Let $\mathbb{Z}_{(p)} \subseteq \mathbb{Q}$ denote the ring of rational p-adic integers (p prime), and let $\zeta \in \mathbb{C}$ be a root of unity of order $p \cdot m$, where $m$ is not divisible by $p$. Then apparently, the ring $\mathbb{Z}_{(p)}[\zeta]$ is a principal ideal domain.
In this paper, the above statement is used implicitly, as if it was true for trivial reasons.
Does somebody know the reason why this is true? Does the statement become wrong if $\zeta$ is an arbitrary root of unity? Thank you in advance!
Best Answer
The useful fact here is that a Dedekind domain with only finitely many primes is a PID. See, for instance, this question. The proof is so simple that I’m abashed not to have seen it.
Anyhow, your ring has only the ideals above $(p)$.