Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\alpha \in A$. How can one compute efficiently the norm of $\alpha$ by hand or by using a calculator?
EDIT[Jul 28, 2012] The question asks an efficient algorithm. There are computer software doing this. I think, however, knowing its algorithm is more enlightening than using it as a blackbox.
EDIT Let $l = 19$. Let $\alpha = 1 + \zeta + \zeta^6$. I computed $N(\alpha) = 191$ by hand. It took me over a half day.