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Given $R$ and $S$ being number rings corresponding to the number fields $K$ and $L$, such that $K \subset L$, show that $\mathfrak Q \cap R$ is nonzero where $\mathfrak Q$ is any nonzero prime ideal of $S$.

This is from Daniel A. Marcus' Number Fields, Chapter $3$, Theorem $20$. It hints that we can use a norm argument. Please help me.

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    $\begingroup$ Hint: Can you show that $N(Q)\in Q$? $\endgroup$ – Mathmo123 Sep 9 '18 at 3:27
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    $\begingroup$ I see. Let $\alpha$ be any nonzero element of $Q$, we can show $N( \alpha ) \in Q$ which is also contained in $R$. $\endgroup$ – yuan Sep 9 '18 at 10:49
  • $\begingroup$ Thank you @Mathmo123. $\endgroup$ – yuan Sep 9 '18 at 10:50
  • $\begingroup$ I don't have the answer on this one. In your shoes, I would try to work out one specific example. For example, $K = \mathbb Q(\sqrt 2)$, $L = \mathbb Q(\root 4 \of 2)$, $\mathfrak Q = \langle \root 4 \of 2 \rangle$. $\endgroup$ – Robert Soupe Sep 10 '18 at 2:08

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