70 CHAPTER V. 



Furthermore, TO, r 1; . . ., r m _i are linearly independent with respect 

 to the GF[p n ], for, if #0, ., x m \ be marks of the latter field such 



that 



h x TO -ir TO _i= 0, 



-i 



0, 



and therefore, since | a^ =(= 0, each 3c/== 0. Hence, when each | e - runs 

 independently through the series of p n marks of the GF[p n ], the 

 expressions X and X' both run through the p nm marks of the GF[p nm \. 

 Every substitution 63) therefore gives rise to a permutation on 

 the marks of that field. 



But we can always find a set of marks AI, A%, . . ., Am of the 

 GF[p nm \ such that 54) will transform the set of marks 1, R, jR 2 , . . ., 

 R m ~ l , linearly independent with respect to the GrF[p n ], into an 

 arbitrary set of m marks of the GF[p nm ], 



(f'-O,... ,-!), 



linearly independent with respect to the G-F[p n ~\. The conditions are 



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



which can be solved for AI, A% 9 . . ., -4^, since the determinant in R 

 is not zero by 72. The resulting substitution 54) will transform 



m 1 ? 1 



the jp nm marks ^^-E' of the (r^[p wm ] into the marks "V^-Z?; all 



?: = o / = o 



distinct; indeed, we have the identity 



t / m 



( ' 



96. EXERCISES ON THE TEXT OF CHAPTER V. 



Ex. 1. Verify that 6 -{- a^ 5 a 4 2 (a arbitrary) represents a sub- 

 stitution on the marks of either the G-F[3 3 ] or of the G-F[2 5 ]. 



Ex. 2. (Hermite). A group of order 168 is generated by the sub- 

 stations ^ M + 6) x > = aQ(x + b} + c (mod7)) 



where 0(V) ^ a? 5 2# 2 and a is a quadratic residue of 7. 



Ex. 3. (Rogers.) In applying Hermite's conditions ( 84) for a 

 substitution quantic, it suffices, when n == 1, to test only the first 



(p 1) powers. This result of Rogers does not generalize immediately 



