Norm in algebraic number fields

Question

Consider an algebraic number field $\mathbb{Q}(\alpha)$ and its ring of integers $O$. If we take any element $\xi \in O$ and we want to calculate its norm $N_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\xi)$, is the norm the constant term of its minimal polynomial?

I know that in general the norm is the constant term of characteristic polynomial. But at least sometimes it seems to be true that it is the constant term of minimal polynomial. So is it true when the element belongs in the ring of integers? If not how do I know when the norm is the constant term of minimal polynomial? ${}$

Answer 1

In a quadratic number field, $N(2) = 4$. However, the minimal polynomial of $2$ is $x-2$.

Answer 2

The constant term of the minimal polynomial of $\xi$ over $\mathbb{Q}$ equals $N_{\mathbb{Q}(\xi)/\mathbb{Q}}(\xi)$. So the answer to your first question is yes if $\mathbb{Q}(\alpha) = \mathbb{Q}(\xi)$. On the other hand we have $N_{E/\mathbb{Q}}(\xi) = (N_{\mathbb{Q}(\xi)/\mathbb{Q}}(\xi))^{|E:\mathbb{Q}(\xi)|}$ for any extension field $E$ of $\mathbb{Q}$ containing $\xi$.