SIMPLE MATRIC ALGEBRA 
§ 7i] 
US 
when m is interpreted as one of the preceding four 
matrices, say R x , since the leading coefficient of 4>{u) = 
R{co) is unity, while the remaining coefficients are now 
polynomials in . . . . , £« with coefficients in F. 
Theorem. The distinct factors irreducible in an 
infinite field F of the left member of either characteristic 
equation of x coincide with the distinct irreducible factors of 
the rank function R{of). 
71. Rank equation of a simple matric algebra. By 
§ 59, any w-rowed square matrix x = (xif) with elements 
in F is a root of 
(31) R(ù}) = (-i) n \xij-8iju\ =0, 0,7 = 1, bij=o{i^j). 
Let the xq be n 2 independent variables of an infinite 
field F. We shall prove that R{of)=o is the rank 
equation. This will follow from the lemma in § 69 if 
we prove that R{u>) is irreducible in F. It suffices to 
prove that its constant term ±|xy\ is irreducible in F. 
In view of the footnote in § 69, this follows from the 
Lemma. The determinant [xfi of n 2 indeterminates 
Xij(i,j = i, . ... , n) is a polynomial f(x I1} x I2 , . ... , 
x nn ) which is irreducible in every field F. 
Suppose that / is a product of two polynomials g and 
h with coefficients in F. Since / is of degree 1 in each 
indeterminate, we may assume that g is of degree o and 
h of degree 1 in No term of the expansion / of \xif\ 
contains the product of x zl by an element x n of the first 
column. Hence g is of degree o in x rz , since otherwise 
x ri Xn would occur in a term of gh=f. Thus h is of 
degree 1 in x n . Since x rc x ri does not occur in a term of 
gh =/, g is of degree o in every x rc .