CLASSIFICATION AND DETERMINATION, etc. 43 



If in an IQ\y v p v ] belonging to the exponent w, we replace x 

 by #*, where A contains only the prime factors of w, the resulting quantic 



decomposes into D2 k ~ l quantics IQ I j>g i * ^ r I belonging to the 



exponent Aw, where 2) is the greatest common divisor of \n and p 1 ^ 1 

 and where 2 fc 1 is the highest power of 2 dividing the numerators of 



each of the fractions ' ' and ^ when reduced to their simplest 



form. 



Ex. 6. (Schonemann). If F(x, a) be an IQ[m, p n ] in which the 

 coefficient of at least one power of x satisfies the equation c pV ~ l = 1 if, 

 and only if, v = n or a multiple of w, the product 



F(x, a)-F(x, a") . . . F(x, a^" 1 ) 

 gives an IQ[mn, p\. 



Ex. 7. (Schonemann). Generalize the theorem of 33 as follows: 

 If p belong to the exponent t modulo e, e being prime, (x e l)/(x l) 

 decomposes modulo p into (e l)/ quantics irreducible modulo p. 



Ex. 8. Prove that x' x + 1 is a PI#[5, 3]. 



Ex. 9. (Pellet). If e be the exponent to which belongs 



the product of the roots of an irreducible congruence of degree v, -F(fl?) 

 (mod p). and if A be a prime divisor of e, then 



1) F(x^) is irreducible modulo p if A does not divide (jp l)/ e ; 



2) F(x*) decomposes into A irreducible factors of degree v if A 

 divides (p l)/c. According as A divides or does not divide e, 

 all of these factors belong or do not belong to the same exponent. 



Ex. 10. Using Jordan's irreducible congruence 

 x*=x + 1 (mod 2), 



show that x belongs to the exponent 73 and x + x*+ # 6 + # 7 -f # 8 to 

 the exponent 7. The product y = x(x -f # 4 -f # 6 -f a; 7 -j- a? 8 ) belongs to the 

 exponent 2 9 1 and is therefore a primitive root of the G-F[2 9 ]. Verify 

 that it satisfies the congruence 



?/+2/ 8 +/+/+2/ 2 +#-f 1=0 (mod 2). 



Ex. 11. If the GF[3*] be defined by i*= i -f 1 (mod 3), the 16 

 PJ[2, 3 2 ] are given by the decomposition of the P/[4, 3] of 54; 

 for example, 



x* x-l = {x 2 (i + 1) x - i} {x*+ (i + 1) x + i - 1}, 

 * 1 ={x 2 (i l)x+i\ [x*+ ix-i + I}. 



