32 CHAPTER III. 



An interesting type of IQ\J>,p n '\ of class p 1 is given by 

 setting every /= except and a p _i; viz., 



Multiplying this by ffo P i and setting F(g) = 0, we find 

 that p is a linear fractional function of . But, by 31, the roots 

 of jF() = may be expressed in the form 



fc fcp w fcp2 fcp(P 1) 



S> 5 > 5 ? ? 5 



Hence its roots are all linear fractional functions of one of them. 

 This result also follows from the fact that 



so that each root is a linear fractional function of /. 



45. Formula 19) expresses the fact that X u becomes X^+i when x 

 is changed into x pH x. Further, if we set X =#, 19) holds true 

 for ^ = 0; viz., . 



^LI EE XQ XQ. 



Hence in order to change x into x pn x in any formula involving 

 the Xfi 9 we have merely to advance the subscripts of each X^ by 

 unity. Applying this operation to formula 21), we have the theorem: 



If F(x), an IQ[p*,p n ], divides Xf""" 1 1 for i<p*- 1, then 

 F(x pn x) decomposes into p n IQ[p", p n ~\, each one being a factor of 

 Xf+1 1 - 1; lut if F(x) divides xl| - 1, then F(x* n -x) decomposes 

 into p n ~ l factors each an IQ[p'+ l , p n ] which divides Xp"~ 1. 



46. As an example under the second part of the last theorem, 

 consider the IQ[p, p n ~\ of class p 1 given at the end of 44. 

 From it we obtain the IQ[p 2 , p n ~\, 



)EE(^-^-o-^-l^ 



where or , Op i, /3 are arbitrary marks of the GrF[p n ~\ such that 



For an IQ\j)*,p\ see Serret, C&urs d'Algebre superieure, II, p. 209. 



Miscellaneous theorems on irreducible quantics, 47 49. 



47. Theorem. An IQ[m,p d ~] is irreducible in the G-F[p nd ] if n 

 be prime to m. 



