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Suppose we have a Dedekind domain $R$ which for the moment we can take to be $\mathcal{O}_K$ for some algebraic number field $K$. Now suppose that $Q_1,Q_2$ are prime ideals that lie over a prime ideal $p$ of $\Bbb{Z}$. This means that $Q_1\cap \Bbb{Z} = p$, $Q_2\cap \Bbb{Z} = p$. Is it true that

$$Q_1Q_2 \cap \Bbb{Z} = p^2?$$

I can see that one containment is true, namely that $p^2 \subseteq Q_1Q_2 \cap \Bbb{Z}$.

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up vote 1 down vote accepted

Try $Q_1=(3,1+\sqrt{-5})$, $Q_2=(3,1-\sqrt{-5})$.

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Yes I see that $Q_1Q_2 \cap \Bbb{Z} = 3$. Thanks for your answer. The reason why I asked this because I thought it could be used to prove my problem here. – user38268 Jan 6 '13 at 4:27

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