# How to compute efficiently the norm of a cyclotomic integer

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.

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@Makoto Note that are freely available software systems for performing computations in algebraic number theory, e.g. Pari. – Bill Dubuque Jul 27 '12 at 22:42
@BillDubuque Yes, I know. I'd like to know its algorithm. – Makoto Kato Jul 27 '12 at 22:58
@Makoto The algorithms are described in Henri Cohen's A Course in Computational Algebraic Number Theory. You can also read the source code. – Bill Dubuque Jul 27 '12 at 23:03
@BillDubuque Thanks. Since I don't have the book at hand, it'd be nice that someone would post the algorithm. It does not need to be the same as that of the book. – Makoto Kato Jul 27 '12 at 23:22
What's the reason for the downvotes? Unless you make it clear, I can't improve my question. – Makoto Kato Jul 28 '12 at 0:10