RANK EQUATION 
III 
69] 
68. Minimum equation of an element of an algebra. 
Let x be an element of an associative algebra A over F. 
If A has a modulus, any polynomial g{<a) with coefficients 
in F which vanishes when w = R x vanishes for co = x by (17) 
and Theorem 1 of § 58, and conversely. Hence the 
minimum equation of R x is the minimum equation of x. 
By the preceding Theorem 2, every root of the former is a 
root of the characteristic equation of R x , which is the 
first characteristic equation 8(00) =0 of x by §60, and 
conversely. The same holds for S x and 8'{of)=o. If 
A lacks a modulus, we employ R x instead of R x and note 
(§ 60) that (17) still hold. 
Theorem. Every root of the minimum equation of an 
element x of any associative algebra is a root of either 
characteristic equation of x and conversely. 
69. Rank equation. By § n the quaternion 
q=(T+&+vj+£k, 
in which <r, £, r], £ are independent real variables, is a 
root of 
CO 2 — 2 (7W+ ( cr 2 + £ 2 + r7 2 + f 2 ) = o , 
and is evidently not a root of an equation of the first 
degree. This quadratic equation is called the rank 
equation of the general real quaternion q since its coeffi 
cients are polynomials in a, £, 77, <T and the coefficient 
of co 2 is unity, and since q is not the root of an equation 
of lower degree whose coefficients have these properties. 
Consider any associative algebra A over a field F. 
Let be a set of basal units of A. Let &, 
be variables ranging independently over F. 
By § 60, the element x = 2of A is a root of coô(w) =0,