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Proposition 4.3.1. Let f(x) = anxn + an-1xn-1 + · · · + a1x + a0 be a polynomial with integer coefficients. If r/s is a rational root of f(x), with (r,s)=1, then r | a0 and s | an.
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) = anxn + an-1xn-1 + · · · + a0be a polynomial with integer coefficients. If there exists a prime number p such that
an-1 an-2 . . . a0 0 (mod p) but
an 0 (mod p) and a0 0 (mod p2),then f(x) is irreducible over the field of rational numbers.
Corollary 4.3.7. If p is prime, then the polynomial
(x) = xp-1 + · · · + x + 1is irreducible over the field of rational numbers.
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