132 
INTEGRAL ALGEBRAIC NUMBERS [chap, ix 
where each Cÿ is a polynomial with integral coefficients 
in the coefficients of /, A, . . . . , K, so that each Cÿ 
is an integer. Transposing the left members, we obtain 
n linear homogeneous equations in co z , . . . . , co n , the 
first step in the solution of which by determinants gives 
Dw z = o, . , Dco n = o, where 
Ci i f C12 . • 
. • c ln 
D = 
C 2 1 C 2 2 f . . 
• • C 2 n 
Cfll Gi2 • • 
. . c nn f 
Hence D = o. Multiplying the expansion of D by 
(— i)”, we get an equation f n + .... =o with integral 
coefficients and leading coefficient unity. Thus / is 
an integral algebraic number, 
83. Reducible polynomials. If we have an identity 
(4) /(»)=/i(x)Mx) 
between three polynomials with rational coefficients 
such that/j and f 2 are of degrees less than the degree of /, 
we call f(x) reducible. If no such identity exists, / is 
called irreducible. 
Theorem 5. A reducible polynomial f(x) with integral 
coefficients and leading coefficient unity is a product of 
two polynomials with integral coefficients and leading 
coefficient unity. 
By hypothesis, we have an identity (4). Let a be 
the coefficient of the highest power of x in f T and write 
f I = ag(x), f 2 = a~ 1 h{x). Then f(x)=g{x)h{x), where g 
and h have rational coefficients and have unity as the 
coefficient of the highest power of x.