Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 4.3 Polynomials with Integer Coefficients

 
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

f(x) = amxm + · · · + a1x + a0,
g(x) = bnxn + · · · + b1x + b0,     and
h(x) = ckxk + · · · + c1x + c0.
If bs and ct are the coefficients of g(x) and h(x) of least index not divisible by p, then as+t is the coefficient of f(x) of least index not divisible by p.

 
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 + · · · + a0

be 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 + 1

is irreducible over the field of rational numbers.

 


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