20 CHAPTER III. 



TJw exponent e to which F(x) belongs must divide p nm 1. 

 For, if pnm ._ ! _ u + r ^ 



where < r < e, then F(x), dividing tf 1, must divide x ke 1 

 and,, by 23 , also #*+ r 1. It must therefore divide their difference, 



x ke (x r -l\ 

 Hence must r be zero. 



Furthermore, e must not divide p nt 1, for t < m\ for, if so, 

 x j 1 and hence also -F(^) would divide x f)Ht - x, so that the degree 

 of F(x) would be a divisor of t. 



An integer which divides a m 1 but not a* 1, t < m, is said 

 to be a proper divisor of a m 1. We may state the result: 



The exponent to which an IQ\m, p n ~\ belongs is a proper divisor 



of (p n y n - 1. 



30. Theorem. - - The number N p n of IQ[m, p n ~] which belong 

 to an exponent e, a proper divisor of(p n ) m 1, i-s (e)/m. 



Let q_i, q-2, . . ., q s be the distinct prime factors of e. Proceed- 

 ing as in 26, we rid x 6 1 of those of its factors which are irre- 

 ducible in the GF[p n ~] and belong to an exponent < e. We obtain 

 the expression / e 



0-i) n \x ^/ 



n\x qi -i) n\x 



which is therefore the product of the irreducible factors of x 1 

 belonging to the exponent e. Each of them is an irreducible factor of 



and hence of degree m or a divisor of m. Since each belongs to an 

 exponent which is a proper divisor of (p n ) m 1, the degree must 

 be m. 



The degree of the above function is clearly 



? V e _L V e V e i. c ^Y e 



K ^ ~t~ ^ ^ t- -j 1 J_ ) 



"" ,.,.,,. _(). 



31. Theorem. - - J/' J^(a;) a^r? g) (a;) belong to and are irreducible 

 in the G-F[p n ~\ and are of the respective degrees m and t, a divisor 

 of m, the roots of the congruence 



10) 9 >(X):-0 / [modF(x)] 



are x 1; xf, xfV..,^/^- 1 ), 



if XL be one root of 10) necessarily belonging to the G-F[p nm ~\. 



