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Let $L/ K$ be a finite extension of number field. Let $\mathcal{P}$ be a prime ideal in $\mathbb{Z}_K= \{ \alpha \in K : \alpha \mbox{ is a algebraic integer}\}$. Prove that $\mathcal{P}\mathbb{Z}_L \neq \mathbb{Z}_L$

Any Hint?

Thanks!

3 Answers 3