Kummer's Lemma and $1+\zeta$

Question

In lecture we were told to think about the following:

Kummer's Lemma: Let $p$ be an odd prime and let $\zeta := e^{2\pi i / p}$. Every unit of $\mathbb{Z}[\zeta]$ is of the form $r\zeta^g$, where $r$ is real and $g$ is an integer.

This theorem says that the units of $\mathbb{Z}[\zeta]$ can be thougth as the points of the lines that pass through the vertices of an equally spaced $p$-gon centered at the origin. (Said vertices are of the form $\zeta^g$.)

However, we know that $1+\zeta$ is a unit. But this number does not lie on one of those lines. What is wrong here?

Edit: This question seems to originate from Stewart & Tall's "Algebraic Number Theory and Fermat's Last Theorem" (4th edition), where it appears as Ex. 11.6 (p. 199).

Answer 1

Let the odd prime number be $p=2k+1$. Consider the number $$ w = \zeta^k(1+\zeta)\ . $$ Then $$ \begin{aligned} \bar w &=\overline{\zeta^k(1+\zeta)} =\bar\zeta^k(1+\bar\zeta) =\zeta^{-k}(1+\zeta^{-1}) \\ & =\zeta^{p-k}\frac 1\zeta(\zeta+1) =\zeta^{k+1}\frac 1\zeta(\zeta+1) =\zeta^k(\zeta+1)=w\ . \end{aligned} $$ So $w$ is a real number, and $1+\zeta=w\;\zeta^{-k}=w\;\zeta^{k+1}$.

Answer 2

Geometrically:

Consider the regular $p$-gon with its center at the origin $O$ and with $A = (0,1)$, and the other points $B_n$ with coordinates corresponding to $\zeta^n$. The lines going through these points and $O$ make angles $\tau n/p$ with the positive $x$-axis, and angles $\tau/2 - \tau n/p$ with the negative $x$-axis. Because $p$ is odd, the two sets of angles together are the angles $\tau n/2p$.

For each of the points $B_n$, the location of the complex point $1+\zeta^n$ can be found by constructing a rhombus using $O$, $A$, and $B_n$ as three vertices. Call this new point $C_n$. (Think the standard picture of vector addition.)

Now $\angle AOB_n$ has a measure equal to $\frac{\tau n}{p}$, meaning $\angle OAC_n$ has half that measure (by the properties of a rhombus). But we've seen that each of the angles $\tau n/2p$ is made by one of the lines passing through the origin and one of the points $B_n$. Therefore $C_n$ must lie on a line passing through $O$ and $B_m$ for some $m \neq n$.