CLASSIFICATION AND DETERMINATION, etc. 25 



The ratio of the Z th term within the parenthesis to the first term is 



where the first and second factors are integers, while the third 

 factor, being > 1 for I > i 2, equals an irreducible fraction with an 

 even numerator. Hence the first term contains 2 to a lower power 

 than the remaining terms in the above parenthesis. In order that 

 pnv_ i sn all be divisible by eA, formula 11) requires that I shall 

 divide the left member of 14). Hence 2 j must divide the first term 

 of the right member and consequently also 2 L ~ l q. Hence the even 

 integer q must contain 2 to the power 1 or j i -f- 1 according 

 as j <! * or j > * Furthermore, by 34, q must contain every odd 

 factor of A. Hence, if v be the least possible integer, 



according as j <; i or j > ?', i. e., according as & = j or fc = i. Hence 



I ml 



2--W=T> "=^^T' 



As at the beginning of 34, we have 



m - 



so that the number of /$[?', jp' 8 ] belonging to the exponent eA 



is 2^- 1 JV. 



By hypothesis, 



- 1 - F^F^as) . . . F a (x)'Q(x\ 



where the irreducible factors of Q(x) in the GF[p n ~\ belong to 

 exponents < e or are of degree < m. The irreducible factors of 

 are therefore of degree < 'km or else belong to exponents 

 Hence the irreducible factors of degree km of the expression 

 must, if they belong to the exponent eA, be factors of Fi(xF), . . ., 

 Since the combined degree of the latter is Nmk = 2 k ~ l vN, and 

 since there are exactly 2 k ~ 1 N irreducible quantics of degree v 

 belonging to the exponent el, it follows that each Fi(x l ~) is the product 

 of 2 k ~ l irreducible quantics of degree v. 



Corollary. - - Since the distinct functions of degree m = 1 which 

 belong to the exponent e = (p n !)/(? are given by the formula 



X Q ad , 



Q being a fixed primitive root in the GrF^j}"-] and a being any integer 

 prime to e, it follows that x l Q ad decomposes in the GF\jp n ~\ into 

 2* i irreducible factors of degree A/2*" 1 belonging to the exponent 



