What are the units of cyclotomic integers?

This question made me realize I had a misconception about the cyclotomic integers: I thought the units were exactly the roots of unity. There are only finitely many units but infinitely many integers so the question is impossible to solve unless there are more units. So what are the units of cyclotomic integers?

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By Dirichlet's unit theorem the only rings of integers with finitely many units are Z itself and the quadratic imaginary integers (i.e. integers in Q(\sqrt{-d})). So you should always expect to find lots of units. – Noah Snyder Aug 24 '10 at 15:33

I don't think there's a straightforward way to describe all the units. If $\zeta$ is a root of unity in the field, then $\frac{\zeta^k-1}{\zeta-1}$ is a unit whenever $k$ is relatively prime to the order of $\zeta$, and therefore products of such elements are units as well. This already gives you infinitely many units, and though they don't necessarily cover all units they do come close (this is a finite-index subgroup of the group of units).

Look up Kummer's Lemma, for instance here, and "Cyclotomic unit".

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Be careful: k should be relatively prime to the order of zeta and not just any number other than the order. For example, if zeta has order 25 you don't want to let k be 5 or 10 or 15 or 20. In that case the ratio you write down in your answer is not a unit. Also, if the order of zeta is not a prime power (or twice a prime power) then already zeta - 1 itself is a unit. – KCd Aug 24 '10 at 18:09
My sloppiness. I'll correct it. – Alon Amit Aug 24 '10 at 18:37

1:By definition, units in a given number field $K$ are the "norm one" integers. That is

$O_K^*$={$u\in O_K|\ |N_{K|\mathbb{Q}}(u)|=1$}.

2:For the structure of the group of units, one has the famous Dirichlet's unit theorem.

3:However, even in the case of cyclotomic fields, there is no explicit formula to produce the whole group of units. But we can describe its subgroup: "cyclotomic units"

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To get the structure of units in cyclotomic fields, one might want to look at Lemma 8.1 in:

Introduction to cyclotomic fields, Lawrence C. Washington.

In short it says that the group of units in a cyclotomic field is generated by cyclotomic units in the field (which includes -1) and the generator of the field.

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