RANK EQUATION 
§ 73] 
117 
£2j, and R 2 {co) is of degree o in the ¡¡zj, D{00) is of degree 
o in both sets and hence involves the single indeterminate 
oj. But 
i?i(w) = co ri -\-CzCc ri ~ r T . . . . , 
where c 1} ... . are homogeneous polynomials in the 
£zj and hence vanish when each ^ = 0. Hence D{co) 
is a divisor <a d of of\ This is impossible since A x has a 
modulus and hence Rz(co) has a constant term not zero 
identically by the corollary in § 69. 
73. Rank equation unaltered by any transformation 
of units. For an associative algebra A with the con 
stants of multiplication 7^, let R(co; £ f , 7^) = o be 
the rank equation which is satisfied by co = x, where 
x = 'E^Ui is the general element of A. Under a trans 
formation of units (§ 61), let x become x' — h^ui, and 
let R become p(co; 7y*). For co — x', both p and R{oo; 
7ijk) are zero; unless they are identical, their differ 
ence is zero for co = x'. Passing back to the initial units, 
we obtain a function of degree <r which is zero for co = x, 
contrary to the definition of r. Hence the rank equation 
is independent of the choice of basal units A 
* Another proof follows from the theorems of §§62 and 70 and the 
fact that each irreducible factor of an invariant is an invariant Com 
pare Bocher, Introduction to Higher Algebra (1907), p. 218.