CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 223 



... (ka + I ft + - = m) 



where Ji(JE), JF,(Z), . . . are the distinct factors of A (JT) which 

 belong to and are irreducible in the GF[p n ]. Designate the roots 

 of F k (E) = 0, and of F t (L) = 0, etc., by the notations 



j T TV" T TP* n T jP n ( l - l \ 



4to> -^i -^o > -^2 = -^o *.! *'' i -^o i - 



Theorem. By a suitable transformation of indices, S can be 

 reduced to a canonical form of the following type: 



ftj = Kifa + rja-!) ( j - 2, . . ., 



+j+ *li a, +j- 1) ( j = 2, . . . , 



$1 = Z/&i , & = Li(&j -f gf y-0 ( J = 2 ; ; &i) 



+j = ^(?j*i+>+ ?* 

 (^ = 0,1,...,?- 



where the 

 indices have the properties: 



1) The indices ??o* (s = 1, . . ., a) are linear homogeneous functions 

 of the initial indices & having as coefficients polynomials in K with 

 coefficients in the GF\_p n ~\\ 



2) TAe indices ^ are conjugate to the ^ , being obtained by re- 

 placing KQ by KI in the coefficients of ^ ; 



3) The indices fo* (s = 1, . . ., |8) are linear homogeneous functions 

 of the indices & w/iose coefficients are polynomials in L Q with coefficients 

 in the ffJftlf]; 



4) T/ie indices ,-, are obtained from the Jo* &2/ VV$MCMQ L by L f ; e^c. 



5) TAe fca indices ^- 8 (i = 0, 1,*. . ., & 1; s 1, . . ., a) may &e 

 replaced by ka linear homogeneous functions yi, of the initial indices (;, 

 wra& coefficients in the 6rF[p n ~], such that S replaces each y it by a 

 linear homogeneous function of the y is with coefficients in the GrF[p*]] 



6) The I fl indices & ma y be replaced by an equal number of linear 

 homogeneous functions zi* of the |,- with coefficients in the 6rJP[_p w ], such 

 that S replaces each by a linear homogeneous function of the z is with 

 coefficients in the field; etc. 



