Rational roots of monic polynomials over ZZ are integers

Suppose \alpha is a rational root of a monic polynomial in \mathbb{Z}[x]. Prove that \alpha is an integer.

By the Rational Root Theorem (Proposition 11 on page 308 of D&F) every rational root of such a polynomial p(x) has the form \alpha = r/s, where r divides the constant term and s the leading term of p(x). Since the leading term of p(x) is 1, s = \pm 1, and so \alpha is an integer.

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