p nr -x== x? nr - x [mod F(x)]. 



Hence, if x pnt x be divisible by F(x) in the 6r-F|j) n ] ? we have 



8) &" r =x [mod JP(a?)]. 



Denote by f(x) any one of the p nm expressions 



c 4- CiX -f c 2 x* 4- + c m -\x m ~^ 

 in which the c's are marks of the G-F[p n ~\. By 24, we derive 



Hence the congruence ,. nr 



t? = g [mod F (x)] 



is satisfied by the^> wm expressions f(x), which are distinct modulo F(x), 

 the latter being an irreducible function of degree m. Since r < m, 

 it follows from 15 that the congruence must be an identity, 

 whence r = 0. 



26. The number N mff n of functions F(x) of degree m belonging 

 to and irreducible in the GrF[p"] may now be readily determined. 

 For brevity, such an irreducible quantic will be designated an 



