§ 6g] RANK EQUATION 113 
zero by the theorem in § in. Since g is not zero identi 
cally, each /,-=o and R(x)= o . 
Lemma. If \{x) =0, where X(co) is a polynomial in 
co whose coefficients are polynomials in . . . . , 
with coefficients in F, then X(co) is exactly divisible by 
F(co) when £ 1} . . . . , £„ are indeterminates. 
For, let g{co) denote the greatest common divisor 
of X(co) and F(co). By V of § 114, there exist poly 
nomials s{co) and /(co) whose coefficients are poly 
nomials in . , , , , £« with coefficients in F and a 
polynomial p in . . . . , with coefficients in F 
such that 
s(co)X(co)-f-t{(a)R(u)=pg{u). 
Hence pg(x)=o. By the paragraph preceding the 
lemma, g(x) = o. Hence the degree of g(co) in co is not 
less than the degree of R(co) in view of the definition of 
the latter. But the degree of the divisor g(w) is not 
greater than that of the dividend R(œ). Hence the 
degrees are equal. Then by IV of §114 with p = i, 
K = 1, R(co) is the product of g(co) by an element of F. 
Since X(co) is divisible by g(co), it is divisible by R(co). 
As noted above, coô(co) is a polynomial having the 
properties assumed for X(co) in the lemma, and hence is 
divisible by R(co). Since the coefficient of the highest 
power of co in coô(co) is ±1, we conclude that that of R(co) 
is a divisor of ±1. Hence q 0 is a number of F and 
may be made equal to be unity by dividing the terms of 
F(co) by it. 
Theorem. Let A be any associative algebra over an 
infinite field F. If . . . . , £„ are independent vari 
ables of F, the element x = is a root of a uniquely