TITLE: When is $\Bbb{Z[\zeta_n]}$ a PID?
QUESTION [10 upvotes]: When is $\Bbb{Z[\zeta_n]}$ a PID?
I was just wondering if $\Bbb{Z[\zeta_n]}$ is PID or not where $\zeta_n$ is $n$th primitive root of unity for arbitrary positive $n$

REPLY [6 votes]: Let $\zeta=e^{2\pi i/23}$. Then an ideal of norm $47$ in $\mathbf{Z}[\zeta]$ 
is not principal. If it were principal then it would lead to an algebraic integer  of norm $47$ in the ring of integers of every intermediate field.
As  the quadratic field in 23rd cyclotomic is imaginary,  this would imply that the positive definite quadratic form for this field will  represent the number $47$. This can be easily checked to be false. So there are non-principal ideals in that cyclotomic field.  This was computed by R Swan in 1969 when settling Noether's question on pure transcendancy of invariant fields. See
https://eudml.org/doc/141959
Later H W Lenstra(1974) showed that this happens for infinitely many primes.