What is a unit of $\mathbb Q(\zeta_7)$ of infinite order?
I understand from the definition of a unit that I'm trying to find a number with unlimited powers of $\zeta_7$ whose product with another number is $1$. However, while I can find ways to make $\pm 1$ on a finite scale (eg. $\zeta_7^6*\zeta_7$, $1+\zeta_7^4 * 1+\zeta_7+\zeta_7^2$, $\zeta_7^{15}*\zeta_7^{20}$) and then add appropriate amounts of them in a series (eg. $\zeta_7^2 *\sum_{n=1}^m(-1)^n*\zeta_7^{7n+5}$ for even $m$), the only way I can seem to come up with that on an infinite scale is something like Grandi's series (letting $m$ in the previous example go to infinity). That doesn't seem like a valid response.
Can someone poke me along in the right direction?