Number theory-question again.
Let $K$ be the biquadratic field $K=\mathbb{Q}[\sqrt{m},\sqrt{n}]$ where $m,n$ are distinct squarefree integers. Let $\mathcal{O}_K$ denote the ring of integers of $K$. If $\alpha \in K$, prove that $\alpha \in \mathcal{O}_K$ if and only if the trace and the norm of $\alpha$ over $\mathbb{Q}(\sqrt{m})$ are algebraic integers.
I've been stuck on this one for quite some time now. The left-right direction is easy, but I'm stuck on the other direction. My "intuition" tells me that I should somehow use that an element of a quadratic extension is integral if and only if its norm and its trace are.
I'd like subtle hints in the right direction. Thanks.
Edit: Some minor corrections for clarity.