26 CHAPTER III. 



eA, provided p n and I are subject to the conditions given in the 

 main theorem. 



37. Since irreducible binomials are lacking in the case treated 

 in the last section, we proceed to set up trinomial IQ[k,p n ~], It is, 

 however, not necessary to suppose that A is a multiple of 4. We 

 suppose merely that 



p*= 2 l t - 1 (t odd, i 5 2) 



and that A is an even integer containing no prime factor not occurring 

 in^-1. Set t, = 2>--% 



so that v is divisible by 2*. If p be a primitive root in the G-F[p n ~\ 

 and if s be any integer prime to A and hence also to v, then x Q* 

 belongs to the exponent (p n !)/<?, where d is the greatest common 

 divisor of s and p n 1, and v is prime to d. Hence ( 36), the 

 binomial x v Q S decomposes into 2 l ~ l irreducible quantics of degree A. 

 We proceed to determine them. 



Since 2 t '~~ 1 and (p n l)/2 are relatively prime, we can determine 

 ( 7, Note) two integers 1 L and 7^ 1 such that 



Multiplying this equation by the even integer s -f (p n 1)/2, we 

 obtain two integers I and Ji for which 



Z2'~ h(p n - 1) = s -f (p n - l)/2. 

 Since the (p n I)/ 2 power of the primitive root $ is 1, we 



aj u_^ = a ai-i + ^ 

 J% ^e G-F[p n ~\ we have flie decomposition 



15) <**-'*+ f* 



where the | ; are marks of the GF[p n ] determined as the roots of the 

 equation . _ 



a 



In fact, by Waring's formula 1 ), the sum of the (2 t '~ 1 ) st powers 

 of the roots u and ^ of the quadratic 



X 2 -^X-1 = 

 is found to be E(%). Expressed otherwise, if % = u , then 



1) Serret, Cours d'Algebre Supdrieure, I, p. 449. 



