$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$

Question

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

1. Does this module have a special name?
2. 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 $n$.

Answer 1

I don't know if you consider this a special name, but the ring $\mathbb{Z}[\zeta]$ is the ring of integers for the number field $\mathbb{Q}(\zeta)$. This is (IMO, anyway) a nontrivial fact, but you can find its proof in Neukirch (see below), or (for the case $n$ prime) in Samuel's ''Algebraic Theory of Numbers''—this link might also be helpful to you. Correspondingly, it is free as a $\mathbb{Z}$-module, and its rank is given by the degree $[\mathbb{Q}(\zeta):\mathbb{Q}] = \varphi(n)$, where $\varphi$ is the Euler totient function. The basis for $\mathbb{Z}[\zeta]$ as a $\mathbb{Z}$-module is given by $1, \zeta, \zeta^{2}, \ldots, \zeta^{d-1}$, where $d = \varphi(n)$. As a reference, see Neukirch's ''Algebraic Number Theory'', page 60, Proposition 10.2.

Answer 2

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