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Problem 4 (Algebraic Numbers)
Let $\mathbb{Z}[x]$ be the set of all polynomials with coefficients in $\mathbb{Z}$, together with the usual properties of polynomial addition and multiplication. Show that $x$ is an algebraic number if and only if $x$ satisfies $q(x)=0$ for some $q(x)\in \mathbb{Z}[x]$.
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