COMPLEX ALGEBRAS 
179 
the coefficients are all zero, while each diagonal element 
is a co. Hence 8{œ) is a product of powers of ¡3— w 
(j = 1, . . . . , h) with exponents ^ 1. By § 70, the 
same is true of the rank function R{œ), in which the 
coefficient of the highest power of co is unity. 
We are now in a position to investigate the sets of 
elements of A with rational co-ordinates which have 
properties R, C, U of §87. To secure the closure property 
C, we assume* that the 7’s in (18) are rational. By prop 
erty R, each coefficient of R{co) =0 is an integer. Since 
its roots are all rational, they are integers. The maxi 
mal set is composed of all elements s in which the are 
integers, while the vj are merely rational. All such 
2’s therefore give the integral elements of A. 
We shall prove that u = i-\-2a p n p is a unit (§ 88) for 
all rational values of the a p . First, 
u(i — a 1 n 1 ) = i — à 2 1 ri 2 1 -{-l 2 = 1 -\-a 12 n 2 -\-l 3 =u 2 , 
where U denotes a linear function of Ui, ni+ x , .... with 
rational coefficients. Similarly, 
u 2 (i — a 12 n 2 ) = i — 1+Æ13W3-H4 • 
Proceeding in this manner, we finally reach the product 
1. Hence 
uv— 1, v= (1 — — a 12 n 2 )(i — a l3 n 3 ) .... 
where the b 2 are rational. Hence u and v are units. 
If n p is of character (i, j), and &,....,& are 
all 5*£<D, 
P 
* This assumption is not necessary for the application we shall 
make in § 104.