228 CHAPTER X. 



where the f^- are linear functions of the , with coefficients involving 

 the imaginary i t - only. As the transformation T does not alter the 

 indices which Y affects, we obtain the desired canonical form. 



215. Consider as an example the substitution in the GF[p n ~\, p n 

 of the form 4Z 1 , 



8: 61- -26, -I., 6J-6i, 6J-6,, 61 -I,, 

 having the characteristic determinant 



where ^T 2 + 1 is irreducible in the field. A root of i* = 1 belongs 

 to the GF[p 2n ] but not to the GF[p n ]. The functions which 8 

 multiplies by i and i are readily found to be respectively 



^ = ^ 4- #2 *#3 + #4, ^2 = *X 4- # 2 + *#3 + #4- 



Introducing A 1; A 2 in place of the indices x 2 , # 3 , S takes the form 



The partial substitution of determinant unity, 



X^ = X^j X^ = X 



multiplies y^ = x^ i x by i and multiplies y% = x t + ix by i. 

 Introducing y l "and y z as new indices in place of x 1 and x, S takes 

 the form 



A< === IrA-^y A 9 - ^A2 



Introducing as new indices, 



_ Q -* _ Q -" 



S takes the canonical form 



^i = ^i> ^' = ^(2/1 + ^1)^ ^2 = -*%; 2/2 = %2 + ^2); 

 where A t and A 2 are conjugate linear functions of | 1? | 8 , | 3 , | 4 , and 

 likewise for ^, ^ 2 . 



216. Theorem. Two linear homogeneous substitutions S and T 

 in the GF[p n ~\ on the indices '% lf | 2 ; > %m have the same canonical 

 form C iff and only if, T is the transformed of S by a linear homo- 

 geneous substitution W in the GFlp**] on the same indices. 



If T= W~~ 1 SW 9 then 8 can be reduced to T by the introduc- 

 tion of new indices defined by the transformation W and therefore S 

 and T have the same canonical form. 



