58 CHAPTER V. 



Likewise , it follows from Y l Y 2 = a k that 



If 49) be irreducible in the GF[p H ], we have ( 31, corollary) 



and therefore n 



nl = %, nl = % 



Hence , n 



(% + fc) p - % + ft- 



It follows that 48) has the solution in the CrF[p n ] 



If 49) be reducible, Y and Y 2 belong to the GF[p n ]. Since k 

 is prime to p n 1, it follows that ^ and r] 2 belong to the GF[p n ]. 



Remark 1. We have shown in 37 that ( J ) 2 r ~ 1 (?; l) com- 

 pletely decomposes in the CrF\p n ] into linear factors, ifp n =2'tl, 

 t odd and i > 1. 



Remark 2. If & be a prime number, <&*(> ) is the only 

 quantic of degree It suitable to represent a substitution on the marks 

 of every GF[p n ] for which p 2n - 1 is not divisible by k (Annals of 

 Mathematics, 1897, pp. 89 91). 



Remark 3. - - The equation 48) is algebraically solvable, having 

 as roots m k m / r\ / -\\ 



where 



and denotes a primitive & tb root of unity. This result is a direct 

 generalization of Cardan's formula for the roots of the reduced cubic 

 and of Valles' solution of the quintic 1 ) 



83. Theorem. 2 ) If d be a divisor of p r \ and v be not a 

 d^ power in the GrF[p n ], the qttantic 



is a SQ[p r ,p], 



We are to prove that <t>() -= j3 has a solution | in the GF[p n ] 

 for |8 chosen arbitrarily in the field. This being evident if ft = 0, 

 we will suppose that ft =\= 0. Writing 



1) Formes imaginaires en Algebre, 1869, vol.1, pp. 90 92. 



2) For the case r = n , this theorem is included in the theorem of 85. 



