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If $\alpha$ and $\beta$ are roots of monic polynomials (not necessarily the same polynomial) in $\mathbb{Z}[x]$, is $\alpha + \beta$?

I know that $\alpha + \beta$ will be the root of some monic polynomial in $\mathbb{Q}[x]$, but I'm not sure if the coefficients will be integers. I've tried to find a counterexample but I've had no success, which leads me to believe it's probably true. I'm not sure how to proceed with a proof.

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Yes, this is true.

One common proof relies on the lemma: If $\alpha$ (an element of some ring $A$ containing $\mathbb{Z}$) is the root of a monic integer polynomial if and only there exists a non-zero, finitely generated $\mathbb{Z}$-module $M \subseteq A$ with $\alpha M \subseteq M$.

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