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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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    What's the reason for the downvotes? Unless you make it clear, I can't improve my question.2012-07-28

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There are standard techniques using determinants, resultants, etc, depending on what representation one is using for elements. Various methods are described below in an excerpt from Henri Cohen's A Course in Computational Algebraic Number Theory - one of the standard references on computational algebraic number theory. There is no need to do this by hand since there are many freely available software systems capable of algebraic number theory, e.g. the system Pari by Cohen's group at Bordeaux, which implements most of the algorithms described in his book.

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    I agree that a computer should be used in experimental number theory. However, I'm more interested in the algorithms than experimental number theory. I apologize if I gave you an impression otherwise.2012-07-28