50 CHAPTER IV. 



We first prove that the numerator expresses the number of sets 

 of m marks A x , A 2 , . . ., A m of the CrF[p nr ] linearly independent with 

 respect to the GrF[p r ]. For A L we may take any one of the p nr ~ 1 

 marks 4= of the GrF[p nr ], for A 2 any one of the p nr p r marks not 

 of the form 0^, where ^ belongs to the GrF[p r ]-, for A 3 any one 

 of the p nr p 2r marks not of the form 0^+ P 2 ^ where Q and p 2 

 belong to the GF[p r ]; etc. 



We next show that the denominator expresses the number of 

 these sets of m independent marks which generate the same additive- 

 group [A!, A 2 , . . ., AJ. In fact,, we may use as a basis -system for 

 the latter any set of m marks A^, A 2 , . . ., A' m chosen as follows. 

 AI may be chosen in p mr 1 ways: 



each y lt - being arbitrary in the GF\_p r ~\ provided not all are simultan- 

 eously zero. A 2 may be chosen in p mr p r ways, viz., 



the y 2t - being taken arbitrarily in the 6rJ^ T (j) r ] but so as to exclude 

 the p r sets of values which make A 2 = pA^, viz., 



where (> runs through the series of marks of the GrF[yf~\; etc. 



70. If the p m marks c^-\ ----- h c m l m of the additive - group 

 [Ai, . . ., A m ] of rank m with respect to the GF\jp\ are multiplied by 

 any particular mark p =j= of the GrF\p n ], the resulting p m marks 

 constitute the additive -group 



^ [Ai, . . ., AJ = [^A 1; . . ., aA m ] 



likewise of rank m with respect to the GF[p\. We will say that 

 [fiA 1; . . ., f*AJ is derived from [Ai, . . ., A m ] by multiplication by p. 

 In particular, we seek those multipliers ft = % which do not alter 

 [Ai, . . ., A TO ], such a mark being called a multiplier of the additive- 

 group [Ai, . . ., A m ]. If Xj and K 2 be multipliers, then will evidently the 

 product 3^ Xg be a multiplier. To prove that ^ = Xj + Xg will also be 

 a multiplier, we observe first that [jnAi, . . ., ^A m ] is an additive- 

 group included within [Ai, . . ., A m ], since ^ and x 2 are multipliers of the 

 latter, and further that it is of rank m if t u =j= 0- Hence 3^-f- >c 2 is 

 a multiplier unless it be zero. Hence the multipliers ^ together with 

 the mark zero constitute an additive, as well as a multiplicative, 

 group and therefore constitute a Galois Field GrF[p k ] included within 



