12 CHAPTER I. 



If MjWg has the period t, we have 



(i^fii) 1 * - ^ - 1, 



whence t is divisible by <? 2 ; similarly, # is divisible by e r But 



(*ll)** I- 



Hence # == e 1 e 2 . 



15. We prove as in algebra the theorem: 



-4n equation of degree k belonging to a field has in the field at 

 most k roots, unless it be an identity, when every mark of the field is 

 a root. 



16. Theorem. - - For every divisor d of s 1, the equation 



X d - 1 = 



has in tJie F[s =^ n ] exactly d roots. 



Setting s 1 = dq, we have the identity 



X 1 -l = (X d -l)(X <| fe- 1 >+X< | fe- 8 >+... + X"+l). 



Since the last factor belongs to the F\s\ and does not vanish 

 for the mark zero, it vanishes for at most d(q 1) marks of the 

 field. But the left side of the identity vanishes for s 1 marks of 

 the field. Hence the factor X d 1 must vanish for at least d marks. 



17. Decompose p n 1 into its prime factors, 



lP-l-i$i$...i$. 

 For each integer i of the series 1, 2, . . . , k, the equation 



has by 16 exactly p h A roots belonging to the f [s = p n \. Of 

 these roots p^"" 1 are also roots of the equation 



and thus belong to exponents less than p^. The remaining roots 

 in number 



belong to the exponent p h .i itself. Any product of the form 



W = U 1 U 2 ...Uk 



will by 14 belong to the exponent p n 1. Forming in every 

 possible way the product w, we obtain 1 ) 



1) This number equals 0(p 1), where <t> (*) denotes the number of 

 integers less than and relatively prime to the positive integer t. See Dirichlet, 

 Vorlesungen u'ber Zahlentheorie, 11. 



