CLASSIFICATION AND DETERMINATION, etc. 35 



Corollary. - //' F(g) be an IQ[m,p] in which the coefficient 

 of m ~- x is not zero, F(l v ) is an IQ\mp, p], 



Examples. - - The following congruences are irreducible: 



(x 2 - x) 2 + (x 2 - x) + 1 = x*+ x + 1 = (mod 2), 



8 - a?) - 1 = x*+ x*+ # 3 4- x-- x - I = (mod 3). 



Primitive roots and primitive irreducible quantics, 50 58. 



50. Theorem. - If E be a primitive root of the GrF[p nm ] and 

 m a divisor of m t any IQ\_m lf p n ~\ belonging to an exponent e may be 

 exhibited as a prodtict 



28) 



wliere t is a multiple of d = (p nm l)/e such that -y -is prime to e. 

 Inversely) if e be a proper divisor of (p n ) m * 1 and t be a multiple of d 

 and -j- be prime to e, the above product gives an IQ\m^ p n ~\ belonging 



to the exponent e. 



Suppose first that cp(X) is an /$[%, p n ~] belonging to the 

 exponent e, where m l is a divisor of m. By 23, <p(X) divides 

 Xr nm -X in the GF[^\ 9 so that any root X l of <p(X) = belongs 

 to the GF[p nm ]. We may therefore set X^ R*. Then, by 31, we 

 have the decomposition 28). Since (f(X) belongs to the exponent e, 

 X^jR' must belong to the exponent e ( 32). Hence t must be a 



multiple of d = (p nm l)/e and -r be prime to e. 



To establish the inverse, we first prove that E t belongs to the 

 exponent e. Since et is assumed to be a multiple of p nm ! 9 we 

 have E et = 1. If R'J= 1, tj is divisible by p nm ~ 1. Set t = dd', so 

 that d' is prime to e. Then must 



jd''(p nm l)/e = (mod p 1). 

 Hence must jd' 9 and therefore, j be divisible by e. Hence 



all belong to the exponent e. Upon raising these marks to the 

 power p n , they are merely permuted. Hence any symmetric function 

 of them, and consequently (p(X) defined by 28), belongs to the GrF[p*\. 

 Furthermore, qp(X) is irreducible in the &JJ\jp f \\ for, if <p l (X} be 

 an irreducible factor of degree m 1 ^!, it belongs to the exponent e. 

 Then by 29, e would be a proper divisor of (p n ) ml 1, so that 

 m l = m^ 



3* 



