An introduction to the modern theory of equations, by Florian Cajori.

138

THEORY OF EQUATIONS

colmmon with zx - 3 x + 2- 2 x-1- 0. Find the H. C. F. of the two
functions and show that all the roots of the first equation satisfy the
second.
Ex. 2. The function x2 + 6x + 7 is irreducible in s(1), and it is not a
divisor of x3 + x x + 3x + 1. Fromr these data show that the two
functions cannot have a common factor.
Ex. 3. The equation ac2 + bx + c = 0 in 2(1) has a root in common
with X3 + 5 x2 + 10 x + 1 = 0. Show that a = b c = 0.
Ex. 4. Prove that two functions in Q2, (x) and f(Z), cannot have a
common factor which is a function of x in 2, if f (x) is irreducible and not
a divisor of p(x).
Ex. 5. If a root of the irreducible equation f (x) = 0 in 2 satisfies the
equation (p(x) = 0 in 0, and if f(x) is of higher degree than (x), then all
the coefficients of p (x) must be zero.
127. Gauss's Lemma. If f (x) has integral coefficients cndc can
be resolved into ratioCnal factors, it can be resolved into rational
factors with integral coefficients.
Consider the two functions,
G(x) c- + a1x + a9X +... (x ) -b + bl + b2 2 + *
Let k be the H. C. F. of the integers aO, ca, a,...; and let 1 be
the H. C. F. of the integers b0o, b2,, **.
Also let k be relatively prime to n, and let I be relatively
prime to n.
We may now write
G(x) E 7 (x), 11(x) l- (X),
where g(x) and h(x) are functions whose denominators are,
respectively, in and n. The numerator of /(x) is an integral
function of x with integral coefficients which have no conllm on
factor, except 1. The same is true of the numerator of
h(x). Hence the smallest denominator of the product g(x) ~ h(x)
is ain.