If we are given an algebraic number field L, and $ \alpha $ is an element of L whose field trace over Q is zero and whose field norm over Q belongs to Z, then does $ \alpha $ necessarily belong to the integral closure of Z in L?  

The Hilbert's theorem 90 states the necessary and sufficient condition for elements to have the trace zero or the norm one, however, if we are given such kind of elements can we know more about them? And the original question is from the quadratic number fields.



trace zero elements in algebraic number fields 
The answer is no. Let $\alpha\in\mathbb{C}$ be a root of $x^3+tx+1$ for $t\in\mathbb{Q}\setminus\mathbb{Z}$, and let $L=\mathbb{Q}(\alpha)$, so that $N\_\mathbb{Q}^L(\alpha)=1$ and $tr\_{\mathbb{Q}}^L(\alpha)=0$. The element $\alpha$ cannot be integral over $\mathbb{Z}$, because its minimal polynomial over $\mathbb{Q}$ is not integral.


