10 CHAPTER I. 



10. Theorem. - - The order of F\s] is a power of p. 

 If U L be a fixed mark =j= ^f of the F[s], the products 



^M! (ft 0, 1,...,|> 1) 



give p distinct marks of the field. If s > p, there exists a mark u 2 

 not of the form c^. Then 



+ C 2 MS, (q, Cg = 0, 1, . . ., jp 1) 



gives p 2 distinct marks. If s > p 2 , there exists a mark u 3 not of the 

 form C L U L -f C 2 U 2 , so that 



' 



gives ^ 3 distinct marks. Proceeding similarly , we must ultimately 

 obtain all the marks of the F[s] expressed by the formula 



CM + C 2 u 2 -\ ---- + c n u n (every c/== 0, 1, . . ., _p 1), 

 not two of these ^> n expressions being equal. Hence s = p n . 



Definition. - A set of marks u ly u^ y . . ., % are said to be 

 linearly independent with respect to the included field F[p\, if the 



equation 



CjWi -f C 2 u 2 H- + Cjfc^ = 0, 



where the c's are marks of the -F[j0] 7 can be satisfied only when 

 every c/ = 0. 



Definition. - - A rational integral function of any number of 

 indet'erminates X 1; X 27 . . ., X k is said to belong to a field if its 

 coefficients are marks of that field. It is irreducible in the field if it 

 is not identically the product of two or more functions belonging 

 to the field ; each function involving some of the indeterminates X { . 

 An equation between functions belonging to a field is said to belong 

 to the field. 



11. Theorem. Any mark u of the F [s =p n ] satisfies an 

 equation of degree k<^n, 



belonging to and irreducible in the F[p]. 



Indeed, a linear relation with coefficients belonging to the F[p] 

 certainly holds between any n -}- 1 marks of the F[p n ] and hence 

 between 



If such a relation holds between the first k -f 1 of these powers of u y 

 u satisfies an equation of degree k. 



