When is $\Bbb{Z[\zeta_n]}$ a PID?

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 $n$

Not always; the first example is with $n=23$, where in the ring $\mathbb{Z}[\zeta_{23}]$, the product $$(1+\zeta_{23}^2+\zeta_{23}^4+\zeta_{23}^5+\zeta_{23}^6+\zeta_{23}^{10}+\zeta_{23}^{11})(1+\zeta_{23}+\zeta_{23}^5+\zeta_{23}^6+\zeta_{23}^7+\zeta_{23}^9+\zeta_{23}^{11})$$ is divisible by the irreducible element $2$ even though neither factor is.

The question of which cyclotomic fields have class number $1$ is fundamentally interesting in its own right and has a famous connection with Fermat's Last Theorem.

Edit: Apparently the full list of cyclotomic fields with class number $1$ is known! Here is the first page of chapter 11 from Washington's Introduction to Cyclotomic Fields:

(Note that the restriction $m\not\equiv2\bmod 4$ is just to get rid of unnecessary duplicate information, since if $m\equiv 2\bmod 4$ then $\mathbb{Q}(\zeta_m)=\mathbb{Q}(\zeta_{m/2})$.)

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.