CLASSIFICATION AND DETERMINATION, etc. 29 



40. For s = 1, there are p 1 factors in the product 21), given 

 by i = l, 2, . . , p 1. The irreducible factors of Xf~ l -l are 

 then said to form together the ^ th class of IQ[p, p"]. Consider first 



which is the product of p n ~ l IQ[p,p n ] of the first class. To 

 decompose it, consider the equation 



It follows at once that 



c p c EE ip n i] (mod p). 



Hence every root r f of 22) belongs to the GF[p n ~\ if, and only if, 

 c be an integer. Setting in 22) 



where A belongs to the GF[p n ~\, we find 



A (x* n x v) EE (mod p). 

 We have therefore in the G-F[p n ~\ the decomposition 1 ) 



p*- 1 



23) A (xP n - x-v)~ II (tfxv Kx - ft), 



where the /3, are the roots of 



A or i/ being determined so that hv is an integer. We have therefore 

 the theorem: The quantic Wx p Kx $ is an IQ[p, p*] if, and 



only if, B = (F- l +F>'- > + ... + p+p+Q. 



Corollary. - - If 1} is an integer not divisible by the prime p, 

 x p x b is irreducible in the GrF[p n ~\ if, and only if, n is not 

 divisible by p; in particular, it is always irreducible modulo p. 



In fact, the condition becomes in this case 



B = nb E|E (mod p). 



41. The decomposition 23) may be given a more explicit form 

 useful below. If /3 be one root of 24), then is also 



25) ft EE aP- a -f p, 



for every mark a in the GF[p rt ~\. Indeed, we have 



P ~ l + +# + &= ""- a + pe"- 1 + + PP + p = Iv. 



1) For the case v = , this decomposition was given without proof by 

 Mathieu, Journal de Maihematiques , (2) vol.6, 1861, p. 280. 



