After having accidentally duplicated this question, I thought I'd follow up with a related question. In an answer to the linked question, Zev Chonoles quotes the first page of Chapter 11 of Washington's Introduction to Cyclotomic Fields which states that the only $\mathbb Z[\zeta_n]$ with $\zeta_n$ a root of unity that are PIDs (with class number 1) are extensions with the following values of $n$:
1, 3, 4, 5, 7, 8,
9, 11, 12, 13, 15, 16,
17, 19, 20, 21, 24, 25,
27, 28, 32, 33, 35, 36,
40, 44, 45, 48, 60, 84.
In addition, values of $n \equiv 2 \mod 4$ are also allowed, because for example, $\mathbb Z[\zeta_{30}] = \mathbb Z[\zeta_{15}]$. So my question is, which are known to be Euclidean Domains? I'm especially interested if $n = 60$ admits a form of the Euclidean division algorithm.
We have the following Theorem, see here, Theorem $5.2$ on page 50:
Theorem 5.2. A cyclotomic field is Euclidean if and only if it is principal ideal domain.
So this leads again back to the duplicate question.