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§ 4.3 Polynomials with Integer Coefficients

 
Proposition 4.3.1. Let f(x) = a_nxⁿ + a_n-1x^n-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 | 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

f(x) = a_mx^m + · · · + a₁x + a₀,

g(x) = b_nxⁿ + · · · + b₁x + b₀,     and

h(x) = c_kx^k + · · · + c₁x + c₀.

 
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_nxⁿ + a_n-1x^n-1 + · · · + a₀

a_n-1 a_n-2 . . . a₀ 0 (mod p) but

a_n 0 (mod p) and a₀ 0 (mod p²),

 
Corollary 4.3.7. If p is prime, then the polynomial

(x) = x^p-1 + · · · + x + 1

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