# Special units of the cyclotomic number field of an odd prime order $l$

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 $k$ be a rational integer not divisible by $l$. How would you prove that $(1 - \zeta^k)/(1 - \zeta)$ is a unit of $A$?

This is a related question.

• I tried to compute the norm of it. – Makoto Kato Jul 25 '12 at 1:40
• That is doable but unnecessary. It will be easier to try to write the inverse as an element of $A$. – Qiaochu Yuan Jul 25 '12 at 1:45
• @MakotoKato Can you show that $(1-\zeta^k)/(1-\zeta) \in A$ ? – Ragib Zaman Jul 25 '12 at 1:47
• @QiaochuYuan I think N($1 - \zeta^k$) = N($1 - \zeta$) is clear. – Makoto Kato Jul 25 '12 at 1:56

Just to verify that this is indeed an element of $A$, and also because it hints at a way of showing that your element is a unit in that ring: a very old identity tells you that $\frac{1 - \zeta^k}{1 - \zeta} = 1 + \zeta + \dots + \zeta^{k - 1}.$ Since $k$ is prime to $l$, you can find an integer $k'$ such that $kk' \equiv 1 \bmod l$. Then writing $\frac{1 - \zeta}{1 - \zeta^k} = \frac{1 - \zeta^{kk'}}{1 - \zeta^k}$ should help quite a bit. These are called, somewhat confusingly, the cyclotomic units. A book on Iwasawa theory should talk about them quite a bit.