224 CHAPTER X. 



For the case = /3 = = 1, we obtained above the canonical form 



( 0,1,...,*-!) 



Ci-Zifci (t-o, i,...,z-i) 



where t? 01 = /"(! , ...,; J ) anc * ty'i = /Xi> -> i5 ^)> an ^, similarly, 

 g,-i are conjugate with g 01 . The new indices therefore have the 

 properties 1) 4). 



We will prove the general theorem by induction, supposing it 

 true for every substitution belonging to the GrF[p n ~] whose char- 

 acteristic determinant has no irreducible factors other than F k (K\ 

 Fi(K\ . . ., and has these to a degree at most a 1, /?, . . . respec- 

 tively. We will prove that the theorem is true for any substitution S 

 for which these factors occur to the degree a, /3, . . . respectively, 

 where a > 1. 



Corresponding to the distinct roots K Q , K v . . ., K k i ofF k (K) = 0, 

 we obtain as above a set of linearly independent conjugate functions 

 A , A A , . . ., Ajfe_i which S multiplies by 7T , K , . . ., K k i respectively. 

 We may introduce these in place of an equal number of the original 

 indices, e. g., | TO _i+i, . ., m- The substitution S then takes the form 



ti = Kih (i = 0, 1, . . ., k - 1) 



or; m k1 



1 1 ^1 ft t I NT1 1 (A 1 O 1\ 



%i / t Pijfy ~T / t Vij^j ^ == 1, /,..., m K). 



coefficients fa belong to the GF[p n ]. Indeed, we may set 



(<-0, !,...,*-!) 



where the X,- are linear functions of the | f with coefficients in the 

 6r-F[jp"]. Since the A* are linearly independent, the X must be 

 linearly independent functions of the &. Since 



the Xj- can be expressed as linear functions of the A,-. Taking the X,- 

 as new indices in place of the A,-, $' takes the form /S"', a substitu- 

 tion on the indices X,- and , with coefficients in the 6rF[jp n ]. But 

 >S" f replaces g t - by 



mJc kl 



for * = 1, . . ., m k. Since these functions belong to the field for 

 arbitrary ^ and X,, the coefficients /3^-, d f j must belong to the field. 



