CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 237 



(not necessarily in the GF[p n ]) to one of the following five types 

 of canonical forms: 



A: x'=}ix, y' 



B: x'=[tx, y=y,v n y, s' = az 



C: x = aXj y' = fly, z = yz 



D: x'=ax, y'=py, z'=p( 



E: x'=ccx, y'=a 



where A satisfies a cubic equation and ft a quadratic equation each 

 belonging to and irreducible in the GF[p n ], while a, /3, y denote 

 marks 4= of the GF[p n ]. 



Upon replacing A by A pW or by W ", we obtain from A a sub- 

 stitution conjugate with A. Any other replacement of A leads to a 

 substitution not conjugate with A ( 102, Corollary), since its 

 characteristic determinant differs from that of A. Hence the type A 



includes -^(p* n p n ) distinct sets of conjugate substitutions, those in 

 different sets being not conjugate under 6r 3 . 



Let S be a substitution of 6r 3 having the canonical form A, 

 where A is a definite mark of the GF[p* n ] not in the GF[p n ~]. If 

 a substitution T of G 3 be commutative with S and if we apply to T 

 the same transformation of indices which reduces S to the form A, 

 then ( 220) T will take the form 



x' = e r x, y' = a rf>n y, J = tf r * 2 n z, 



where a is a primitive root of the GF[p* n ] and r is some positive 

 integer <^p 3n 1. Hence S is commutative with exactly p 3n 1 

 substitutions of G 3 , so that S is one of JV-j-(> 3 " 1) conjugate 

 substitutions within 6r 3 . The total number of substitutions of G 3 

 reducible to the canonical forms A is therefore 



a) (p*"-p}(p**-p)(p*"-p*). 



Type B includes y (p* n p n ) ( p n 1) distinct sets of conjugate 



substitutions. In fact, the replacement of ft by pP n leads to a sub- 

 stitution conjugate with B, while any other replacement of ft or any 

 change in a leads to a substitution not conjugate with B. A sub- 

 stitution of 6r 3 commutative with a particular substitution reducible 

 to a type B has the canonical form 



