This question seems not very hard, but it is starting to embarrass me. So I thought I can use your ideas to solve it, and I would be thankful in advance.
Let $K/\mathbb{Q}$ be a number field of degree $n$ with the discriminant $D$. For $\alpha\in\mathcal{O}_K$ define $||\alpha||:=\max\{|\sigma(\alpha)|:\mbox{over all euclidean place} \}$
I want to show that there exists an $0\neq\alpha\in\mathcal{O}_K$ such that $Tr_{K/\mathbb{Q}}(\alpha)=0$ and $||\alpha||\ll|D|^{\frac{1}{2(n-1)}}$, where the implied constant depends only on $n$.
I was trying to use geometry of number, then it is easy to find an $0\neq\alpha\in\mathcal{O}_K$ such that and $||\alpha||\ll|D|^{\frac{1}{2(n-1)}}$ but I could not prove this $\alpha$ can be choosen so that $Tr(\alpha)=0$.