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Let $\alpha$ be an algebraic integer and let $f$ be a monic polynomial over $\mathbb{Z}$ of least degree having $\alpha$ as a root. Prove that $f$ is irreducible.

I am having so many troubles with this question. I have no clue where to start, any help is much appreciated!

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Hint: Suppose $f$ is reducible. Then $f(x)=g(x)h(x)$ for some non-constant polynomials $g$ and $h$. And $f(\alpha) = g(\alpha)h(\alpha) = 0$, so...

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