io8 CHARACTERISTIC, RANK EQUATIONS [chap, vii 
This is zero for every y in A if and only if 
n 
Hence x = 'Z%iUi^o is properly nilpotent in A if and 
only if relations (26) hold (with not all 
zero). 
Theorem. Let the n-rowed square matrix (r#), in 
which nj is the trace of UiUj, he of rank* r. An algebra A 
over a non-modular field has no properly nilpotent elements 
{and hence is semi-simple) if and only if r=n. Also, A 
has a maximal nilpotent invariant sub-algebra N of order 
v if and only if v = n — r> o. The value of r depends solely 
upon the constants of multiplication of A. 
The reader is now in a position to follow the proof in 
chapter viii of the principal theorem on algebras. 
For an important application to the arithmetic of 
algebras, we shall need the explicit expression for t s j, 
which is the trace of u s uj = Hy s jiUi and hence is the sum 
of the diagonal elements of the first matrix of the element 
obtained from # = Sby replacing & by yA 
diagonal element of the first matrix of x is given by (6) 
withy = k. Hence 
n 
* A matrix is said to be of rank r if at least one f-rowed minor is 
not zero, while every (r+i)-rowed minor is zero. Then r of the £, in 
(26) are expressible uniquely in terms of the remaining n—r, which are 
arbitrary. See Dickson’s First Course in the Theory of Equations (1922), 
p, 116.