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Proposition 4.3.1.
Let f(x)
= a_{n}x^{n}
+ a_{n-1}x^{n-1}
+ · · · +
a_{1}x +
a_{0}
be a polynomial with integer coefficients.
If r/s is a rational root of f(x), with (r,s)=1, then
r | a_{0} and s | a_{n}.
Definition 4.3.2.
A polynomial with integer coefficients is called
primitive
if the greatest common divisor of all of its coefficients is 1.
Lemma 4.3.3.
Let p be a prime number, and let f(x) = g(x)h(x), where
Theorem 4.3.4. [Gauss's Lemma]
The product of two primitive polynomials is itself primitive.
Theorem 4.3.5.
A polynomial with integer coefficients that can be factored into polynomials
with rational coefficients can also be factored into polynomials of the same
degree with integer coefficients.
Theorem 4.3.6. [Eisenstein's Irreducibility Criterion]
Let
f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + · · · + a_{0}
be a polynomial with integer coefficients. If there exists a prime number p such thata_{n-1} a_{n-2} . . . a_{0} 0 (mod p) but
a_{n} 0 (mod p) and a_{0} 0 (mod p^{2}),
then f(x) is irreducible over the field of rational numbers.
Corollary 4.3.7.
If p is prime, then the polynomial
(x) = x^{p-1} + · · · + x + 1
is irreducible over the field of rational numbers.
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