112 CHARACTERISTIC, RANK EQUATIONS [chap, vu 
where 8(œ) is the first characteristic determinant of x 
and is a polynomial in co whose coefficients are poly 
nomials in . . . . , with coefficients in F. 
Hence there exists a least positive integer r such that 
x is a root of an equation of degree r, 
(29) C 0 Ct) r -\~CiCO r ~ I -{- .... =0, 
with or without a constant term according as A has or 
lacks a modulus, where each q is a polynomial in £, 
with coefficients in F, while c 0 is not zero 
identically. 
When are indeterminates, c 0 , c S} ... . 
have a greatest common divisor g by Theorem V of 
§114. Write Ci=gqi. Then (29) becomes gR(co)=o, 
where 
(3°) R{co)=q 0 œ r +qiU r ~ I + 
Here q 0 , q I} . . . . have no common divisor other than 
a number of F, and q 0 is not zero identically. These 
properties remain true when we interpret 
as independent variables of F, provided F be an infinite 
field as we shall assume henceforth.* 
By means of x = Sand the multiplication table 
(1) of the units Ui, we may express R(x) in the form 
'ZfiUi, where/,• is a polynomial in & with co 
efficients in F. Since gR{x)—o , each gfi=o . By HI of 
§112, the corresponding function gfi of indeterminates 
£1,...., £„ is zero identically, so that one factor is 
* For, if /, g, h are polynomials in .... , £„ with coefficients 
in F, and if f—gh when the £’s are indeterminates, evidently f=gh 
when the £’s are independent variables in F. What we need is the 
converse, and it is true by III of §112.