Question on ideal whose radical is maximal is primary

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An ideal whose radical is maximal: https://math.stackexchange.com/q/649179
It's said that $α=αm^n+sαβ∈Q$. How do we know that $α∈Q$?

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By hypothesis, $m^n\in Q$ and $\alpha\beta\in Q$, so any linear combination of these lies in $Q$, that's all.