# When is $\mathbb Z[\zeta_n]$ a Euclidean Domain?

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.

• @hurkyl Good catch. I've edited the question. – hatch22 Sep 18 '17 at 17:36
• @Hurkyl I suspect what is meant is the ring of integers of those fields. While $\mathbb Z[\zeta_n]$ is certainly contained in the ring of integers, can you confirm that the containment is actually equality? – Aaron Sep 18 '17 at 17:40
• – André 3000 Sep 18 '17 at 17:46

We have the following Theorem, see here, Theorem $5.2$ on page 50:
• The thesis you reference, Theorem 5.2, rests his conclusion on Weinberger's result which is, of course, conditional on GRH. An unconditional proof of the result that A cyclotomic field is Euclidean if and only if it is principal ideal domain appears in Harper, $\mathbb{Z}[\sqrt{14}]$ is Euclidean, cms.math.ca/10.4153/CJM-2004-003-9 – Malcolm Mar 29 '18 at 13:23