$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$ Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$.


*

*Does this module have a special name?

*Does a basis exist for every $n$? And if so, is there an algorithm to find a basis given an $n$?


I was just playing around with this, and noticed that for $n=3$ we have e.g. the basis $(1,\zeta)$ because $1+\zeta = -\zeta^2$. For $n=4$ we obviously have the basis $(1,i)$, but I was unable to generalize this for an arbitrary $n$.

The ring $\mathbb Z[\zeta]$ is called the ring of cyclotomic integers.
