If $\displaystyle K$ is a number field with ring of integers $\displaystyle \mathcal{O}_K$, then prove that if $\displaystyle P$ is a prime ideal of $\displaystyle \mathcal{O}_K$ then $\displaystyle P \cap \mathbb{Z}$ is a prime ideal of $\displaystyle \mathbb{Z}$.

It's easy to show that it's an ideal of $\displaystyle \mathbb{Z}$ but I'm struggling to show that it must be prime; any help would be very useful!