52 



CHAPTER IV. 



72. Theorem. The marks A t , A 2 , . . ., A m of tlie GF[p >im ] are 

 Imearly independent with respect to tlie GF[p n ] if and only if the following 

 determinant 1 ) is not s&ro in the GF[p n \: 



First, if AI, A 2 , . . ., A m be Imearly dependent, i. e., if a relation 



Mi 4- y2^H ----- 1- rm^ m = 



holds, the coefficients ^ heing marks of the GF\p n \ not all zero, 

 then will the determinant | A | vanish. 



Secondly, if the determinant vanish, set 



TO 1 



where E is a primitive root of the GF[p nm ] and therefore satisfies 

 an equation of degree m belonging to and irreducible in the GrF[p n ~\, 

 and where the (i ;j belong to the latter field. Then 



I 1 - 



A > 



; t-0, ...,- 



where the determinant in Pi, when written in full, is 2 ) 

 1 1 ...1 



R Rf . RP o,.. , m i 



n 



o>< 



and therefore is not zero in the GF[p nm ]. Hence, if | A i = 0, then 

 must |^|=0, so that between the A/ exists a linear relation with 

 coefficients belonging to the G-F[p n ] and not all zero. 



73. If A be a mark of the GF[p nm ], the marks 



are said to be conjugate with respect to the GF[p n ~\. Any symmetric 

 function of them is unaltered upon being raised to the power p n and 



1) Its decomposition into linear factors is given by Moore, U A two -fold 

 generalization of Format's theorem", Bull. Amer. Math. Soc., vol. 2 (1896). 



2) Baltzer, Determinanten , p. 85. 



