## Extended Fermat's Last Theorem

I know that $x^4 + y^4 = z^4$ has no obvious solutions in $\mathbb{Z}[i]$.
I know that $x^3 + y^3 = z^3$ has no obvious solutions in $\mathbb{Z}[\omega]$

I am curious, define $\zeta = e^{2\pi i/n}$.
Does $x^n + y^n = z^n$, $n\geq 2$, have ever solutions in $\mathbb{Z}[\zeta]$?

I ask this because I know that Kummer's dream of approaching FLT falls apart for some primes.