Finding rings of integers is in general a tricky problem. For example, \(\mathbb{Z}[\sqrt{3}]\) is not the ring of integers of \(\mathbb{Q}(\sqrt{3})\).
Perhaps the best way forwards with your question is this.
Let \(K=\mathbb{Q}(w)\) be a number field and \(c\in K\).
Then K is a finitedimensional vector space over \(\mathbb{Q}\) and the function \(T:K\rightarrow K\) given by \(T(x)=cx\) is linear.
So, by choosing a basis for K over \(\mathbb{Q}\), you can represent T as a matrix, and the norm of c (with respect to K over \(\mathbb{Q}\)) is the determinant of that matrix.
Last fiddled with by Nick on 20180105 at 10:34
Reason: Fixed typo
