18 CHAPTER II. PROOF OF THE EXISTENCE OF THE GF[p^ etc. 



27. Continuing the investigation, let 

 m = q^q^ . . . q r s * , 



i> #2, . - ., q s being the distinct prime factors of m. For brevity, 

 we use the symbol .. -, __ p nt__ 



We proceed to prove the formula, due to Dedekind for n = 1, 



In this expression, the term 



in which the product extends over the C 8 ,k combinations q,^, . . ., qj k 

 of the integers qi, . . ., q s taken ~k together, occurs in the numerator 

 or in the denominator according as / is even or odd. Each IQ[m,p n ~] 

 occurs once as a factor in 77 = [m] but divides no other T7&; it is 

 therefore a simple factor of the fraction. If there be any factor of 

 the fraction having the degree m l < m, we denote it by F(x). By 

 25, m must be a divisor of m. Denote by q if q% } . . ., q Sl the prime 

 factors entering in m to a higher power than in m . Then m 1 divides 



- but not L (j = Si 4- 1, Si -f 2, . . ., 5). It follows that, 

 gi 22... 2*,. 2; 



if ~k > Si, J7& does not contain .F(#) of degree m^ while, for k < s, 

 II k contains F(x) as often as ~k integers can be selected from q lf q 2 , . . ., q,j 

 viz., G Sl .it times. Hencne F(x) occurs in the numerator and denomi- 

 nator of our fraction to the respective degrees, 



These numbers are equal, since their difference equals (1 l)- 9 i = 

 It follows that every irreducible factor of our expression is an IQ 

 The number of the latter multiplied by the degree m of must equal 

 the degree of the fraction, so that 



This number cannot be zero; for, upon dividing by the last 

 term, which is the lowest power of p entering into the expression, 

 we would then obtain unity expressed as the algebraic sum of a 

 series of powers of the prime number p with exponents > 1. It 

 follows that the number of J[w,^ w ] is 5> 1- [See Ex. 2 below]. 



28. Let F(x) be an IQ[m,p~\. As in 6, the totality of rational 

 functions of x belonging to the GF[p n ~\ can be separated into p nm 



