ARITHMETIC OF AN ALGEBRA [chap, x 
1 78 
where the two linear sets on the right have only zero in 
common. Every element of Qj is of character 
O’, 7). Then 
N 2 =eN 2 e = 'Le$H j , N = B x +N 2 , B t = SQ, 
summed for i, 7 = 1, . . . . , h. Hence the elements of 
B x are linear combinations of elements each having a 
definite character. The same is true of B 2 in 
N 2 — B 2 -\-N 3 , etc. In view also of § 100 we may there 
fore choose a normalized set of basal units of N each 
having a definite character. 
Theorem i. Any complex algebra A of the first 
category has a set of basal units e x , ...., eu,n x , .... , 
n g , where each n p is nil potent and has a definite character, 
while 
(18) e1 e,, ein p n p , n p ej n p , n p n p ^y pPT n T , 
summed for t — i, . . . . , g; r>p, t> a; such that 
n p , n a , n T have the respective characters (i, j), (7, /), (i, l). 
All further products of two units are zero. 
To find the first characteristic determinant 6(co) of 
the general element z = x+y of A, where 
x= .... A~^h, y=v I n I -\- .... A~v g n g , 
we proceed as in the footnote to § 60. If n a is of char 
acter (7, -), 
zej = £j6j +lin. func. of «j, . , . . , n g ; 
zn a — ^n a A-lin. func. of n a+1 , n a + 2 , 
Transposing the left members after replacing z by o>, 
we obtain linear equations in the units such that the 
elements below the main diagonal of the determinant of