MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 51 



the fundamental GF[p n ]. It is called the multiplier Galois Field of 

 the additive -group [A t , . . ., A m j. Every GF[pf] included within the 

 GF[p k ] is called a multiplier Galois Field of the additive -group. 

 By 23, k f is a divisor of k and k a divisor of n. 



The additive- group [A 1? . . ., A m ] of rank m with respect to the GF[p] 

 may be exhibited as an additive- group [A^, ..., A' m '] of rank m' = m/k' 

 with respect to any multiplier GF[p k '~\. 



In proof , let yi, y2> , TV run independently through the series 

 of marks of the GF[p k "\. Taking Aj to be any particular mark I =f= 

 in [AJ, . . ., A m ], the p k ' marks y^ are all distinct and all belong 

 to [Ai, . . ., A m ]. Taking A' 2 any mark in [Ai, . . ., Aj different from 

 the ftAj, the p* k> marks ftA^-f^Ag are all distinct and all belong 

 to [Ai, . . ., A OT ]. Proceeding similarly, we obtain ultimately a set 

 of p m ' k> distinct marks y 1 ^,( -f y 8 A 2 H ----- h y m 'tin' giving all the marks 

 of [Ai, . . ., A m ]. In particular, p*=*pF m ', so that &' divides m. 



Corollary I. Since k is a particular k', k divides m. 



Corollary II. Within the GF[p n ~] the number A(p,n,m,k) of 

 additive -groups of rank m with respect to the GF[p] which have 

 the GF[p k ] as a multiplier Galois Field equals the total number of 

 additive -groups of rank m/k with respect to the GF\j> k ~\: 



A(V fcU (pn-^( p n- pk)(p n- p ^ . . . (pn- p m-k } 



L W ' !_ w- A r/i- ^2*) . . . (pm- pm-k) 



71. If A' be a divisor of # and n and if AI, h z , . . ., ^ are the 

 prime factors occurring in both m and w to a higher power than 

 in fc, there are in the G-F[p n ~\ exactly 



m 



A (p, ^ m, 



additive -groups of rank M with respect to the GF[p\ which have 

 the &F[p k ] as the multiplier Galois Field. 



Indeed, from the A(p,n,m,k) additive -groups having the GF[yF\ 

 as a multiplier Galois Field, we must eliminate those having a larger 

 multiplier Galois Field. It suffices to eliminate those having the 

 #JF[p*V], for i = 1, 2, . . ., or t, as a multiplier Galois Field. But 

 the A(p^n,m^kh 1 ) additive -groups with the GF[p kh >] and the 

 A(p,n,m,kh 2 ) additive -groups with the &F[p***] are not distinct but 

 have in common A(p,n 9 m,lc hji 2 ) additive -groups each with the 

 GF[^ hti '^\ as a multiplier Galois Field. By this principle, we readily 

 determine the number of distinct additive -groups among the sets of 

 A(p,-n,m,1ch t ) with the GF[jf h i~\ as multiplier Galois Fields. Sub- 

 tracting this number from A(p,n,m,lG), we obtain the required number. 



4* 



