## 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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