MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 49 



Hence, if -- 1 be a square , we have 



pn _ 1^2 + 2 + 4^ N+S+l-(p-l). 

 If - 1 be a not- square , we have 



Additive -groups In the GF[p*\ and their multiplier Galois Fields 1 }, 



68-71. 



68. A set of m marks A 1; A 2 , . . ., A m belonging to the GF[p n ] 

 and linearly independent with respect to the GF[p] give rise to 

 p m distinct marks of the larger field, 



39) Cj^H- c 2 A 2 H ----- h c TO A m (every d= 0, 1, . . ., p 1). 



Indeed, an identity between two of the marks 39) would contradict 

 the linear independence of A 1? A 2 , . . ., A^. Since the sum of any 

 two of these p m marks 39) may be expressed as one of the set, they 

 are said to form an additive- group [A 1? A 2 , . . ., AJ of rank m with 

 respect to the GF[p\ and the marks A^ A 2 , . . ., A m are said to form 

 its basis -system. In particular, the GF[p n ] may be exhibited as an 

 additive -group of rank n ( 10). 



These conceptions are capable of the following direct generali- 

 zation. Any m marks A^ A 2 , . . ., A TO of the GF[p nr ] are called 

 linearly independent with respect to the GF[p r ~\ if the equation 



^1*1+ %^H ----- (~ ymA m = 0, 



in which the y/ are marks of the GF\_p r ~\, can be satisfied only in 

 case every y = 0. [See 72]. A system of m linearly independent 

 marks gives rise to p rm distinct marks of the GF[p nr ] 



yi^l-fftAgH ----- 1" Vm^m 



by letting th y/s run independently through the series of the marks 

 of the GF[p r ~\. These p rm marks are said to form an additive-group 

 [A D A 2 , . . ., A m ] of rank m with respect to the GF[p r ~], the marks 

 Aj, . . ., A m forming its basis -system. 



If A m _t_i be any mark of GF[p nr ] not in the additive -group 

 [A 1? . . ., A m ] of rank m with respect to the GF[p r ], then the m -f- 1 

 marks A 1? . . ., A m , A m _|_i are linearly independent with respect to 

 the GF[p r ] and therefore define an additive -group [A 1? . . ., A m , A^ + J 

 of rank m -\- 1 with respect to the GF[p r ]. 



69. Theorem. - Within the GF[p nr ] the number of additive- 

 groups [Aj, . . ., A ro ] of rank m with respect to the GF[p r ~\ is 



pr) . . . ( pn r _ p (m - 1) r) 

 pr) . . . (pmr p(m l)r) ' 



1) Moore, Mathematical Papers, Congress of 1893, p. 214, p. 216; Math. 

 Ann. vol. 55, 12. 



DlCKSON, Linear Groups. 4 



