View Single Post
 2018-01-05, 09:30 #2 Nick     Dec 2012 The Netherlands 5×353 Posts 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 finite-dimensional 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 2018-01-05 at 10:34 Reason: Fixed typo