22 CHAPTER El. 



Note. If a be a primitive root of e, then a -h fce(& = 0, 1, 2, . . .) 

 are also primitive roots of e. By the theorem of Dirichlet, this 

 arithmetical progression contains an infinity of prime numbers. With 

 respect to any such prime p, V is irreducible modulo p. A fortiori, 

 F is algebraically irreducible. 



Determination of IQ[m,p n ] whose degree m contains no prime 

 factors other than those of p n 1, 3438. 



34. Theorem. -- Let Fi(x),F*(x), . . ., F N (x) denote the IQ\m,p n '\ 

 which "belong to an exponent 



e = (p nm - l)/d, 



and let I ~be an integer relatively prime to d and containing no prime 

 factors other than those occurring in p nm 1. With the exception of 

 the case in which I is a multiple of 4 while p"" 1 is of the form 41 1, 

 all of the IQ[lm } p n 1 which belong to the exponent el are given by 

 the N quantics Fi(af), . . ., F N (x l \ 



By definition, I contains no prime factor other than those 

 occurring in e. Hence el and e contain exactly the same prime 



factors, so that .... ... 



< < 



By 30, we have el 



If we suppose satisfied the conditions (obtained below) under which el 

 shall be a proper divisor of (_p n ) TO * 1 ? we will have 



Since e divides p nm I, the irreducible factors of x e 1 are of degree 



< m ( 25). Hence, in the notation of the theorem, 



x e - 1 = Fi(x)F*(x) . . . F N (x) Q(x) 



where the irreducible factors of Q(x) either belong to an exponent 



< e or else are of degree < m. Therefore 



where every irreducible factor of Q(x?) is of degree < Im or else 

 belongs to an exponent < el. Since there are exactly N irreducible 

 factors of degree ml which belong to the exponent el, they must 

 be identical with Fi(x*), . . ., F N (x^). 



Calling v the least integer such that p nv 1 is divisible by el, 

 we seek the conditions under which v = ml. Since m is by hypo- 

 thesis the least integer for which p nm 1 is divisible by e, v must 

 be a multiple of m. For, if v = qm -f- r, 0<Jr<m ? then e divides 



^ &n( J p nmq _ ^ and t ence a l so their difference ^mq(nr_ 



