114 CHARACTERISTIC, RANK EQUATIONS [chap, vn 
determined rank equation R{co) =o in which the coefficient 
of the highest power oo r is unity, while the remaining 
coefficients are polynomials in with coeffi 
cients in F. Also, x is not a root of any equation of degree 
<r all of whose coefficients are such polynomials. 
The integer r is called the rank of algebra A. 
Corollary. If A has a modulus e, the constant term 
c of R(co) is not zero identically. 
For, R{u>) divides 5(w), so that c divides 5(o)=A(x), 
But A(e) = i by the footnote in § 58. 
The theorem fails for finite fields. Consider the 
algebra A = (u I} u„ uf) over the field composed of the 
two classes of residues of integers modulo 2, where 
uj = Ui, UiUj = o{j y^i). The modulus of A is e = Hui. 
Either characteristic determinant is 
A=(£ x —w)(£ a —w) (£ 3 —co). 
Evidently every element x of A is a root of co 2 = co. 
Now 
A= (co —w 2 )(co+i+^ I +| 2 +^ 3 ) + p (mod 2), 
where 
P=5co-^ 3 , i = I+S&+S&&. 
Thus sx — £ 1 £ 2 % 3 e = o for every x in A. Another such 
linear equation satisfied by x is ax = o where cr = (1 — £.) 
(i-y(i-y. 
70. Let x be an element of A whose co-ordinates 
£1, . . . . , are independent variables in F. As in 
§ 68, the rank equation R{co)=o of x is the minimum 
equation of matrices R x and S x (or of R* and S x if A has 
no modulus). The discussion* in § 67 is seen to hold 
* An indirect proof of the lemma consists in seeing that it is a trans 
lation of that in § 69.