# The prime ideal in a number ring

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.

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