104 CHARACTERISTIC, RANK EQUATIONS [chap, vii 
By chapter xi, we may extend F to a field F' in which 
f{co) *g(co) decomposes into linear functions of co: 
/(oj) = (a I —co) .... (a n — co), g(u) =/3(co —fa) .... (co —&). 
If 7 is the w-rowed unit matrix, we have in F' 
g{m) = p{m-pj) . . . . (m-l3 r I). 
Passing to determinants, we get 
\g{m) | = /3»| m-pj\ .... \m-^ r I \ = P n fM . . . . /(/3 r ). 
But, by the initial formulae, 
f(Pj) — (a-z — fij) .... (a„—j3j), 
g{ a k) = P(a-k — Pi) • • • • (a k — P r ). 
Hence 
( 2 3) U(w) )=g(ai) . . . . g(a n ). 
Let £ be a variable in the field F' and write h{co, £) 
for g{co) — £. Then h{m, £)=g{m) — £7, so that the 
characteristic determinant of g(w) is the determinant 
of h{m, £). Applying (23) with the polynomial g(m) 
replaced by h{m, £), we see that the determinant of the 
latter is equal to the product 
Ha-i, £).... h{a n , £) = [g(a,)-£] .... [g(a„)-£]. 
Equating the latter to zero, we therefore obtain the char 
acteristic equation of matrix g(m). Hence its roots are 
g(a 1), . . . . , g(a n ). 
64. Roots of the characteristic equation of g{x). 
Theorem. Let g(co) he a polynomial of the type in 
§ 60. Let F t he an extension of the field F such that the 
first {or second) characteristic equation of the element x of