CHAPTER IH. CLASSIFICATION AND DETERMINATION, etc. 19 



distinct classes of residues modulo F(x), each being represented by 

 one of the p nm residues 



a 4- a\x + a-2% 2 H ----- h flm-i^" 1 (X s i n th 



Proceeding as in 6 ; we find that these classes of residues form 

 the 6rF|jj w "']. We can therefore construct the GF[p r "} in as many 

 ways as we can express r as the product of two positive integers n, m, 

 viz., by using an IQ[m,p*]. From the theorem at the beginning of 

 26 it follows that the GF[p nm ^ is contained in the G-F[^ nm ] if, 

 and only if, m L divides m. 



EXERCISES. 



Ex. 1. Granting the existence of the G-F[p n ], the existence of 

 the G-F[p n f\, q being prime, follows by 26. By induction, the GF[p r ] 

 exists for r arbitrary. 



Ex. 2. Obtain for the number of IQ[m,p n ] given in 27 the 

 following limits: 



pnm pn = == $ (m) pnm pn 



Hint: Expand each power of p n into a series in log p n and apply 



Ex. 3. By decomposing modulo 2 the expression (# 2 x)/(x 2 #), 

 obtain the three IQ [4, 2] given in the left members below. Defining the 

 6r.F[2 2 ] by means of the irreducible congruence 



* 2 -H-j-1^0 (mod 2), 

 obtain the six /$[2, 2 2 ] by means of the following decompositions: 



x* -f x -f 1 = (x* + oo + i) (x 2 + x 4- ^ 2 ), 



CHAPTER HI. 



CLASSIFICATION AND DETERMINATION OF IRREDUCIBLE 



QUANTICS. 



29. Definition. - - An J(?[%J>*k as F(x), is said to belong to an 

 exponent e if e be the least positive integer for which F(x) divides 

 x* 1 in the GF[p n ~\. [Compare 32.] 



