Intersection of an ideal in a subring

Question

Let $P=\left\langle 7,5 +\sqrt{-10}\right\rangle$ be an ideal in $\mathbb{Z}[\sqrt{-10}]$. Is it true that $P \cap \mathbb{Z} = 7\mathbb{Z}$ ? I am kind of stuck at this. Well $7 \mathbb{Z} \subset P$ can be clearly seen. But please help me with the other side.

Thanks in advance.

Answer 1

If you know that $I=(7,5+\sqrt{-10})$ is not the unit ideal then the result is easy, $(7)\subseteq I\cap\mathbb{Z}\neq \mathbb{Z}$ and $(7)$ is a maximal ideal thus $(7)= I\cap\mathbb{Z}$.

One way of showing that $I$ is not the unit ideal is that

$$(7,5+\sqrt{-10})(7,5-\sqrt{-10})=(49,35)=(7).$$