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.