Since the ej are basal units of algebra C, 
C 
(8) WjWk = (i,k = i, . . . . , c). 
t—1 
We may express (6) in the form 
(9) ai=ui+vi (i= 1, . . . . , c), 
where V{ is in Ah. Since Ah is invariant in A Iy 
O'iQ'k ~ w №k~\~W , ik , 
where fia and n\k below are in Ah. Hence, by (8) and (9), 
^ ^ ) Mik Mik 7ikt v t • 
But the product a^k of two elements of A can be 
expressed in one and only one way as a linear combina 
tion, with coefficients in F, of the basal units of A, which 
are composed of those of N and a z , . . . . , a c . Hence 
the 7ikt are numbers of F. 
But F z was derived from F by the adjunction of a 
finite number of roots of equations with coefficients in 
F. Hence F 1 = F(£ 1 , £ 2 , . . . .), where 
are linearly independent with respect to F. We may 
therefore write 
~Vi = VioF Vj 2 ^ 2 + . . . . , 
where the Vij are in Ah Write 
Zj di~\~ Vio , B (Zl, Z 2 , . • • • , Z(,) ,